{"id":113,"date":"2025-04-09T17:10:41","date_gmt":"2025-04-09T17:10:41","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-7-linear-inequalities-and-absolute-value-inequalities-college-algebra-2e-openstax\/"},"modified":"2026-03-19T15:56:34","modified_gmt":"2026-03-19T15:56:34","slug":"2-7-linear-inequalities-and-absolute-value-inequalities","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-7-linear-inequalities-and-absolute-value-inequalities\/","title":{"raw":"2.7 Linear Inequalities and Absolute Value Inequalities","rendered":"2.7 Linear Inequalities and Absolute Value Inequalities"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_b888ed1c-c7d6-4bc4-a647-2fb628a05ad5\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"ui-has-child-title\" data-type=\"abstract\"><header>\r\n<h2 data-type=\"title\">Learning Objectives<\/h2>\r\n<\/header><section>\r\n<p id=\"para-00001\">In this section, you will:<\/p>\r\n\r\n<ul id=\"list-00001\">\r\n \t<li>Use interval notation<\/li>\r\n \t<li>Use properties of inequalities.<\/li>\r\n \t<li>Solve inequalities in one variable algebraically.<\/li>\r\n \t<li>Solve absolute value inequalities.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<div id=\"Figure_02_07_001\" class=\"os-figure\">\r\n<figure class=\"medium\" data-id=\"Figure_02_07_001\"><span id=\"fs-id1521249\" data-type=\"media\" data-alt=\"Several red winner\u2019s ribbons lie on a white table.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Several red winner\u2019s ribbons lie on a white table.\" width=\"423\" height=\"282\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/d749a0eb956b42942e5bf0aa047f58fa1b4a6989\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">1<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1795255\">It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.<\/p>\r\n\r\n<section id=\"fs-id2781560\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Interval Notation<\/h2>\r\n<p id=\"fs-id981594\">Indicating the solution to an inequality such as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>can be achieved in several ways.\r\n<p id=\"fs-id2384427\">We can use a number line as shown in <a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Figure_02_07_002\">Figure 2<\/a><strong>.<\/strong> The blue ray begins at<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.\r\n<div id=\"Figure_02_07_002\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_07_002\"><span id=\"fs-id3094790\" data-type=\"media\" data-alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"508\" height=\"47\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/7292fc5c016c86fabdb7accf5e59376bacf91556\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">2<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id2049812\">We can use <span id=\"term-00001\" class=\"no-emphasis\" data-type=\"term\">set-builder notation<\/span>:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>}<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>}<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>which translates to \u201call real numbers <em data-effect=\"italics\">x <\/em>such that <em data-effect=\"italics\">x <\/em>is greater than or equal to 4.\u201d Notice that braces are used to indicate a set.\r\n<p id=\"fs-id1962920\">The third method is <span id=\"term-00002\" data-type=\"term\">interval notation<\/span>, in which solution sets are indicated with parentheses or brackets. The solutions to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>are represented as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.\r\n<p id=\"fs-id1417925\">The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be \u201cequaled.\u201d A few examples of an <span id=\"term-00003\" data-type=\"term\">interval<\/span>, or a set of numbers in which a solution falls, are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or all numbers between\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22122<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>6<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>6<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>including\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22122<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>but not including\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>6<\/mn><mo>;<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>6<\/mn><mo>;<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>all real numbers between, but not including\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22121<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22121<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>0<\/mn><mo>;<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>0<\/mn><mo>;<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>all real numbers less than and including\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>1.<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>1.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math><a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Table_02_07_01\">Table 1<\/a> outlines the possibilities.\r\n<div id=\"Table_02_07_01\" class=\"os-table\">\r\n<table data-id=\"Table_02_07_01\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\" data-align=\"center\">Set Indicated<\/th>\r\n<th scope=\"col\" data-align=\"center\">Set-Builder Notation<\/th>\r\n<th scope=\"col\" data-align=\"center\">Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>All real numbers between <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b<\/em>, but not including <em data-effect=\"italics\">a <\/em>or <em data-effect=\"italics\">b<\/em><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers greater than <em data-effect=\"italics\">a<\/em>, but not including <em data-effect=\"italics\">a<\/em><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mi>a<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mi>a<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em data-effect=\"italics\">b<\/em>, but not including <em data-effect=\"italics\">b<\/em><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers greater than <em data-effect=\"italics\">a<\/em>, including <em data-effect=\"italics\">a<\/em><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2265<\/mo><mi>a<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2265<\/mo><mi>a<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>[<\/mo> <mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>[<\/mo> <mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em data-effect=\"italics\">b<\/em>, including <em data-effect=\"italics\">b<\/em><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>]<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>]<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em data-effect=\"italics\">a <\/em>and<em data-effect=\"italics\"> b<\/em>, including <em data-effect=\"italics\">a<\/em><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>[<\/mo> <mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>[<\/mo> <mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, including <em data-effect=\"italics\">b<\/em><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>]<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>]<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b<\/em>, including <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b<\/em><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>[<\/mo> <mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>]<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>[<\/mo> <mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>]<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em data-effect=\"italics\">a<\/em> or greater than <em data-effect=\"italics\">b<\/em><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>a<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>or<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>a<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>or<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>a<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u222a<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>b<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>a<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u222a<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>b<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers<\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is all real numbers<\/mtext>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is all real numbers<\/mtext>\r\n<\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span>\r\n<span class=\"os-number\">1<\/span><\/div>\r\n<\/div>\r\n<div id=\"Example_02_07_01\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">1<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1940764\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1767073\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Using Interval Notation to Express All Real Numbers Greater Than or Equal to <em data-effect=\"italics\">a<\/em><\/h3>\r\n<p id=\"fs-id3169488\">Use interval notation to indicate all real numbers greater than or equal to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22122.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id3039360\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1295224\">Use a bracket on the left of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22122<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and parentheses after infinity:\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>The bracket indicates that\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22122<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>is included in the set with all real numbers greater than\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22122<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>to infinity.\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2383200\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#1<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_02_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2682208\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1517140\">Use interval notation to indicate all real numbers between and including<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22123<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22123<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>5.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>5.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_07_02\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">2<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1543726\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1411353\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Using Interval Notation to Express All Real Numbers Less Than or Equal to <em data-effect=\"italics\">a <\/em>or Greater Than or Equal to <em data-effect=\"italics\">b<\/em><\/h3>\r\n<p id=\"fs-id1395714\">Write the interval expressing all real numbers less than or equal to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22121<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22121<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>or greater than or equal to\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>1.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>1.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1943180\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1736769\">We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mo>\u2212<\/mo><mi>\u221e<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mi>\u221e<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and ends at\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22121<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22121<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>which is written as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2513420\">The second interval must show all real numbers greater than or equal to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>which is written as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>However, we want to combine these two sets. We accomplish this by inserting the union symbol,\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u222a<\/mo><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u222a<\/mo><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>between the two intervals.\r\n<div id=\"fs-id3035449\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>\u222a<\/mo><mrow><mo>[<\/mo> <mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>\u22121<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>\u222a<\/mo><mrow><mo>[<\/mo> <mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow> <mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1592740\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#2<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_07_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id906336\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2644263\">Express all real numbers less than<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22122<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>or greater than or equal to 3 in interval notation.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2734136\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Properties of Inequalities<\/h2>\r\n<p id=\"fs-id1500576\">When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the <span id=\"term-00004\" class=\"no-emphasis\" data-type=\"term\">addition property<\/span> and the <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">multiplication property<\/span> to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.<\/p>\r\n\r\n<div id=\"fs-id2425325\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"A General Note\"><span class=\"os-title-label\" data-type=\"\">Properties of Inequalities<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"fs-id3176836\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mstyle><mi>A<\/mi><mi>d<\/mi><mi>d<\/mi><mi>i<\/mi><mi>t<\/mi><mi>i<\/mi><mi>o<\/mi><mi>n<\/mi><\/mstyle><mspace width=\"0.8\"><\/mspace><mtext><\/mtext><mstyle><mi>P<\/mi><mi>r<\/mi><mi>o<\/mi><mi>p<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>y<\/mi><\/mstyle>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mstyle><mi>M<\/mi><mi>u<\/mi><mi>l<\/mi><mi>t<\/mi><mi>i<\/mi><mi>p<\/mi><mi>l<\/mi><mi>i<\/mi><mi>c<\/mi><mi>a<\/mi><mi>t<\/mi><mi>i<\/mi><mi>o<\/mi><mi>n<\/mi><\/mstyle><mspace width=\"0.8\"><\/mspace><mtext><\/mtext><mstyle><mi>P<\/mi><mi>r<\/mi><mi>o<\/mi><mi>p<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>y<\/mi><\/mstyle>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>and<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mi>c<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>and<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mstyle><mi>A<\/mi><mi>d<\/mi><mi>d<\/mi><mi>i<\/mi><mi>t<\/mi><mi>i<\/mi><mi>o<\/mi><mi>n<\/mi><\/mstyle><mspace width=\"0.8\"><\/mspace><mtext><\/mtext><mstyle><mi>P<\/mi><mi>r<\/mi><mi>o<\/mi><mi>p<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>y<\/mi><\/mstyle>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mstyle><mi>M<\/mi><mi>u<\/mi><mi>l<\/mi><mi>t<\/mi><mi>i<\/mi><mi>p<\/mi><mi>l<\/mi><mi>i<\/mi><mi>c<\/mi><mi>a<\/mi><mi>t<\/mi><mi>i<\/mi><mi>o<\/mi><mi>n<\/mi><\/mstyle><mspace width=\"0.8\"><\/mspace><mtext><\/mtext><mstyle><mi>P<\/mi><mi>r<\/mi><mi>o<\/mi><mi>p<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>y<\/mi><\/mstyle>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>and<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mi>c<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>and<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1340108\">These properties also apply to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>a<\/mi><mo>\u2264<\/mo><mi>b<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>a<\/mi><mo>\u2264<\/mo><mi>b<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>&gt;<\/mo><mi>b<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>&gt;<\/mo><mi>b<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>a<\/mi><mo>\u2265<\/mo><mi>b<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>a<\/mi><mo>\u2265<\/mo><mi>b<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_07_03\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">3<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2453957\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2644438\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Demonstrating the Addition Property<\/h3>\r\n<p id=\"fs-id1245519\">Illustrate the addition property for inequalities by solving each of the following:<\/p>\r\n\r\n<ol id=\"fs-id3085981\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>6<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>6<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d2<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id890147\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1477754\">The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.<\/p>\r\n\r\n<ol id=\"fs-id1419199\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span><span data-type=\"newline\">\r\n<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>+<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn><mo>+<\/mo><mn>15<\/mn><mo> <\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Add 15 to both sides<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>19<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>+<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn><mo>+<\/mo><mn>15<\/mn><mo> <\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Add 15 to both sides<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>19<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span><span data-type=\"newline\">\r\n<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mn>6<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>6<\/mn><mo>+<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4em\"><\/mspace><mtext>Add 1 to both sides<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mn>7<\/mn><mo>\u2265<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mn>6<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>6<\/mn><mo>+<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4em\"><\/mspace><mtext>Add 1 to both sides<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mn>7<\/mn><mo>\u2265<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d2<\/span><span data-type=\"newline\">\r\n<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4em\"><\/mspace><mtext>Subtract 7 from both sides<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4em\"><\/mspace><mtext>Subtract 7 from both sides<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2269039\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#3<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_07_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2677967\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1698684\">Solve:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3<\/mn><mi>x<\/mi><mn>\u22122<\/mn><mo>&lt;<\/mo><mn>1.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><mi>x<\/mi><mn>\u22122<\/mn><mo>&lt;<\/mo><mn>1.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_07_04\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">4<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1525452\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2507109\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Demonstrating the Multiplication Property<\/h3>\r\n<p id=\"fs-id1895873\">Illustrate the multiplication property for inequalities by solving each of the following:<\/p>\r\n\r\n<ol id=\"fs-id3061147\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>6<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d2<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>10<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>10<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1238058\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<ol id=\"fs-id1496121\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span><span data-type=\"newline\">\r\n<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.5em\"><\/mspace><mn>3<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>&lt;<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.5em\"><\/mspace><mn>3<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>&lt;<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span><span data-type=\"newline\">\r\n<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiply by <\/mtext><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Reverse the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiply by <\/mtext><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Reverse the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d2<\/span><span data-type=\"newline\">\r\n<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2.5em\"><\/mspace><mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>10<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3.5em\"><\/mspace><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiply by <\/mtext><mo>\u2212<\/mo><mn>1.<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4.5em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Reverse the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2.5em\"><\/mspace><mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>10<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3.5em\"><\/mspace><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiply by <\/mtext><mo>\u2212<\/mo><mn>1.<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4.5em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Reverse the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1730534\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#4<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_07_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2422478\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1396214\">Solve:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>\u2265<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>\u2265<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1249131\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Inequalities in One Variable Algebraically<\/h2>\r\n<p id=\"fs-id2710126\">As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.<\/p>\r\n\r\n<div id=\"Example_02_07_05\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">5<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2417012\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1407961\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving an Inequality Algebraically<\/h3>\r\n<p id=\"fs-id2444209\">Solve the inequality:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>13<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>10<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>13<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>10<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2437055\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1449306\">Solving this inequality is similar to solving an equation up until the last step.<\/p>\r\n\r\n<div id=\"fs-id2628929\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\".5em\"><\/mspace><mn>13<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>10<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>13<\/mn><mo>\u2212<\/mo><mn>17<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>\u22124<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Move variable terms to one side of the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.8em\"><\/mspace><mn>\u221217<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>\u221217<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Isolate the variable term<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3.1em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Dividing both sides by <\/mtext><mn>\u221217<\/mn><mspace width=\"0.8\"><\/mspace><mtext>reverses the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\".5em\"><\/mspace><mn>13<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>10<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>13<\/mn><mo>\u2212<\/mo><mn>17<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>\u22124<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Move variable terms to one side of the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.8em\"><\/mspace><mn>\u221217<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>\u221217<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Isolate the variable term<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3.1em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Dividing both sides by <\/mtext><mn>\u221217<\/mn><mspace width=\"0.8\"><\/mspace><mtext>reverses the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1846594\">The solution set is given by the interval<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or all real numbers less than and including 1.\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1572093\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#5<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_07_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1457850\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1790143\">Solve the inequality and write the answer using interval notation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mo>\u2212<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>&lt;<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>&lt;<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_07_06\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">6<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2441066\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1416866\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving an Inequality with Fractions<\/h3>\r\n<p id=\"fs-id1435934\">Solve the following inequality and write the answer in interval notation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2917362\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1408780\">We begin solving in the same way we do when solving an equation.<\/p>\r\n\r\n<div id=\"fs-id1513950\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2.8em\"><\/mspace><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\".9em\"><\/mspace><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Put variable terms on one side<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>9<\/mn>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Write fractions with common denominator<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2.5em\"><\/mspace><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>17<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4.2em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>17<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiplying by a negative number reverses the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4.2em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mfrac>\r\n<mrow>\r\n<mn>15<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>34<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2.8em\"><\/mspace><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\".9em\"><\/mspace><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Put variable terms on one side<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>9<\/mn>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Write fractions with common denominator<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2.5em\"><\/mspace><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>17<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4.2em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>8<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>12<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>17<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiplying by a negative number reverses the inequality<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"4.2em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mfrac>\r\n<mrow>\r\n<mn>15<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>34<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1556088\">The solution set is the interval<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mfrac>\r\n<mrow>\r\n<mn>15<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>34<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo> <mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mfrac>\r\n<mrow>\r\n<mn>15<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mn>34<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2497523\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#6<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_07_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2436750\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1774157\">Solve the inequality and write the answer in interval notation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2264<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>8<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2264<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>8<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2947575\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Understanding Compound Inequalities<\/h2>\r\n<p id=\"fs-id1574863\">A <span id=\"term-00006\" data-type=\"term\">compound inequality<\/span> includes two inequalities in one statement. A statement such as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>4<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>6<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>4<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>6<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>means\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>4<\/mn><mo>&lt;<\/mo><mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>4<\/mn><mo>&lt;<\/mo><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>\u2264<\/mo><mn>6.<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2264<\/mo><mn>6.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.\r\n<div id=\"Example_02_07_07\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">7<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2501975\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1516646\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Compound Inequality<\/h3>\r\n<p id=\"fs-id2503657\">Solve the compound inequality:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2382216\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2442855\">The first method is to write two separate inequalities:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6.<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>We solve them independently.\r\n<div id=\"fs-id2434911\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mn>2<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mn>2<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1290697\">Then, we can rewrite the solution as a compound inequality, the same way the problem began.<\/p>\r\n\r\n<div id=\"fs-id1922364\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1583611\">In interval notation, the solution is written as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>[<\/mo> <mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo><mn>2<\/mn>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo><mn>2<\/mn>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id1722369\">The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.<\/p>\r\n\r\n<div id=\"fs-id2020141\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Isolate the variable term, and subtract 2 from all three parts<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Divide through all three parts by 2<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Isolate the variable term, and subtract 2 from all three parts<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Divide through all three parts by 2<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2390439\">We get the same solution:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>[<\/mo> <mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo><mn>2<\/mn>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo><mn>2<\/mn>\r\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1215697\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#7<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_07_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2434068\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1941577\">Solve the compound inequality:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>4<\/mn><mo>&lt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/mn><mo>\u2264<\/mo><mn>10.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>4<\/mn><mo>&lt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/mn><mo>\u2264<\/mo><mn>10.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_07_08\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">8<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1500526\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1919530\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Compound Inequality with the Variable in All Three Parts<\/h3>\r\n<p id=\"fs-id1243509\">Solve the compound inequality with variables in all three parts:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>10.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>10.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2048755\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1191836\">Let's try the first method. Write two inequalities<strong>:<\/strong><\/p>\r\n\r\n<div id=\"fs-id3239342\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>10<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mn>3<\/mn><mo>&gt;<\/mo><mn>6<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>\u221210<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mn>5<\/mn><mo>&gt;<\/mo><mn>6<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.8em\"><\/mspace><mn>2<\/mn><mi>x<\/mi><mo>&gt;<\/mo><mn>\u22128<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>\r\n<mo>&gt;<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2.5em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mn>\u22124<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.3em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.5em\"><\/mspace><mn>\u22124<\/mn><mo>&lt;<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>10<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mn>3<\/mn><mo>&gt;<\/mo><mn>6<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>\u221210<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mn>5<\/mn><mo>&gt;<\/mo><mn>6<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.8em\"><\/mspace><mn>2<\/mn><mi>x<\/mi><mo>&gt;<\/mo><mn>\u22128<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>\r\n<mo>&gt;<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2.5em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mn>\u22124<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.3em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.5em\"><\/mspace><mn>\u22124<\/mn><mo>&lt;<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1954589\">The solution set is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22124<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22124<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or in interval notation\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo><\/mo><mn>\u22124<\/mn><mo>,<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo><\/mo><mn>\u22124<\/mn><mo>,<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>6<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line. See <a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Figure_02_07_003\">Figure 3<\/a><strong>.<\/strong>\r\n<div id=\"Figure_02_07_003\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_07_003\"><span id=\"fs-id1929508\" data-type=\"media\" data-alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\" width=\"487\" height=\"69\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/f664946114f8c591904e1c6d4310aad6242d6b10\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">3<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2700910\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#8<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_07_08\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2957500\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1441254\">Solve the compound inequality:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>5<\/mn><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>5<\/mn><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1161758\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Absolute Value Inequalities<\/h2>\r\n<p id=\"fs-id1338469\">As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo>,<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo>,<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>has an absolute value of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>as it is <em data-effect=\"italics\">x <\/em>units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.\r\n<p id=\"fs-id1230083\">An <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">absolute value inequality<\/span> is an equation of the form<\/p>\r\n\r\n<div id=\"fs-id3107252\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>&lt;<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>\u2264<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><mo>&gt;<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>or <\/mtext><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><mo>\u2265<\/mo><mi>B<\/mi><mo>,<\/mo>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>&lt;<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>\u2264<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>&gt;<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>or <\/mtext><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>\u2265<\/mo><mi>B<\/mi><mo>,<\/mo><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1184764\">Where <em data-effect=\"italics\">A<\/em>, and sometimes <em data-effect=\"italics\">B<\/em>, represents an algebraic expression dependent on a variable <em data-effect=\"italics\">x. <\/em>Solving the inequality means finding the set of all<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><em data-effect=\"italics\">-<\/em>values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.\r\n<p id=\"fs-id1581445\">There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.<\/p>\r\n<p id=\"fs-id1199584\">Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of <em data-effect=\"italics\">x-<\/em>values such that the distance between<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and 600 is less than or equal to 200. We represent the distance between\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and 600 as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and therefore,\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>200<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>200<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>or\r\n<div id=\"fs-id1681298\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>\u2212200<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mn>200<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>\u2212200<\/mn><mo>+<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><mo>+<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mn>200<\/mn><mo>+<\/mo><mn>600<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>400<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>800<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>\u2212200<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mn>200<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>\u2212200<\/mn><mo>+<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><mo>+<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mn>200<\/mn><mo>+<\/mo><mn>600<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>400<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>800<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1976398\">This means our returns would be between $400 and $800.<\/p>\r\n<p id=\"fs-id1323840\">To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.<\/p>\r\n\r\n<div id=\"fs-id2930959\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"A General Note\"><span class=\"os-title-label\" data-type=\"\">Absolute Value Inequalities <\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1520813\">For an algebraic expression <em data-effect=\"italics\">X,<\/em> and<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>k<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>k<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>an <span id=\"term-00008\" data-type=\"term\">absolute value inequality<\/span> is an inequality of the form\r\n<div id=\"fs-id2016621\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is equivalent to\u00a0<\/mtext><mo>\u2212<\/mo><mi>k<\/mi><mo>&lt;<\/mo><mi>X<\/mi><mo>&lt;<\/mo><mi>k<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>&gt;<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is equivalent to\u00a0<\/mtext><mi>X<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>or\u00a0<\/mtext><mi>X<\/mi><mo>&gt;<\/mo><mi>k<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is equivalent to\u00a0<\/mtext><mo>\u2212<\/mo><mi>k<\/mi><mo>&lt;<\/mo><mi>X<\/mi><mo>&lt;<\/mo><mi>k<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>&gt;<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is equivalent to\u00a0<\/mtext><mi>X<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>or\u00a0<\/mtext><mi>X<\/mi><mo>&gt;<\/mo><mi>k<\/mi><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1277147\">These statements also apply to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mi>k<\/mi><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mi>k<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mi>k<\/mi><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mi>k<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_07_09\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">9<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2433767\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2020938\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Determining a Number within a Prescribed Distance<\/h3>\r\n<p id=\"fs-id1758539\">Describe all values<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>within a distance of 4 from the number 5.\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1570653\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2293280\">We want the distance between<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and 5 to be less than or equal to 4. We can draw a number line, such as in <a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Figure_02_07_004\">Figure 4<\/a><strong>,<\/strong> to represent the condition to be satisfied.\r\n<div id=\"Figure_02_07_004\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_07_004\"><span id=\"fs-id1762406\" data-type=\"media\" data-alt=\"A number line with one tick mark in the center labeled: 5. The tick marks on either side of the center one are not marked. Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A number line with one tick mark in the center labeled: 5. The tick marks on either side of the center one are not marked. Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.\" width=\"481\" height=\"80\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/921f3c827d871d2af31e89f140164ac0c046fed6\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">4<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1181980\">The distance from<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>to 5 can be represented using an absolute value symbol,\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Write the values of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>that satisfy the condition as an absolute value inequality.\r\n<div id=\"fs-id3273982\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1334091\">We need to write two inequalities as there are always two solutions to an absolute value equation.<\/p>\r\n\r\n<div id=\"fs-id1690088\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>\u2264<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>\u2265<\/mo><mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.8em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.8em\"><\/mspace><mi>x<\/mi><mo>\u2265<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>\u2264<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>\u2265<\/mo><mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.8em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.8em\"><\/mspace><mi>x<\/mi><mo>\u2265<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2021289\">If the solution set is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>\u2264<\/mo><mn>9<\/mn> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2264<\/mo><mn>9<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>then the solution set is an interval including all real numbers between and including 1 and 9.\r\n<p id=\"fs-id1288018\">So<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>is equivalent to\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>9<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow> <\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\r\n<mn>1<\/mn><mo>,<\/mo><mn>9<\/mn>\r\n<\/mrow> <mo>]<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>in interval notation.\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2957557\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#9<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_07_09\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1811321\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1811322\">Describe all <em data-effect=\"italics\">x-<\/em>values within a distance of 3 from the number 2.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_07_10\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">10<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1428981\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1428983\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving an Absolute Value Inequality<\/h3>\r\n<p id=\"fs-id1802343\">Solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow><mo>|<\/mo>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<mo>|<\/mo><mo>\u2264<\/mo><mn>3<\/mn>\r\n\r\n&nbsp;\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>\u2264<\/mo><mn>3<\/mn><\/annotation-xml><\/semantics><\/math>.\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1571578\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<div id=\"fs-id1753466\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow><mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><\/mtd><\/mtr><\/mtable>\r\n\r\n<mo>|<\/mo>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn> <mo>|<\/mo><mo>\u2264<\/mo><mn>3<\/mn>\r\n\r\n\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><\/mtd><\/mtr>\r\n\r\n<mn>\u22123<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2264<\/mo><mn>3<\/mn>\r\n\r\n\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><\/mtd><\/mtr>\r\n\r\n<mn>\u22122<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>4<\/mn>\r\n\r\n\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><\/mtd><\/mtr>\r\n\r\n<mo stretchy=\"false\">[<\/mo><mn>\u22122<\/mn><mo>,<\/mo><mn>4<\/mn><mo stretchy=\"false\">]<\/mo>\r\n\r\n&nbsp;\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><mo>|<\/mo>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn> <mo>|<\/mo><mo>\u2264<\/mo><mn>3<\/mn>\r\n\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><mn>\u22123<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2264<\/mo><mn>3<\/mn>\r\n\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><mn>\u22122<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>4<\/mn>\r\n\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><mo stretchy=\"false\">[<\/mo><mn>\u22122<\/mn><mo>,<\/mo><mn>4<\/mn><mo stretchy=\"false\">]<\/mo>\r\n\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_07_11\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">11<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1690263\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2947783\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Using a Graphical Approach to Solve Absolute Value Inequalities<\/h3>\r\n<p id=\"fs-id2947788\">Given the equation<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>|<\/mo>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn> <mo>|<\/mo><mo>+<\/mo><mn>3<\/mn><mo>,<\/mo>\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac><mo>|<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>|<\/mo><mo>+<\/mo><mn>3<\/mn><mo>,<\/mo><\/annotation-xml><\/semantics><\/math>determine the <em data-effect=\"italics\">x<\/em>-values for which the <em data-effect=\"italics\">y<\/em>-values are negative.\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id3130412\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id3130414\">We are trying to determine where<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow><mi>y<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo>&nbsp;\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>y<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/annotation-xml><\/semantics><\/math>which is when\r\n\r\n<math display=\"inline\"><semantics><mrow><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>|<\/mo>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn> <mo>|<\/mo><mo>+<\/mo><mn>3<\/mn><mo>&lt;<\/mo><mn>0.<\/mn>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac><mo>|<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>|<\/mo><mo>+<\/mo><mn>3<\/mn><mo>&lt;<\/mo><mn>0.<\/mn><\/annotation-xml><\/semantics><\/math>We begin by isolating the absolute value.\r\n<div id=\"fs-id3156619\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow><mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><\/mtd><\/mtr><\/mtable>\r\n\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>|<\/mo>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<mo>|<\/mo><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\r\n\r\n<mtd columnalign=\"left\"><\/mtd>\r\n\r\n<mspace width=\"2em\"><\/mspace><mtext>Multiply both sides by \u20132, and reverse the inequality<\/mtext><mo>.<\/mo>\r\n\r\n\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mspace width=\"1.5em\"><\/mspace>\r\n<mo>|<\/mo>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<mo>|<\/mo><mo>&gt;<\/mo><mn>6<\/mn><\/mtd><\/mtr>\r\n\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\"><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>|<\/mo>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<mo>|<\/mo><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\r\n\r\n&nbsp;\r\n\r\n<\/mtd>\r\n<mtd columnalign=\"left\"><mspace width=\"2em\"><\/mspace><mtext>Multiply both sides by \u20132, and reverse the inequality<\/mtext><mo>.<\/mo>\r\n\r\n&nbsp;\r\n\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mspace width=\"1.5em\"><\/mspace>\r\n<mo>|<\/mo>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<mo>|<\/mo><mo>&gt;<\/mo><mn>6<\/mn>&nbsp;\r\n\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id3260232\">Next, we solve for the equality<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mo>|<\/mo>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<mo>|<\/mo><mo>=<\/mo><mn>6.<\/mn>&nbsp;\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>|<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>|<\/mo><mo>=<\/mo><mn>6.<\/mn><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id2762668\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.9em\"><\/mspace><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>11<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>or<\/mtext><mspace width=\"2em\"><\/mspace>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.9em\"><\/mspace><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>11<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.9em\"><\/mspace><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>11<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>or<\/mtext><mspace width=\"2em\"><\/mspace>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.9em\"><\/mspace><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>11<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2644216\">Now, we can examine the graph to observe where the <em data-effect=\"italics\">y-<\/em>values are negative. We observe where the branches are below the <em data-effect=\"italics\">x-<\/em>axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>11<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>11<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and that the graph opens downward. See <a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Figure_02_07_007\">Figure 5<\/a><strong>.<\/strong>\r\n<div id=\"Figure_02_07_007\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_07_007\"><span id=\"fs-id2777886\" data-type=\"media\" data-alt=\"A coordinate plan with the x-axis ranging from -5 to 5 and the y-axis ranging from -4 to 4. The function y = -1\/2|4x \u2013 5| + 3 is graphed. An open circle appears at the point -0.25 and an arrow\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A coordinate plan with the x-axis ranging from -5 to 5 and the y-axis ranging from -4 to 4. The function y = -1\/2|4x \u2013 5| + 3 is graphed. An open circle appears at the point -0.25 and an arrow\" width=\"396\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/f34a03da829882c6e85c8ae73414515a4fd1bb0c\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">5<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1969441\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#10<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_07_10\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1951872\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1951873\">Solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mo>\u2212<\/mo><mn>2<\/mn><mrow><mo>|<\/mo> <mrow>\r\n<mi>k<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mo>\u2212<\/mo><mn>6.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mn>2<\/mn><mrow><mo>|<\/mo> <mrow>\r\n<mi>k<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mo>\u2212<\/mo><mn>6.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1904611\" class=\"media-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Media\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Media\"><span class=\"os-title-label\">Media<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1904617\">Access these online resources for additional instruction and practice with linear inequalities and absolute value inequalities.<\/p>\r\n\r\n<ul id=\"fs-id1904622\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/intervalnotn\" target=\"_blank\" rel=\"noopener nofollow\">Interval notation<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/solvelinineq\" target=\"_blank\" rel=\"noopener nofollow\">How to solve linear inequalities<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/solveineq\" target=\"_blank\" rel=\"noopener nofollow\">How to solve an inequality<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/absvaleq\" target=\"_blank\" rel=\"noopener nofollow\">Absolute value equations<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/compndineqs\" target=\"_blank\" rel=\"noopener nofollow\">Compound inequalities<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/absvalineqs\" target=\"_blank\" rel=\"noopener nofollow\">Absolute value inequalities<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.7 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id2529020\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1901196\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1901201\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1901202\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1901201-solution\">1<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1901203\">When solving an inequality, explain what happened from Step 1 to Step 2:<\/p>\r\n<p id=\"fs-id1901206\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>Step 1<\/mtext>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>&gt;<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>Step 2<\/mtext>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>Step 1<\/mtext>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>&gt;<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>Step 2<\/mtext>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"3em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1904125\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1904126\" data-type=\"problem\"><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1904128\">When solving an inequality, we arrive at:<\/p>\r\n<p id=\"fs-id1904131\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mn>2<\/mn><mo>&lt;<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mn>2<\/mn><mo>&lt;<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id3274019\">Explain what our solution set is.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3274022\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3274023\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3274022-solution\">3<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3274024\">When writing our solution in interval notation, how do we represent all the real numbers?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2304131\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2304132\" data-type=\"problem\"><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2304133\">When solving an inequality, we arrive at:<\/p>\r\n<p id=\"fs-id2304136\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mn>2<\/mn><mo>&gt;<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"1.2em\"><\/mspace><mn>2<\/mn><mo>&gt;<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id1886074\">Explain what our solution set is.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1886077\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1886078\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1886077-solution\">5<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1886080\">Describe how to graph<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>y<\/mi><mo>=<\/mo><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>y<\/mi><mo>=<\/mo><mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2925477\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id2925482\">For the following exercises, solve the inequality. Write your final answer in interval notation.<\/p>\r\n\r\n<div id=\"fs-id2925485\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2925486\" data-type=\"problem\"><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2925487\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>\u2264<\/mo><mn>9<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>\u2264<\/mo><mn>9<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2925506\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2925508\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2925506-solution\">7<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2925509\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>\u2265<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>\u2265<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2783754\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2783755\" data-type=\"problem\"><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2783756\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2783780\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2783781\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2783780-solution\">9<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2783782\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2523115\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2523116\" data-type=\"problem\"><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2523117\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>5<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mn>5<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2372045\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2372046\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2372045-solution\">11<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2372047\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22125<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22125<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2996112\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2996114\" data-type=\"problem\"><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2996115\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1705070\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1705071\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1705070-solution\">13<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1705072\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mn>8<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>\u2265<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>10<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mn>8<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>\u2265<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mrow>\r\n<mn>10<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2786106\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2786107\" data-type=\"problem\"><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2786108\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1886125\">For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.<\/p>\r\n\r\n<div id=\"fs-id1886129\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1886130\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1886129-solution\">15<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1886131\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>9<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>\u22126<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>9<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>\u22126<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1929693\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1929694\" data-type=\"problem\"><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1929695\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>7<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1929727\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1929728\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1929727-solution\">17<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1929729\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>|<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>11<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>|<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>11<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1768168\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1768169\" data-type=\"problem\"><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1768170\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>+<\/mo><mn>1<\/mn><mo>\u2264<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>+<\/mo><mn>1<\/mn><mo>\u2264<\/mo><mn>6<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1768206\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1768207\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1768206-solution\">19<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2742909\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>+<\/mo><mn>4<\/mn><mo>\u2265<\/mo><mn>10<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>+<\/mo><mn>4<\/mn><mo>\u2265<\/mo><mn>10<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1909611\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1909612\" data-type=\"problem\"><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1909614\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>\u22122<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>13<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>\u22122<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>13<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1909646\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1909647\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1909646-solution\">21<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1909648\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>|<\/mo><mo>&lt;<\/mo><mn>\u22124<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>|<\/mo><mo>&lt;<\/mo><mn>\u22124<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3062896\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3062897\" data-type=\"problem\"><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3062898\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>\u22121<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3062925\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3062926\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3062925-solution\">23<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3062928\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>2<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1663908\">For the following exercises, describe all the <em data-effect=\"italics\">x<\/em>-values within or including a distance of the given values.<\/p>\r\n\r\n<div id=\"fs-id1663916\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1663917\" data-type=\"problem\"><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1663918\">Distance of 5 units from the number 7<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1663921\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1663922\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1663921-solution\">25<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1663923\">Distance of 3 units from the number 9<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3085702\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3085704\" data-type=\"problem\"><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3085705\">Distance of 10 units from the number 4<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3085708\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3085709\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3085708-solution\">27<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3085710\">Distance of 11 units from the number 1<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id3085740\">For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.<\/p>\r\n\r\n<div id=\"fs-id3085744\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3085746\" data-type=\"problem\"><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3085747\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22124<\/mn><mo>&lt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>\u2264<\/mo><mn>18<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22124<\/mn><mo>&lt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>\u2264<\/mo><mn>18<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2272698\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2272699\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2272698-solution\">29<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2272700\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2413129\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2413130\" data-type=\"problem\"><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2413131\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>7<\/mn><mo>+<\/mo><mi>y<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>7<\/mn><mo>+<\/mo><mi>y<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2413161\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2413162\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2413161-solution\">31<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2413163\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>&lt;<\/mo><mn>\u221211<\/mn><mspace width=\"0.8\"><\/mspace><mtext>or <\/mtext><mspace width=\"0.8\"><\/mspace><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>&lt;<\/mo><mn>\u221211<\/mn><mspace width=\"0.8\"><\/mspace><mtext>or <\/mtext><mspace width=\"0.8\"><\/mspace><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>6<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1779673\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1779674\" data-type=\"problem\"><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1779676\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1779700\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1779705\">For the following exercises, graph the function. Observe the points of intersection and shade the <em data-effect=\"italics\">x<\/em>-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.<\/p>\r\n\r\n<div id=\"fs-id1779714\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1779716\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1779714-solution\">33<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1779717\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>2<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1710959\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1710960\" data-type=\"problem\"><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1710961\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>5<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1710991\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1710992\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1710991-solution\">35<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1710993\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3021007\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3021008\" data-type=\"problem\"><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3021009\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>7<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>7<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3021037\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3021038\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3021037-solution\">37<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3021039\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2826233\">For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the <em data-effect=\"italics\">y<\/em>-values of the lines.<\/p>\r\n\r\n<div id=\"fs-id2826242\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2826244\" data-type=\"problem\"><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2826245\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>&lt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>&lt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2826268\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2826269\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2826268-solution\">39<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2826270\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1911262\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1911263\" data-type=\"problem\"><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1911264\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1911288\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1911289\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1911288-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1911290\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3059337\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3059338\" data-type=\"problem\"><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3059339\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&lt;<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&lt;<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id3071965\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id3071970\">For the following exercises, write the set in interval notation.<\/p>\r\n\r\n<div id=\"fs-id3071973\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3071974\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3071973-solution\">43<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3071975\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\r\n<mn>\u22121<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mrow><mn>3<\/mn> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/mrow>\r\n<\/mrow> <\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\r\n<mn>\u22121<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mrow><mn>3<\/mn> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/mrow>\r\n<\/mrow> <\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2831815\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2831816\" data-type=\"problem\"><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2831817\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mrow><mn>7<\/mn> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/mrow>\r\n<\/mrow> <\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\r\n<mi>x<\/mi><mo>\u2265<\/mo><mrow><mn>7<\/mn> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/mrow>\r\n<\/mrow> <\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2831853\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2831854\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2831853-solution\">45<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2831856\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\r\n<mi>x<\/mi><mo>&lt;<\/mo><mrow><mn>4<\/mn> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/mrow>\r\n<\/mrow> <\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\r\n<mi>x<\/mi><mo>&lt;<\/mo><mrow><mn>4<\/mn> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/mrow>\r\n<\/mrow> <\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1752303\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1752304\" data-type=\"problem\"><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1752305\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mrow><mo>|<\/mo><mrow>\r\n<mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is all real numbers<\/mtext><mrow><mrow><\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/mrow>\r\n<\/mrow> <\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>{<\/mo> <mrow>\r\n<mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mrow><mo>|<\/mo><mrow>\r\n<mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is all real numbers<\/mtext><mrow><mrow><\/mrow> <mo>}<\/mo><\/mrow>\r\n<\/mrow><\/mrow>\r\n<\/mrow> <\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1752345\">For the following exercises, write the interval in set-builder notation.<\/p>\r\n\r\n<div id=\"fs-id1752348\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1752349\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1752348-solution\">47<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3275472\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3275537\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3275538\" data-type=\"problem\"><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3275539\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2743879\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2743880\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2743879-solution\">49<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2743881\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">[<\/mo><mn>\u22123<\/mn><mo>,<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">[<\/mo><mn>\u22123<\/mn><mo>,<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2743944\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2743945\" data-type=\"problem\"><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2306896\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">[<\/mo><mn>\u22124<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mo>\u222a<\/mo><mo stretchy=\"false\">[<\/mo><mn>9<\/mn><mo>,<\/mo><mi>\u221e<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo stretchy=\"false\">[<\/mo><mn>\u22124<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mo>\u222a<\/mo><mo stretchy=\"false\">[<\/mo><mn>9<\/mn><mo>,<\/mo><mi>\u221e<\/mi><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2306932\">For the following exercises, write the set of numbers represented on the number line in interval notation.<\/p>\r\n\r\n<div id=\"fs-id2306937\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2306938\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2306937-solution\">51<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2306940\" class=\"first-element\" data-type=\"media\" data-alt=\"A number line with two tick marks labeled: -2 and 1 respectively. There is an open circle around the -2 and a dot on the 1 with an arrow connecting the two.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A number line with two tick marks labeled: -2 and 1 respectively. There is an open circle around the -2 and a dot on the 1 with an arrow connecting the two.\" width=\"327\" height=\"49\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/c4578c3c93b430f4b6c1dadcdbc17cb89773ab31\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2306978\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2519857\" data-type=\"problem\"><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2519859\" class=\"first-element\" data-type=\"media\" data-alt=\"Number line with two tick marks labeled: -1 and 2 respectively. There is an open circle around the tick mark labeled -1 and a line that extends leftward from the circle. There is a dot around the tick mark labeled 2 and a line that extends rightward from the dot.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Number line with two tick marks labeled: -1 and 2 respectively. There is an open circle around the tick mark labeled -1 and a line that extends leftward from the circle. There is a dot around the tick mark labeled 2 and a line that extends rightward from the dot.\" width=\"327\" height=\"49\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/0d0ccffea4f593111ec4c247e4d2bfc48cdc71a2\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2519870\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2519871\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2519870-solution\">53<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2519872\" class=\"first-element\" data-type=\"media\" data-alt=\"Number line with one tick mark labeled 4. There is a dot on the tick mark and an arrow extending leftward from the dot.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Number line with one tick mark labeled 4. There is a dot on the tick mark and an arrow extending leftward from the dot.\" width=\"326\" height=\"47\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/209c1341f665f4bdeb785beb3ae583bf6cfdf2d0\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2519914\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id2519919\">For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall (2<sup>nd<\/sup> CALC 5:intersection, 1<sup>st<\/sup> curve, enter, 2<sup>nd<\/sup> curve, enter, guess, enter). Copy a sketch of the graph and shade the <em data-effect=\"italics\">x<\/em>-axis for your solution set to the inequality. Write final answers in interval notation.<\/p>\r\n\r\n<div id=\"fs-id2919704\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2919706\" data-type=\"problem\"><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2919707\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>5<\/mn><mo>&lt;<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>5<\/mn><mo>&lt;<\/mo><mn>2<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2919739\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2919740\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2919739-solution\">55<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2919741\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>4<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1787455\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1787456\" data-type=\"problem\"><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1787457\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mn>2<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1787491\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2958611\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1787491-solution\">57<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2958612\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>3<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2958690\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2958691\" data-type=\"problem\"><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2958692\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>5<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2663030\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<div id=\"fs-id2663035\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2663036\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2663035-solution\">59<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2663037\">Solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2968133\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2968134\" data-type=\"problem\"><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2968135\">Solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>12<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>12<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2968163\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2968164\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2968163-solution\">61<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2968165\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2264<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2260<\/mo><mn>\u22127<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>\u2260<\/mo><mn>\u22127<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2750208\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2750209\" data-type=\"problem\"><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2750210\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>130<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3000<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>130<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3000<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>is a profit formula for a small business. Find the set of <em data-effect=\"italics\">x<\/em>-values that will keep this profit positive.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2791668\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id2791673\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2791674\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2791673-solution\">63<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2791675\">In chemistry the volume for a certain gas is given by<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>V<\/mi><mo>=<\/mo><mn>20<\/mn><mi>T<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>V<\/mi><mo>=<\/mo><mn>20<\/mn><mi>T<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">V<\/em> is measured in cc and <em data-effect=\"italics\">T<\/em> is temperature in \u00baC. If the temperature varies between 80\u00baC and 120\u00baC, find the set of volume values.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2936685\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2936686\" data-type=\"problem\"><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2936687\">A basic cellular package costs $20\/mo. for 60 min of calling, with an additional charge of $.30\/min beyond that time.. The cost formula would be<\/p>\r\n\r\n<math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mspace width=\"0.5em\"><\/mspace><mi>C<\/mi><mo>=<\/mo><mn>20<\/mn><mo>+<\/mo><mn>.30<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>60<\/mn><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mspace width=\"0.5em\"><\/mspace><mi>C<\/mi><mo>=<\/mo><mn>20<\/mn><mo>+<\/mo><mn>.30<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>60<\/mn><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>If you have to keep your bill no greater than $50, what is the maximum calling minutes you can use?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_b888ed1c-c7d6-4bc4-a647-2fb628a05ad5\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"ui-has-child-title\" data-type=\"abstract\">\n<header>\n<h2 data-type=\"title\">Learning Objectives<\/h2>\n<\/header>\n<section>\n<p id=\"para-00001\">In this section, you will:<\/p>\n<ul id=\"list-00001\">\n<li>Use interval notation<\/li>\n<li>Use properties of inequalities.<\/li>\n<li>Solve inequalities in one variable algebraically.<\/li>\n<li>Solve absolute value inequalities.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<div id=\"Figure_02_07_001\" class=\"os-figure\">\n<figure class=\"medium\" data-id=\"Figure_02_07_001\"><span id=\"fs-id1521249\" data-type=\"media\" data-alt=\"Several red winner\u2019s ribbons lie on a white table.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Several red winner\u2019s ribbons lie on a white table.\" width=\"423\" height=\"282\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/d749a0eb956b42942e5bf0aa047f58fa1b4a6989\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/div>\n<\/div>\n<p id=\"fs-id1795255\">It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.<\/p>\n<section id=\"fs-id2781560\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Interval Notation<\/h2>\n<p id=\"fs-id981594\">Indicating the solution to an inequality such as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>can be achieved in several ways.<\/p>\n<p id=\"fs-id2384427\">We can use a number line as shown in <a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Figure_02_07_002\">Figure 2<\/a><strong>.<\/strong> The blue ray begins at<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.<\/p>\n<div id=\"Figure_02_07_002\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_07_002\"><span id=\"fs-id3094790\" data-type=\"media\" data-alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"508\" height=\"47\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/7292fc5c016c86fabdb7accf5e59376bacf91556\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">2<\/span><\/div>\n<\/div>\n<p id=\"fs-id2049812\">We can use <span id=\"term-00001\" class=\"no-emphasis\" data-type=\"term\">set-builder notation<\/span>:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn>\n<\/mrow> <mo>}<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn>\n<\/mrow> <mo>}<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>which translates to \u201call real numbers <em data-effect=\"italics\">x <\/em>such that <em data-effect=\"italics\">x <\/em>is greater than or equal to 4.\u201d Notice that braces are used to indicate a set.<\/p>\n<p id=\"fs-id1962920\">The third method is <span id=\"term-00002\" data-type=\"term\">interval notation<\/span>, in which solution sets are indicated with parentheses or brackets. The solutions to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>4<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>are represented as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>[<\/mo> <mrow>\n<mn>4<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\n<mn>4<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.<\/p>\n<p id=\"fs-id1417925\">The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be \u201cequaled.\u201d A few examples of an <span id=\"term-00003\" data-type=\"term\">interval<\/span>, or a set of numbers in which a solution falls, are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>[<\/mo> <mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mn>6<\/mn>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or all numbers between<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22122<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>6<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>6<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>including<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22122<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>but not including<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>6<\/mn><mo>;<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>6<\/mn><mo>;<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><mn>0<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><mn>0<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>all real numbers between, but not including<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22121<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22121<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>0<\/mn><mo>;<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>0<\/mn><mo>;<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>1<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>1<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>all real numbers less than and including<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>1.<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>1.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math><a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Table_02_07_01\">Table 1<\/a> outlines the possibilities.<\/p>\n<div id=\"Table_02_07_01\" class=\"os-table\">\n<table data-id=\"Table_02_07_01\">\n<thead>\n<tr>\n<th scope=\"col\" data-align=\"center\">Set Indicated<\/th>\n<th scope=\"col\" data-align=\"center\">Set-Builder Notation<\/th>\n<th scope=\"col\" data-align=\"center\">Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>All real numbers between <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b<\/em>, but not including <em data-effect=\"italics\">a <\/em>or <em data-effect=\"italics\">b<\/em><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td>All real numbers greater than <em data-effect=\"italics\">a<\/em>, but not including <em data-effect=\"italics\">a<\/em><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mi>a<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mi>a<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em data-effect=\"italics\">b<\/em>, but not including <em data-effect=\"italics\">b<\/em><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td>All real numbers greater than <em data-effect=\"italics\">a<\/em>, including <em data-effect=\"italics\">a<\/em><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2265<\/mo><mi>a<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2265<\/mo><mi>a<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>[<\/mo> <mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>[<\/mo> <mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em data-effect=\"italics\">b<\/em>, including <em data-effect=\"italics\">b<\/em><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow> <mo>]<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow> <mo>]<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em data-effect=\"italics\">a <\/em>and<em data-effect=\"italics\"> b<\/em>, including <em data-effect=\"italics\">a<\/em><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>[<\/mo> <mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>[<\/mo> <mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, including <em data-effect=\"italics\">b<\/em><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>&lt;<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow> <mo>]<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow> <mo>]<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b<\/em>, including <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b<\/em><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>a<\/mi><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>[<\/mo> <mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow> <mo>]<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>[<\/mo> <mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi>\n<\/mrow> <mo>]<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em data-effect=\"italics\">a<\/em> or greater than <em data-effect=\"italics\">b<\/em><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>a<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>or<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mi>a<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>or<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mi>b<\/mi>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>a<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u222a<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>b<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>a<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u222a<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>b<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<tr>\n<td>All real numbers<\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is all real numbers<\/mtext>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mo>|<\/mo><mi>x<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is all real numbers<\/mtext>\n<\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/div>\n<\/div>\n<div id=\"Example_02_07_01\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1940764\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1767073\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Using Interval Notation to Express All Real Numbers Greater Than or Equal to <em data-effect=\"italics\">a<\/em><\/h3>\n<p id=\"fs-id3169488\">Use interval notation to indicate all real numbers greater than or equal to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22122.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id3039360\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1295224\">Use a bracket on the left of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22122<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and parentheses after infinity:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>[<\/mo> <mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>The bracket indicates that<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22122<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>is included in the set with all real numbers greater than<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22122<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>to infinity.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2383200\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#1<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_02_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2682208\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1517140\">Use interval notation to indicate all real numbers between and including<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22123<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22123<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>5.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>5.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_07_02\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">2<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1543726\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1411353\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Using Interval Notation to Express All Real Numbers Less Than or Equal to <em data-effect=\"italics\">a <\/em>or Greater Than or Equal to <em data-effect=\"italics\">b<\/em><\/h3>\n<p id=\"fs-id1395714\">Write the interval expressing all real numbers less than or equal to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22121<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22121<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>or greater than or equal to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>1.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>1.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1943180\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1736769\">We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mo>\u2212<\/mo><mi>\u221e<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mi>\u221e<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and ends at<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22121<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22121<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>which is written as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>\u22121<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>\u22121<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2513420\">The second interval must show all real numbers greater than or equal to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>1<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>1<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>which is written as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>[<\/mo> <mrow>\n<mn>1<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\n<mn>1<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>However, we want to combine these two sets. We accomplish this by inserting the union symbol,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u222a<\/mo><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u222a<\/mo><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>between the two intervals.<\/p>\n<div id=\"fs-id3035449\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>\u22121<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow><mo>\u222a<\/mo><mrow><mo>[<\/mo> <mrow>\n<mn>1<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>\u22121<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow><mo>\u222a<\/mo><mrow><mo>[<\/mo> <mrow>\n<mn>1<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow> <mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1592740\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#2<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_07_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id906336\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2644263\">Express all real numbers less than<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22122<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22122<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>or greater than or equal to 3 in interval notation.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2734136\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Properties of Inequalities<\/h2>\n<p id=\"fs-id1500576\">When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the <span id=\"term-00004\" class=\"no-emphasis\" data-type=\"term\">addition property<\/span> and the <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">multiplication property<\/span> to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.<\/p>\n<div id=\"fs-id2425325\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"A General Note\"><span class=\"os-title-label\" data-type=\"\">Properties of Inequalities<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"fs-id3176836\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mstyle><mi>A<\/mi><mi>d<\/mi><mi>d<\/mi><mi>i<\/mi><mi>t<\/mi><mi>i<\/mi><mi>o<\/mi><mi>n<\/mi><\/mstyle><mspace width=\"0.8\"><\/mspace><mtext><\/mtext><mstyle><mi>P<\/mi><mi>r<\/mi><mi>o<\/mi><mi>p<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>y<\/mi><\/mstyle>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mstyle><mi>M<\/mi><mi>u<\/mi><mi>l<\/mi><mi>t<\/mi><mi>i<\/mi><mi>p<\/mi><mi>l<\/mi><mi>i<\/mi><mi>c<\/mi><mi>a<\/mi><mi>t<\/mi><mi>i<\/mi><mi>o<\/mi><mi>n<\/mi><\/mstyle><mspace width=\"0.8\"><\/mspace><mtext><\/mtext><mstyle><mi>P<\/mi><mi>r<\/mi><mi>o<\/mi><mi>p<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>y<\/mi><\/mstyle>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>and<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mi>c<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>and<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mstyle><mi>A<\/mi><mi>d<\/mi><mi>d<\/mi><mi>i<\/mi><mi>t<\/mi><mi>i<\/mi><mi>o<\/mi><mi>n<\/mi><\/mstyle><mspace width=\"0.8\"><\/mspace><mtext><\/mtext><mstyle><mi>P<\/mi><mi>r<\/mi><mi>o<\/mi><mi>p<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>y<\/mi><\/mstyle>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mstyle><mi>M<\/mi><mi>u<\/mi><mi>l<\/mi><mi>t<\/mi><mi>i<\/mi><mi>p<\/mi><mi>l<\/mi><mi>i<\/mi><mi>c<\/mi><mi>a<\/mi><mi>t<\/mi><mi>i<\/mi><mi>o<\/mi><mi>n<\/mi><\/mstyle><mspace width=\"0.8\"><\/mspace><mtext><\/mtext><mstyle><mi>P<\/mi><mi>r<\/mi><mi>o<\/mi><mi>p<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>y<\/mi><\/mstyle>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>and<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mi>c<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>If <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mo>&lt;<\/mo><mi>b<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>and<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.8\"><\/mspace><mtext>then <\/mtext><mspace width=\"0.8\"><\/mspace><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mi>b<\/mi><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1340108\">These properties also apply to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>a<\/mi><mo>\u2264<\/mo><mi>b<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>a<\/mi><mo>\u2264<\/mo><mi>b<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>&gt;<\/mo><mi>b<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>&gt;<\/mo><mi>b<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>a<\/mi><mo>\u2265<\/mo><mi>b<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>a<\/mi><mo>\u2265<\/mo><mi>b<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_07_03\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">3<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2453957\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2644438\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Demonstrating the Addition Property<\/h3>\n<p id=\"fs-id1245519\">Illustrate the addition property for inequalities by solving each of the following:<\/p>\n<ol id=\"fs-id3085981\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>6<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>6<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d2<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<details id=\"fs-id890147\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1477754\">The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.<\/p>\n<ol id=\"fs-id1419199\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>+<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn><mo>+<\/mo><mn>15<\/mn><mo> <\/mo>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Add 15 to both sides<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>19<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>+<\/mo><mn>15<\/mn><mo>&lt;<\/mo><mn>4<\/mn><mo>+<\/mo><mn>15<\/mn><mo> <\/mo>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Add 15 to both sides<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>19<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mn>6<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>6<\/mn><mo>+<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4em\"><\/mspace><mtext>Add 1 to both sides<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mn>7<\/mn><mo>\u2265<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mn>6<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>6<\/mn><mo>+<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4em\"><\/mspace><mtext>Add 1 to both sides<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mn>7<\/mn><mo>\u2265<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4em\"><\/mspace><mtext>Subtract 7 from both sides<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mo>&gt;<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4em\"><\/mspace><mtext>Subtract 7 from both sides<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2269039\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#3<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_07_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2677967\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1698684\">Solve:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3<\/mn><mi>x<\/mi><mn>\u22122<\/mn><mo>&lt;<\/mo><mn>1.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><mi>x<\/mi><mn>\u22122<\/mn><mo>&lt;<\/mo><mn>1.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_07_04\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">4<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1525452\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2507109\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Demonstrating the Multiplication Property<\/h3>\n<p id=\"fs-id1895873\">Illustrate the multiplication property for inequalities by solving each of the following:<\/p>\n<ol id=\"fs-id3061147\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>6<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>5<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d2<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>10<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>10<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<details id=\"fs-id1238058\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<ol id=\"fs-id1496121\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.5em\"><\/mspace><mn>3<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>&lt;<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.5em\"><\/mspace><mn>3<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>&lt;<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiply by <\/mtext><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Reverse the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiply by <\/mtext><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Reverse the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2.5em\"><\/mspace><mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>10<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3.5em\"><\/mspace><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiply by <\/mtext><mo>\u2212<\/mo><mn>1.<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4.5em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Reverse the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2.5em\"><\/mspace><mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>10<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3.5em\"><\/mspace><mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiply by <\/mtext><mo>\u2212<\/mo><mn>1.<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4.5em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Reverse the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1730534\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#4<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_07_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2422478\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1396214\">Solve:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>\u2265<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>\u2265<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1249131\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Inequalities in One Variable Algebraically<\/h2>\n<p id=\"fs-id2710126\">As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.<\/p>\n<div id=\"Example_02_07_05\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">5<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2417012\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1407961\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving an Inequality Algebraically<\/h3>\n<p id=\"fs-id2444209\">Solve the inequality:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>13<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>10<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>13<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>10<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2437055\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1449306\">Solving this inequality is similar to solving an equation up until the last step.<\/p>\n<div id=\"fs-id2628929\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\".5em\"><\/mspace><mn>13<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>10<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>13<\/mn><mo>\u2212<\/mo><mn>17<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>\u22124<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Move variable terms to one side of the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.8em\"><\/mspace><mn>\u221217<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>\u221217<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Isolate the variable term<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3.1em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Dividing both sides by <\/mtext><mn>\u221217<\/mn><mspace width=\"0.8\"><\/mspace><mtext>reverses the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\".5em\"><\/mspace><mn>13<\/mn><mo>\u2212<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>10<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>13<\/mn><mo>\u2212<\/mo><mn>17<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>\u22124<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Move variable terms to one side of the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.8em\"><\/mspace><mn>\u221217<\/mn><mi>x<\/mi><mo>\u2265<\/mo><mn>\u221217<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Isolate the variable term<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3.1em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Dividing both sides by <\/mtext><mn>\u221217<\/mn><mspace width=\"0.8\"><\/mspace><mtext>reverses the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1846594\">The solution set is given by the interval<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>1<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>1<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or all real numbers less than and including 1.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1572093\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#5<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_07_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1457850\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1790143\">Solve the inequality and write the answer using interval notation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mo>\u2212<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>&lt;<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>1.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>&lt;<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>1.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_07_06\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">6<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2441066\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1416866\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving an Inequality with Fractions<\/h3>\n<p id=\"fs-id1435934\">Solve the following inequality and write the answer in interval notation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2917362\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1408780\">We begin solving in the same way we do when solving an equation.<\/p>\n<div id=\"fs-id1513950\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2.8em\"><\/mspace><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\".9em\"><\/mspace><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Put variable terms on one side<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>9<\/mn>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>8<\/mn>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Write fractions with common denominator<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2.5em\"><\/mspace><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>17<\/mn>\n<\/mrow>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4.2em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<mrow>\n<mn>17<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiplying by a negative number reverses the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4.2em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mfrac>\n<mrow>\n<mn>15<\/mn>\n<\/mrow>\n<mrow>\n<mn>34<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2.8em\"><\/mspace><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\".9em\"><\/mspace><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Put variable terms on one side<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>9<\/mn>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>8<\/mn>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Write fractions with common denominator<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2.5em\"><\/mspace><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>17<\/mn>\n<\/mrow>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2265<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4.2em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>8<\/mn>\n<\/mfrac>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>12<\/mn>\n<\/mrow>\n<mrow>\n<mn>17<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Multiplying by a negative number reverses the inequality<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"4.2em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mfrac>\n<mrow>\n<mn>15<\/mn>\n<\/mrow>\n<mrow>\n<mn>34<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1556088\">The solution set is the interval<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mfrac>\n<mrow>\n<mn>15<\/mn>\n<\/mrow>\n<mrow>\n<mn>34<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow> <mo>]<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo> <mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mfrac>\n<mrow>\n<mn>15<\/mn>\n<\/mrow>\n<mrow>\n<mn>34<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow> <mo>]<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2497523\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#6<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_07_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2436750\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1774157\">Solve the inequality and write the answer in interval notation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2264<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>8<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2264<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>8<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2947575\" data-depth=\"1\">\n<h2 data-type=\"title\">Understanding Compound Inequalities<\/h2>\n<p id=\"fs-id1574863\">A <span id=\"term-00006\" data-type=\"term\">compound inequality<\/span> includes two inequalities in one statement. A statement such as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>4<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>6<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>4<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>6<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>means<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>4<\/mn><mo>&lt;<\/mo><mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>4<\/mn><mo>&lt;<\/mo><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>\u2264<\/mo><mn>6.<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2264<\/mo><mn>6.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.<\/p>\n<div id=\"Example_02_07_07\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">7<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2501975\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1516646\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Compound Inequality<\/h3>\n<p id=\"fs-id2503657\">Solve the compound inequality:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2382216\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2442855\">The first method is to write two separate inequalities:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6.<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6.<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>We solve them independently.<\/p>\n<div id=\"fs-id2434911\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>1<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mn>2<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>\u2264<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>1<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mn>2<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>\u2264<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1290697\">Then, we can rewrite the solution as a compound inequality, the same way the problem began.<\/p>\n<div id=\"fs-id1922364\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1583611\">In interval notation, the solution is written as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>[<\/mo> <mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>,<\/mo><mn>2<\/mn>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>,<\/mo><mn>2<\/mn>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id1722369\">The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.<\/p>\n<div id=\"fs-id2020141\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>1<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Isolate the variable term, and subtract 2 from all three parts<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Divide through all three parts by 2<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>1<\/mn><mo>\u2264<\/mo><mn>2<\/mn><mi>x<\/mi><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Isolate the variable term, and subtract 2 from all three parts<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>\u2264<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Divide through all three parts by 2<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2390439\">We get the same solution:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>[<\/mo> <mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>,<\/mo><mn>2<\/mn>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>,<\/mo><mn>2<\/mn>\n<\/mrow> <mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1215697\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#7<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_07_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2434068\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1941577\">Solve the compound inequality:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>4<\/mn><mo>&lt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/mn><mo>\u2264<\/mo><mn>10.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>4<\/mn><mo>&lt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/mn><mo>\u2264<\/mo><mn>10.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_07_08\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">8<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1500526\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1919530\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Compound Inequality with the Variable in All Three Parts<\/h3>\n<p id=\"fs-id1243509\">Solve the compound inequality with variables in all three parts:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>10.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo>+<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>10.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2048755\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1191836\">Let&#8217;s try the first method. Write two inequalities<strong>:<\/strong><\/p>\n<div id=\"fs-id3239342\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>10<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mn>3<\/mn><mo>&gt;<\/mo><mn>6<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>\u221210<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mn>5<\/mn><mo>&gt;<\/mo><mn>6<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.8em\"><\/mspace><mn>2<\/mn><mi>x<\/mi><mo>&gt;<\/mo><mn>\u22128<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>\n<mo>&gt;<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2.5em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mn>\u22124<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.3em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.5em\"><\/mspace><mn>\u22124<\/mn><mo>&lt;<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>10<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mn>3<\/mn><mo>&gt;<\/mo><mn>6<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>\u221210<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mn>5<\/mn><mo>&gt;<\/mo><mn>6<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.8em\"><\/mspace><mn>2<\/mn><mi>x<\/mi><mo>&gt;<\/mo><mn>\u22128<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>\n<mo>&gt;<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2.5em\"><\/mspace><mi>x<\/mi><mo>&gt;<\/mo><mn>\u22124<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.3em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.5em\"><\/mspace><mn>\u22124<\/mn><mo>&lt;<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1954589\">The solution set is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22124<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22124<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or in interval notation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>(<\/mo>\n<mrow>\n<mo><\/mo><mn>\u22124<\/mn><mo>,<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo>\n<mrow>\n<mo><\/mo><mn>\u22124<\/mn><mo>,<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>6<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line. See <a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Figure_02_07_003\">Figure 3<\/a><strong>.<\/strong><\/p>\n<div id=\"Figure_02_07_003\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_07_003\"><span id=\"fs-id1929508\" data-type=\"media\" data-alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\" width=\"487\" height=\"69\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/f664946114f8c591904e1c6d4310aad6242d6b10\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">3<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2700910\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#8<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_07_08\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2957500\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1441254\">Solve the compound inequality:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>5<\/mn><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>5<\/mn><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1161758\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Absolute Value Inequalities<\/h2>\n<p id=\"fs-id1338469\">As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>x<\/mi><mo>,<\/mo><mn>0<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>x<\/mi><mo>,<\/mo><mn>0<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>has an absolute value of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>as it is <em data-effect=\"italics\">x <\/em>units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.<\/p>\n<p id=\"fs-id1230083\">An <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">absolute value inequality<\/span> is an equation of the form<\/p>\n<div id=\"fs-id3107252\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>&lt;<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>\u2264<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><mo>&gt;<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>or <\/mtext><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><mo>\u2265<\/mo><mi>B<\/mi><mo>,<\/mo>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>&lt;<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>\u2264<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>&gt;<\/mo><mi>B<\/mi><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>or <\/mtext><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>|<\/mo><mi>A<\/mi><mo>|<\/mo><mo>\u2265<\/mo><mi>B<\/mi><mo>,<\/mo><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1184764\">Where <em data-effect=\"italics\">A<\/em>, and sometimes <em data-effect=\"italics\">B<\/em>, represents an algebraic expression dependent on a variable <em data-effect=\"italics\">x. <\/em>Solving the inequality means finding the set of all<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><em data-effect=\"italics\">&#8211;<\/em>values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.<\/p>\n<p id=\"fs-id1581445\">There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.<\/p>\n<p id=\"fs-id1199584\">Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of <em data-effect=\"italics\">x-<\/em>values such that the distance between<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and 600 is less than or equal to 200. We represent the distance between<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and 600 as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and therefore,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>200<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>200<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>or<\/p>\n<div id=\"fs-id1681298\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd>\n<mrow>\n<mn>\u2212200<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mn>200<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mn>\u2212200<\/mn><mo>+<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><mo>+<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mn>200<\/mn><mo>+<\/mo><mn>600<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mn>400<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>800<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd>\n<mrow>\n<mn>\u2212200<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mn>200<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mn>\u2212200<\/mn><mo>+<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>600<\/mn><mo>+<\/mo><mn>600<\/mn><mo>\u2264<\/mo><mn>200<\/mn><mo>+<\/mo><mn>600<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mn>400<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>800<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1976398\">This means our returns would be between $400 and $800.<\/p>\n<p id=\"fs-id1323840\">To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.<\/p>\n<div id=\"fs-id2930959\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"A General Note\"><span class=\"os-title-label\" data-type=\"\">Absolute Value Inequalities <\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1520813\">For an algebraic expression <em data-effect=\"italics\">X,<\/em> and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>k<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>k<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>an <span id=\"term-00008\" data-type=\"term\">absolute value inequality<\/span> is an inequality of the form<\/p>\n<div id=\"fs-id2016621\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is equivalent to\u00a0<\/mtext><mo>\u2212<\/mo><mi>k<\/mi><mo>&lt;<\/mo><mi>X<\/mi><mo>&lt;<\/mo><mi>k<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>&gt;<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is equivalent to\u00a0<\/mtext><mi>X<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>or\u00a0<\/mtext><mi>X<\/mi><mo>&gt;<\/mo><mi>k<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is equivalent to\u00a0<\/mtext><mo>\u2212<\/mo><mi>k<\/mi><mo>&lt;<\/mo><mi>X<\/mi><mo>&lt;<\/mo><mi>k<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>&gt;<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is equivalent to\u00a0<\/mtext><mi>X<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mi>k<\/mi><mspace width=\"0.8\"><\/mspace><mtext>or\u00a0<\/mtext><mi>X<\/mi><mo>&gt;<\/mo><mi>k<\/mi><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1277147\">These statements also apply to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mi>k<\/mi><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mi>k<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mi>k<\/mi><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>X<\/mi> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mi>k<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_07_09\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">9<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2433767\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2020938\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Determining a Number within a Prescribed Distance<\/h3>\n<p id=\"fs-id1758539\">Describe all values<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>within a distance of 4 from the number 5.<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1570653\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2293280\">We want the distance between<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>and 5 to be less than or equal to 4. We can draw a number line, such as in <a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Figure_02_07_004\">Figure 4<\/a><strong>,<\/strong> to represent the condition to be satisfied.<\/p>\n<div id=\"Figure_02_07_004\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_07_004\"><span id=\"fs-id1762406\" data-type=\"media\" data-alt=\"A number line with one tick mark in the center labeled: 5. The tick marks on either side of the center one are not marked. Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A number line with one tick mark in the center labeled: 5. The tick marks on either side of the center one are not marked. Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.\" width=\"481\" height=\"80\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/921f3c827d871d2af31e89f140164ac0c046fed6\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">4<\/span><\/div>\n<\/div>\n<p id=\"fs-id1181980\">The distance from<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>to 5 can be represented using an absolute value symbol,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>.<\/mo> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>.<\/mo> <\/mrow><\/annotation-xml><\/semantics><\/math>Write the values of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi> <\/mrow><\/annotation-xml><\/semantics><\/math>that satisfy the condition as an absolute value inequality.<\/p>\n<div id=\"fs-id3273982\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1334091\">We need to write two inequalities as there are always two solutions to an absolute value equation.<\/p>\n<div id=\"fs-id1690088\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>\u2264<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>\u2265<\/mo><mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.8em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mn>9<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.8em\"><\/mspace><mi>x<\/mi><mo>\u2265<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>\u2264<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>and<\/mtext><mspace width=\"2em\"><\/mspace>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>\u2265<\/mo><mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.8em\"><\/mspace><mi>x<\/mi><mo>\u2264<\/mo><mn>9<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.8em\"><\/mspace><mi>x<\/mi><mo>\u2265<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2021289\">If the solution set is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>\u2264<\/mo><mn>9<\/mn> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2264<\/mo><mn>9<\/mn> <\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>then the solution set is an interval including all real numbers between and including 1 and 9.<\/p>\n<p id=\"fs-id1288018\">So<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>is equivalent to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>[<\/mo> <mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>9<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow> <\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>[<\/mo> <mrow>\n<mn>1<\/mn><mo>,<\/mo><mn>9<\/mn>\n<\/mrow> <mo>]<\/mo><\/mrow> <\/mrow><\/annotation-xml><\/semantics><\/math>in interval notation.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2957557\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#9<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_07_09\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1811321\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1811322\">Describe all <em data-effect=\"italics\">x-<\/em>values within a distance of 3 from the number 2.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_07_10\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">10<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1428981\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1428983\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving an Absolute Value Inequality<\/h3>\n<p id=\"fs-id1802343\">Solve<\/p>\n<p><math display=\"inline\"><semantics><mrow><mo>|<\/mo>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<mo>|<\/mo><mo>\u2264<\/mo><mn>3<\/mn>\n&nbsp;\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>\u2264<\/mo><mn>3<\/mn><\/annotation-xml><\/semantics><\/math>.<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1571578\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<div id=\"fs-id1753466\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow><mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><\/mtd><\/mtr><\/mtable>\n<mo>|<\/mo>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn> <mo>|<\/mo><mo>\u2264<\/mo><mn>3<\/mn>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><\/mtd><\/mtr>\n<mn>\u22123<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2264<\/mo><mn>3<\/mn>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><\/mtd><\/mtr>\n<mn>\u22122<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>4<\/mn>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><\/mtd><\/mtr>\n<mo stretchy=\"false\">[<\/mo><mn>\u22122<\/mn><mo>,<\/mo><mn>4<\/mn><mo stretchy=\"false\">]<\/mo>\n&nbsp;\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><mo>|<\/mo>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn> <mo>|<\/mo><mo>\u2264<\/mo><mn>3<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><mn>\u22123<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2264<\/mo><mn>3<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><mn>\u22122<\/mn><mo>\u2264<\/mo><mi>x<\/mi><mo>\u2264<\/mo><mn>4<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><mo stretchy=\"false\">[<\/mo><mn>\u22122<\/mn><mo>,<\/mo><mn>4<\/mn><mo stretchy=\"false\">]<\/mo>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_07_11\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">11<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1690263\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2947783\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Using a Graphical Approach to Solve Absolute Value Inequalities<\/h3>\n<p id=\"fs-id2947788\">Given the equation<\/p>\n<p><math display=\"inline\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>|<\/mo>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn> <mo>|<\/mo><mo>+<\/mo><mn>3<\/mn><mo>,<\/mo>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac><mo>|<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>|<\/mo><mo>+<\/mo><mn>3<\/mn><mo>,<\/mo><\/annotation-xml><\/semantics><\/math>determine the <em data-effect=\"italics\">x<\/em>-values for which the <em data-effect=\"italics\">y<\/em>-values are negative.<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id3130412\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id3130414\">We are trying to determine where<\/p>\n<p><math display=\"inline\"><semantics><mrow><mi>y<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo>&nbsp;\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mi>y<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/annotation-xml><\/semantics><\/math>which is when<\/p>\n<p><math display=\"inline\"><semantics><mrow><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>|<\/mo>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn> <mo>|<\/mo><mo>+<\/mo><mn>3<\/mn><mo>&lt;<\/mo><mn>0.<\/mn>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac><mo>|<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>|<\/mo><mo>+<\/mo><mn>3<\/mn><mo>&lt;<\/mo><mn>0.<\/mn><\/annotation-xml><\/semantics><\/math>We begin by isolating the absolute value.<\/p>\n<div id=\"fs-id3156619\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow><mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><\/mtd><\/mtr><\/mtable>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>|<\/mo>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<mo>|<\/mo><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\n<mtd columnalign=\"left\"><\/mtd>\n<mspace width=\"2em\"><\/mspace><mtext>Multiply both sides by \u20132, and reverse the inequality<\/mtext><mo>.<\/mo>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mspace width=\"1.5em\"><\/mspace>\n<mo>|<\/mo>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<mo>|<\/mo><mo>&gt;<\/mo><mn>6<\/mn><\/mtd><\/mtr>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n&nbsp;\n&nbsp;\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\"><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>|<\/mo>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<mo>|<\/mo><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\n&nbsp;\n<\/mtd>\n<mtd columnalign=\"left\"><mspace width=\"2em\"><\/mspace><mtext>Multiply both sides by \u20132, and reverse the inequality<\/mtext><mo>.<\/mo>\n&nbsp;\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mspace width=\"1.5em\"><\/mspace>\n<mo>|<\/mo>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<mo>|<\/mo><mo>&gt;<\/mo><mn>6<\/mn>&nbsp;\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id3260232\">Next, we solve for the equality<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mo>|<\/mo>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<mo>|<\/mo><mo>=<\/mo><mn>6.<\/mn>&nbsp;\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>|<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>|<\/mo><mo>=<\/mo><mn>6.<\/mn><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id2762668\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.9em\"><\/mspace><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>11<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>or<\/mtext><mspace width=\"2em\"><\/mspace>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.9em\"><\/mspace><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mn>11<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.9em\"><\/mspace><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>11<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>or<\/mtext><mspace width=\"2em\"><\/mspace>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.9em\"><\/mspace><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mn>11<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2644216\">Now, we can examine the graph to observe where the <em data-effect=\"italics\">y-<\/em>values are negative. We observe where the branches are below the <em data-effect=\"italics\">x-<\/em>axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mn>11<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mn>11<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and that the graph opens downward. See <a class=\"autogenerated-content\" href=\"2-7-linear-inequalities-and-absolute-value-inequalities#Figure_02_07_007\">Figure 5<\/a><strong>.<\/strong><\/p>\n<div id=\"Figure_02_07_007\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_07_007\"><span id=\"fs-id2777886\" data-type=\"media\" data-alt=\"A coordinate plan with the x-axis ranging from -5 to 5 and the y-axis ranging from -4 to 4. The function y = -1\/2|4x \u2013 5| + 3 is graphed. An open circle appears at the point -0.25 and an arrow\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A coordinate plan with the x-axis ranging from -5 to 5 and the y-axis ranging from -4 to 4. The function y = -1\/2|4x \u2013 5| + 3 is graphed. An open circle appears at the point -0.25 and an arrow\" width=\"396\" height=\"372\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/f34a03da829882c6e85c8ae73414515a4fd1bb0c\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">5<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1969441\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#10<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_07_10\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1951872\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1951873\">Solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mo>\u2212<\/mo><mn>2<\/mn><mrow><mo>|<\/mo> <mrow>\n<mi>k<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mo>\u2212<\/mo><mn>6.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><mn>2<\/mn><mrow><mo>|<\/mo> <mrow>\n<mi>k<\/mi><mo>\u2212<\/mo><mn>4<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mo>\u2212<\/mo><mn>6.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1904611\" class=\"media-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Media\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Media\"><span class=\"os-title-label\">Media<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1904617\">Access these online resources for additional instruction and practice with linear inequalities and absolute value inequalities.<\/p>\n<ul id=\"fs-id1904622\">\n<li><a href=\"http:\/\/openstax.org\/l\/intervalnotn\" target=\"_blank\" rel=\"noopener nofollow\">Interval notation<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/solvelinineq\" target=\"_blank\" rel=\"noopener nofollow\">How to solve linear inequalities<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/solveineq\" target=\"_blank\" rel=\"noopener nofollow\">How to solve an inequality<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/absvaleq\" target=\"_blank\" rel=\"noopener nofollow\">Absolute value equations<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/compndineqs\" target=\"_blank\" rel=\"noopener nofollow\">Compound inequalities<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/absvalineqs\" target=\"_blank\" rel=\"noopener nofollow\">Absolute value inequalities<\/a><\/li>\n<\/ul>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.7 Section Exercises<\/span><\/h2>\n<section id=\"fs-id2529020\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1901196\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1901201\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1901202\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1901201-solution\">1<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1901203\">When solving an inequality, explain what happened from Step 1 to Step 2:<\/p>\n<p id=\"fs-id1901206\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>Step 1<\/mtext>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>&gt;<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>Step 2<\/mtext>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>Step 1<\/mtext>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>&gt;<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>Step 2<\/mtext>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"3em\"><\/mspace><mi>x<\/mi><mo>&lt;<\/mo><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1904125\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1904126\" data-type=\"problem\"><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1904128\">When solving an inequality, we arrive at:<\/p>\n<p id=\"fs-id1904131\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mn>2<\/mn><mo>&lt;<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mn>2<\/mn><mo>&lt;<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id3274019\">Explain what our solution set is.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3274022\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3274023\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3274022-solution\">3<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3274024\">When writing our solution in interval notation, how do we represent all the real numbers?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2304131\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2304132\" data-type=\"problem\"><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2304133\">When solving an inequality, we arrive at:<\/p>\n<p id=\"fs-id2304136\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mn>2<\/mn><mo>&gt;<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"1.2em\"><\/mspace><mn>2<\/mn><mo>&gt;<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id1886074\">Explain what our solution set is.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1886077\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1886078\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1886077-solution\">5<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1886080\">Describe how to graph<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>y<\/mi><mo>=<\/mo><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>y<\/mi><mo>=<\/mo><mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2925477\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id2925482\">For the following exercises, solve the inequality. Write your final answer in interval notation.<\/p>\n<div id=\"fs-id2925485\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2925486\" data-type=\"problem\"><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2925487\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>\u2264<\/mo><mn>9<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>\u2264<\/mo><mn>9<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2925506\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2925508\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2925506-solution\">7<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2925509\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>\u2265<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>\u2265<\/mo><mn>7<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2783754\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2783755\" data-type=\"problem\"><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2783756\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2783780\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2783781\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2783780-solution\">9<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2783782\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2265<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2523115\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2523116\" data-type=\"problem\"><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2523117\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>5<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2264<\/mo><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mn>5<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2372045\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2372046\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2372045-solution\">11<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2372047\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22125<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22125<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2996112\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2996114\" data-type=\"problem\"><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2996115\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1705070\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1705071\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1705070-solution\">13<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1705072\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<mn>8<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>\u2265<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>10<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<mn>8<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>\u2265<\/mo><mfrac>\n<mn>3<\/mn>\n<mrow>\n<mn>10<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2786106\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2786107\" data-type=\"problem\"><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2786108\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>\u2264<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>\u2264<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>5<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1886125\">For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.<\/p>\n<div id=\"fs-id1886129\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1886130\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1886129-solution\">15<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1886131\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>9<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>\u22126<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>9<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>\u22126<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1929693\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1929694\" data-type=\"problem\"><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1929695\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>7<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1929727\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1929728\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1929727-solution\">17<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1929729\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>|<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>11<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>|<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>11<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1768168\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1768169\" data-type=\"problem\"><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1768170\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>+<\/mo><mn>1<\/mn><mo>\u2264<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>+<\/mo><mn>1<\/mn><mo>\u2264<\/mo><mn>6<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1768206\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1768207\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1768206-solution\">19<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2742909\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>+<\/mo><mn>4<\/mn><mo>\u2265<\/mo><mn>10<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>+<\/mo><mn>4<\/mn><mo>\u2265<\/mo><mn>10<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1909611\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1909612\" data-type=\"problem\"><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1909614\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>\u22122<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>13<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>\u22122<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>13<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1909646\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1909647\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1909646-solution\">21<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1909648\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>|<\/mo><mo>&lt;<\/mo><mn>\u22124<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>|<\/mo><mo>&lt;<\/mo><mn>\u22124<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3062896\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3062897\" data-type=\"problem\"><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3062898\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>\u22121<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3062925\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3062926\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3062925-solution\">23<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3062928\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>2<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1663908\">For the following exercises, describe all the <em data-effect=\"italics\">x<\/em>-values within or including a distance of the given values.<\/p>\n<div id=\"fs-id1663916\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1663917\" data-type=\"problem\"><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1663918\">Distance of 5 units from the number 7<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1663921\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1663922\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1663921-solution\">25<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1663923\">Distance of 3 units from the number 9<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3085702\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3085704\" data-type=\"problem\"><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3085705\">Distance of 10 units from the number 4<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3085708\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3085709\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3085708-solution\">27<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3085710\">Distance of 11 units from the number 1<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id3085740\">For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.<\/p>\n<div id=\"fs-id3085744\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3085746\" data-type=\"problem\"><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3085747\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22124<\/mn><mo>&lt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>\u2264<\/mo><mn>18<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22124<\/mn><mo>&lt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>\u2264<\/mo><mn>18<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2272698\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2272699\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2272698-solution\">29<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2272700\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2413129\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2413130\" data-type=\"problem\"><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2413131\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>7<\/mn><mo>+<\/mo><mi>y<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><mo>&lt;<\/mo><mn>7<\/mn><mo>+<\/mo><mi>y<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2413161\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2413162\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2413161-solution\">31<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2413163\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>&lt;<\/mo><mn>\u221211<\/mn><mspace width=\"0.8\"><\/mspace><mtext>or <\/mtext><mspace width=\"0.8\"><\/mspace><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>6<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>&lt;<\/mo><mn>\u221211<\/mn><mspace width=\"0.8\"><\/mspace><mtext>or <\/mtext><mspace width=\"0.8\"><\/mspace><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>\u2265<\/mo><mn>6<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1779673\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1779674\" data-type=\"problem\"><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1779676\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1779700\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1779705\">For the following exercises, graph the function. Observe the points of intersection and shade the <em data-effect=\"italics\">x<\/em>-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.<\/p>\n<div id=\"fs-id1779714\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1779716\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1779714-solution\">33<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1779717\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>|<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>|<\/mo><mo>&gt;<\/mo><mn>2<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1710959\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1710960\" data-type=\"problem\"><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1710961\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>5<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1710991\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1710992\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1710991-solution\">35<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1710993\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2264<\/mo><mn>4<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3021007\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3021008\" data-type=\"problem\"><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3021009\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>7<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>7<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3021037\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3021038\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3021037-solution\">37<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3021039\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2826233\">For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the <em data-effect=\"italics\">y<\/em>-values of the lines.<\/p>\n<div id=\"fs-id2826242\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2826244\" data-type=\"problem\"><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2826245\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>&lt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>&lt;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2826268\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2826269\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2826268-solution\">39<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2826270\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>&gt;<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1911262\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1911263\" data-type=\"problem\"><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1911264\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1911288\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1911289\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1911288-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1911290\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&gt;<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3059337\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3059338\" data-type=\"problem\"><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3059339\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&lt;<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>&lt;<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id3071965\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id3071970\">For the following exercises, write the set in interval notation.<\/p>\n<div id=\"fs-id3071973\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3071974\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3071973-solution\">43<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3071975\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\n<mn>\u22121<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mrow><mn>3<\/mn> <mo>}<\/mo><\/mrow>\n<\/mrow><\/mrow>\n<\/mrow> <\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\n<mn>\u22121<\/mn><mo>&lt;<\/mo><mi>x<\/mi><mo>&lt;<\/mo><mrow><mn>3<\/mn> <mo>}<\/mo><\/mrow>\n<\/mrow><\/mrow>\n<\/mrow> <\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2831815\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2831816\" data-type=\"problem\"><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2831817\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\n<mi>x<\/mi><mo>\u2265<\/mo><mrow><mn>7<\/mn> <mo>}<\/mo><\/mrow>\n<\/mrow><\/mrow>\n<\/mrow> <\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\n<mi>x<\/mi><mo>\u2265<\/mo><mrow><mn>7<\/mn> <mo>}<\/mo><\/mrow>\n<\/mrow><\/mrow>\n<\/mrow> <\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2831853\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2831854\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2831853-solution\">45<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2831856\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\n<mi>x<\/mi><mo>&lt;<\/mo><mrow><mn>4<\/mn> <mo>}<\/mo><\/mrow>\n<\/mrow><\/mrow>\n<\/mrow> <\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mi>x<\/mi><mrow><mo>|<\/mo><mrow>\n<mi>x<\/mi><mo>&lt;<\/mo><mrow><mn>4<\/mn> <mo>}<\/mo><\/mrow>\n<\/mrow><\/mrow>\n<\/mrow> <\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1752303\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1752304\" data-type=\"problem\"><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1752305\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mrow><mo>|<\/mo><mrow>\n<mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is all real numbers<\/mtext><mrow><mrow><\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/mrow>\n<\/mrow> <\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>{<\/mo> <mrow>\n<mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mrow><mo>|<\/mo><mrow>\n<mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mspace width=\"0.8\"><\/mspace><mtext>is all real numbers<\/mtext><mrow><mrow><\/mrow> <mo>}<\/mo><\/mrow>\n<\/mrow><\/mrow>\n<\/mrow> <\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1752345\">For the following exercises, write the interval in set-builder notation.<\/p>\n<div id=\"fs-id1752348\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1752349\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1752348-solution\">47<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3275472\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>6<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mi>\u221e<\/mi><mo>,<\/mo><mn>6<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3275537\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3275538\" data-type=\"problem\"><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3275539\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo><mi>\u221e<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2743879\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2743880\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2743879-solution\">49<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2743881\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">[<\/mo><mn>\u22123<\/mn><mo>,<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">[<\/mo><mn>\u22123<\/mn><mo>,<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2743944\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2743945\" data-type=\"problem\"><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2306896\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo stretchy=\"false\">[<\/mo><mn>\u22124<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mo>\u222a<\/mo><mo stretchy=\"false\">[<\/mo><mn>9<\/mn><mo>,<\/mo><mi>\u221e<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo stretchy=\"false\">[<\/mo><mn>\u22124<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><mo>\u222a<\/mo><mo stretchy=\"false\">[<\/mo><mn>9<\/mn><mo>,<\/mo><mi>\u221e<\/mi><mo stretchy=\"false\">)<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2306932\">For the following exercises, write the set of numbers represented on the number line in interval notation.<\/p>\n<div id=\"fs-id2306937\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2306938\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2306937-solution\">51<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2306940\" class=\"first-element\" data-type=\"media\" data-alt=\"A number line with two tick marks labeled: -2 and 1 respectively. There is an open circle around the -2 and a dot on the 1 with an arrow connecting the two.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A number line with two tick marks labeled: -2 and 1 respectively. There is an open circle around the -2 and a dot on the 1 with an arrow connecting the two.\" width=\"327\" height=\"49\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/c4578c3c93b430f4b6c1dadcdbc17cb89773ab31\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2306978\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2519857\" data-type=\"problem\"><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2519859\" class=\"first-element\" data-type=\"media\" data-alt=\"Number line with two tick marks labeled: -1 and 2 respectively. There is an open circle around the tick mark labeled -1 and a line that extends leftward from the circle. There is a dot around the tick mark labeled 2 and a line that extends rightward from the dot.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Number line with two tick marks labeled: -1 and 2 respectively. There is an open circle around the tick mark labeled -1 and a line that extends leftward from the circle. There is a dot around the tick mark labeled 2 and a line that extends rightward from the dot.\" width=\"327\" height=\"49\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/0d0ccffea4f593111ec4c247e4d2bfc48cdc71a2\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2519870\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2519871\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2519870-solution\">53<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id2519872\" class=\"first-element\" data-type=\"media\" data-alt=\"Number line with one tick mark labeled 4. There is a dot on the tick mark and an arrow extending leftward from the dot.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Number line with one tick mark labeled 4. There is a dot on the tick mark and an arrow extending leftward from the dot.\" width=\"326\" height=\"47\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/209c1341f665f4bdeb785beb3ae583bf6cfdf2d0\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2519914\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id2519919\">For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall (2<sup>nd<\/sup> CALC 5:intersection, 1<sup>st<\/sup> curve, enter, 2<sup>nd<\/sup> curve, enter, guess, enter). Copy a sketch of the graph and shade the <em data-effect=\"italics\">x<\/em>-axis for your solution set to the inequality. Write final answers in interval notation.<\/p>\n<div id=\"fs-id2919704\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2919706\" data-type=\"problem\"><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2919707\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>5<\/mn><mo>&lt;<\/mo><mn>2<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>5<\/mn><mo>&lt;<\/mo><mn>2<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2919739\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2919740\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2919739-solution\">55<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2919741\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>4<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1787455\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1787456\" data-type=\"problem\"><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1787457\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mn>2<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>&gt;<\/mo><mn>2<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1787491\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2958611\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1787491-solution\">57<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2958612\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>3<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>&lt;<\/mo><mn>3<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2958690\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2958691\" data-type=\"problem\"><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2958692\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2265<\/mo><mn>5<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2663030\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<div id=\"fs-id2663035\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2663036\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2663035-solution\">59<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2663037\">Solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2968133\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2968134\" data-type=\"problem\"><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2968135\">Solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>12<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>x<\/mi><mo>&gt;<\/mo><mn>12<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2968163\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2968164\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2968163-solution\">61<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2968165\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2264<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>\u2264<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2260<\/mo><mn>\u22127<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>\u2260<\/mo><mn>\u22127<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2750208\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2750209\" data-type=\"problem\"><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2750210\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>130<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3000<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>p<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>130<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3000<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>is a profit formula for a small business. Find the set of <em data-effect=\"italics\">x<\/em>-values that will keep this profit positive.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2791668\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id2791673\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2791674\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2791673-solution\">63<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2791675\">In chemistry the volume for a certain gas is given by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>V<\/mi><mo>=<\/mo><mn>20<\/mn><mi>T<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>V<\/mi><mo>=<\/mo><mn>20<\/mn><mi>T<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">V<\/em> is measured in cc and <em data-effect=\"italics\">T<\/em> is temperature in \u00baC. If the temperature varies between 80\u00baC and 120\u00baC, find the set of volume values.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2936685\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2936686\" data-type=\"problem\"><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2936687\">A basic cellular package costs $20\/mo. for 60 min of calling, with an additional charge of $.30\/min beyond that time.. The cost formula would be<\/p>\n<p><math display=\"block\"><semantics><mrow>\n<mrow>\n<mspace width=\"0.5em\"><\/mspace><mi>C<\/mi><mo>=<\/mo><mn>20<\/mn><mo>+<\/mo><mn>.30<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>60<\/mn><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mspace width=\"0.5em\"><\/mspace><mi>C<\/mi><mo>=<\/mo><mn>20<\/mn><mo>+<\/mo><mn>.30<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>60<\/mn><mo stretchy=\"false\">)<\/mo><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>If you have to keep your bill no greater than $50, what is the maximum calling minutes you can use?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":158,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-113","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/113","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":2,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/113\/revisions"}],"predecessor-version":[{"id":324,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/113\/revisions\/324"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/51"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/113\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=113"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=113"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=113"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}