{"id":137,"date":"2025-04-09T17:15:58","date_gmt":"2025-04-09T17:15:58","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-6-absolute-value-functions-college-algebra-2e-openstax\/"},"modified":"2025-08-19T19:06:31","modified_gmt":"2025-08-19T19:06:31","slug":"3-6-absolute-value-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-6-absolute-value-functions\/","title":{"raw":"3.6 Absolute Value Functions","rendered":"3.6 Absolute Value Functions"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_2e387575-c04f-40e1-8895-195affae8fdb\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Graph an absolute value function.<\/li>\r\n \t<li>Solve an absolute value equation.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_616\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-616 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.4-300x200.jpg\" alt=\"\" width=\"300\" height=\"200\" \/> Figure 1. \"Colorado National Monument Entrance\" by Danial Schwen is licensed under CC BY-SA 4.0[\/caption]\r\n\r\n<div>\r\n<div>\r\n\r\nColorado is famous for its diverse landscape and amazing views. Suppose you are traveling from Denver to Grand Junction to visit the beautiful Colorado National Monument. As a student who is taking College Algebra, you cannot resist doing some mathematics before your trip. You want to calculate how many miles you will be driving. Since we can measure the distance in both directions, it is useful to consider distance as an absolute value function. In this section, we will continue our investigation of absolute value functions.\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137426078\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Understanding Absolute Value<\/h2>\r\n<p id=\"fs-id1165135449691\">Recall that in its basic form\u00a0[latex] f(x)=|x|, [\/latex] the absolute value function is one of our toolkit functions. The <span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">absolute value<\/span> function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Absolute Value Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe absolute value function can be defined as a piecewise function\r\n<p style=\"text-align: center;\">[latex] f(x) = |x| = \\begin{cases}x &amp; \\text{if} &amp; x \\geq 0 \\\\-x &amp; \\text{if} &amp; x &lt; 0\\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Using Absolute Value to Determine Resistance<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nElectrical parts <span class=\"TextRun SCXW60696911 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW60696911 BCX2\">of a cellphone<\/span><\/span> come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often\u00a0[latex] \\pm1\\%, \\pm5\\% [\/latex] or [latex] \\pm10\\% [\/latex]\r\n\r\nSuppose we have a resistor rated at 680 ohms,\u00a0[latex] \\pm5\\%. [\/latex] Use the absolute value function to express the range of possible values of the actual resistance.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can find that 5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance\u00a0[latex] R [\/latex] in ohms,\r\n<p style=\"text-align: center;\">[latex] |R-680|\\le34 [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nStudent who score within 20 points of 80 will <span class=\"TextRun SCXW127968588 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW127968588 BCX2\">pass a test for a summer course at CC<\/span><\/span>A. Write this as a distance from 80 using absolute value notation.\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135186288\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing an Absolute Value Function<\/h2>\r\n<p id=\"fs-id1165135570012\">The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the <span id=\"term-00003\" class=\"no-emphasis\" data-type=\"term\">origin<\/span> in Figure 2.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_617\" align=\"aligncenter\" width=\"290\"]<img class=\"size-medium wp-image-617\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-2-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" \/> Figure 2[\/caption]\r\n<p id=\"fs-id1165135639350\">Figure 3 shows the graph of\u00a0[latex] y=2|x+3|+4. [\/latex] The graph of\u00a0[latex] y=|x| [\/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at\u00a0[latex] (3, 4) [\/latex] for this transformed function.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_618\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-618\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-3-300x223.jpeg\" alt=\"\" width=\"300\" height=\"223\" \/> Figure 3[\/caption]\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Writing an Equation for an Absolute Value Function Given a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite an equation for the function graphed in Figure 4.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_619\" align=\"aligncenter\" width=\"290\"]<img class=\"size-medium wp-image-619\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-4-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" \/> Figure 4[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure 5.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_620\" align=\"aligncenter\" width=\"290\"]<img class=\"size-medium wp-image-620\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-5-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" \/> Figure 5[\/caption]\r\n\r\nWe also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance as shown in Figure 6.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_621\" align=\"aligncenter\" width=\"290\"]<img class=\"size-medium wp-image-621\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-6-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" \/> Figure 6[\/caption]\r\n\r\nFrom this transformation we can write the equation\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcll}f(x) &amp;=&amp; 2|x-3|-2, &amp; \\quad \\text{treating the stretch as } a \\, \\text{vertical stretch, or} \\\\f(x) &amp;=&amp; |2(x-3)|-2, &amp; \\quad \\text{treating the stretch as } a \\, \\text{horizontal compression}.\\end{array} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nNote that these equations are algebraically equivalent\u2014the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression. Note also that if the vertical stretch factor is negative, there is also a reflection about the x-axis.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q:<\/strong> <strong>If we couldn\u2019t observe the stretch of the function from the graphs, could we algebraically determine it?<\/strong>\r\n\r\n<em data-effect=\"italics\">A: Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for\u00a0[latex] x [\/latex] and [latex] f(x). [\/latex]<\/em>\r\n<p style=\"text-align: center;\">[latex] f(x)=a|x-3|-2 [\/latex]<\/p>\r\n<em data-effect=\"italics\">Now substituting in the point [latex] (1, 2) [\/latex]<\/em>\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl} 2 &amp;=&amp; a|1-3|-2 \\\\ 4 &amp;=&amp; 2a \\\\ a &amp;=&amp; 2 \\end{array} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite the equation for the absolute value function that is horizontally shifted left 2 units, is vertically reflected, and vertically shifted up 3 units.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q:<\/strong> <strong>Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?\r\n<\/strong>\r\n<p id=\"fs-id1165137581861\"><em data-effect=\"italics\">A: Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.\r\n<\/em><\/p>\r\n<p id=\"fs-id1165137444543\"><em data-effect=\"italics\">No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure 7).<\/em><\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_622\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-622\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-7-300x95.jpeg\" alt=\"\" width=\"300\" height=\"95\" \/> Figure 7. (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165133257286\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving an Absolute Value Equation<\/h2>\r\n<p id=\"fs-id1165137401775\">In Other Type of Equations, we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as\u00a0[latex] 8=|2x-6|, [\/latex] we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.<\/p>\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl rcl}2x - 6 &amp;=&amp; 8 &amp; \\quad \\text{or} \\quad &amp; 2x - 6 &amp;=&amp; -8 \\\\2x &amp;=&amp; 14 &amp; &amp; 2x &amp;=&amp; -2 \\\\x &amp;=&amp; 7 &amp; &amp; x &amp;=&amp; -1\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137641126\">Knowing how to solve problems involving <span id=\"term-00004\" class=\"no-emphasis\" data-type=\"term\">absolute value functions<\/span> is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\r\n<p id=\"fs-id1165137937577\">An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcll} |x| &amp;=&amp; 4, \\\\ |2x-1| &amp;=&amp; 3 &amp; \\text{or} \\\\ |5x+2| &amp;=&amp; -4 &amp;=&amp; 9\\end{array} [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Solutions to Absolute Value Equations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor real numbers\u00a0[latex] A [\/latex] and\u00a0[latex] B [\/latex] an equation of the form\u00a0[latex] |A|=B, [\/latex] with\u00a0[latex] B\\ge0, [\/latex] will have solutions when\u00a0[latex] A=B [\/latex] or\u00a0[latex] A=-B. [\/latex] If\u00a0[latex] B&lt;0, [\/latex] the equation\u00a0[latex] |A|=B [\/latex] has no solution.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given the formula for an absolute value function, find the horizontal intercepts of its graph<\/strong>.\r\n<ol>\r\n \t<li>Isolate the absolute value term.<\/li>\r\n \t<li>Use\u00a0[latex] |A|=B [\/latex] to write\u00a0[latex] A=B [\/latex] or\u00a0[latex] -A=B [\/latex] assuming [latex] B&gt;0. [\/latex]<\/li>\r\n \t<li>Solve for [latex] x. [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Finding the Zeros of an Absolute Value Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the function\u00a0[latex] f(x)=|4x+1|-7, [\/latex] find the values of\u00a0[latex] x [\/latex] such that [latex] f(x)=0. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcllllll}0 &amp;=&amp; |4x+1|-7 &amp;&amp;&amp;&amp;&amp;\\text{Substitute 0 for } f(x). \\\\7 &amp;=&amp; |4x+1| &amp;&amp;&amp;&amp;&amp;\\text{Isolate the absolute value on one side of the equation}. \\\\ \\\\ \\\\ \\\\ 7 &amp;=&amp; 4x+1 &amp;\\text{or}&amp; \\quad -7 &amp;=&amp; 4x+1 &amp;\\text{Break into two separate equations and solve}. \\\\6 &amp;=&amp; 4x &amp;&amp; -8 &amp;=&amp; 4x &amp; \\\\ \\\\ \\\\ x &amp;=&amp; \\frac{6}{4} = 1.5 &amp;&amp; x &amp;=&amp; \\frac{-8}{4} = -2 &amp;\\end{array} [\/latex]<\/p>\r\nThe function outputs 0 when\u00a0[latex] x=\\frac{3}{2} [\/latex] or\u00a0[latex] x=-2. [\/latex] See Figure 8.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_623\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-623\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-8-300x259.jpeg\" alt=\"\" width=\"300\" height=\"259\" \/> Figure 8[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the function\u00a0[latex] f(x)=|2x-1|-3, [\/latex] find the values of\u00a0[latex] x [\/latex] such that [latex] f(x)=0. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Should we always expect two answers when solving<\/strong> [latex] |A|=B? [\/latex]\r\n\r\nA:\u00a0<em data-effect=\"italics\">No. We may find one, two, or even no answers. For example, there is no solution to<\/em> [latex] 2+|3x-5|=1. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135571678\" data-depth=\"1\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess these online resources for additional instruction and practice with absolute value.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=X_gqB9bVOVE&amp;feature=youtu.be\/\">Graphing Absolute Value Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=zmlay7PlM-E&amp;feature=youtu.be\/\">Graphing Absolute Value Functions 2<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/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\">3.6 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165135255406\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165137406985\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165137734873\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135510060\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137734873-solution\">1<\/a><span class=\"os-divider\">. <\/span>How do you solve an absolute value equation?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137593210\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131968049\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>How can you tell whether an absolute value function has two <em data-effect=\"italics\">x<\/em>-intercepts without graphing the function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133103957\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133103959\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133103957-solution\">3<\/a><span class=\"os-divider\">. <\/span>When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135264708\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135149122\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>How can you use the graph of an absolute value function to determine the <em data-effect=\"italics\">x<\/em>-values for which the function values are negative?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134273549\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<div id=\"fs-id1165137841613\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137841615\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137841613-solution\">5<\/a><span class=\"os-divider\">. <\/span>Describe all numbers\u00a0[latex] x [\/latex] that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135445894\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135445896\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span>Describe all numbers\u00a0[latex] x [\/latex] that are at a distance of\u00a0[latex] \\frac{1}{2} [\/latex] from the number\u00a0[latex] -4. [\/latex] Express this set of numbers using absolute value notation.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137542576\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137648320\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137542576-solution\">7<\/a><span class=\"os-divider\">. <\/span>Describe the situation in which the distance that point\u00a0[latex] x [\/latex] is from 10 is at least 15 units. Express this set of numbers using absolute value notation.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134057540\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137464076\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span>Find all function values\u00a0[latex] f(x) [\/latex] such that the distance from\u00a0[latex] f(x) [\/latex] to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137589731\">For the following exercises, find the \u00a0and intercepts of the graphs of each function.<\/p>\r\n\r\n<div id=\"fs-id1165134401702\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135362510\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134401702-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=4|x-3|+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137824535\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137824537\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-3|x-2|-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134371172\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134371174\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134371172-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-2|x+1|+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137656160\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137656163\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-5|x+2|+15 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"eip-613\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"eip-946\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"eip-613-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2|x-1|-6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"eip-240\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"eip-147\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=|-2x+1|-13 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"eip-452\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"eip-163\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"eip-452-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-|x-9|+16 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137602379\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165133047532\">For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.<\/p>\r\n\r\n<div id=\"fs-id1165137891404\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137817696\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] y=|x-1| [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137679099\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137679101\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137679099-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] y=|x+1| [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135422938\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135422940\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] y=|x|+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137406944\">For the following exercises, graph the given functions by hand.<\/p>\r\n\r\n<div id=\"fs-id1165135332726\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135332729\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135332726-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] y=|x|-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137601710\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137601713\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] y=-|x| [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137735265\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137431347\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137735265-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] y=-|x|-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137731590\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137603675\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] y=-|x-3|-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137394585\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137394587\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137394585-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-|x-1|-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135335986\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137651575\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-|x+3|+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137705796\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137705798\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137705796-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2|x+3|+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137427199\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137619904\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=3|x-2|+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137715460\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137715462\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137715460-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=|2x-4|-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135258293\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135258295\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=|3x+9|+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137464095\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137464097\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137464095-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=-|x-1|-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137443657\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137911316\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-|x+4|-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137803326\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137803328\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137803326-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{1}{2}|x+4|-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137897208\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<div id=\"fs-id1165137749758\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137749760\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Use a graphing utility to graph\u00a0[latex] f(x)=10|x-2| [\/latex] on the viewing window\u00a0[latex] [0, 4]. [\/latex] Identify the corresponding range. Show the graph.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137413783\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137434783\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137413783-solution\">33<\/a><span class=\"os-divider\">. <\/span>Use a graphing utility to graph\u00a0[latex] f(x)=-100|x|+100 [\/latex] on the viewing window\u00a0[latex] [-5, 5]. [\/latex] Identify the corresponding range. Show the graph.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137831208\">For the following exercises, graph each function using a graphing utility. Specify the viewing window.<\/p>\r\n\r\n<div id=\"fs-id1165137762283\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135464843\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-0.1|0.1(0.2-x)|+0.3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"eip-id1165134039354\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"eip-id1165134039356\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"eip-id1165134039354-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=4\\cdot 10^9|x-(5\\cdot 10^9)|+2\\cdot 10^9 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137419467\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1165137901338\">For the following exercises, solve the inequality.<\/p>\r\n\r\n<div id=\"fs-id1165137434569\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137434570\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>If possible, find all values of\u00a0[latex] a [\/latex] such that there are no [latex] x- [\/latex]intercepts for [latex] f(x)=2|x+1|+a. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137639316\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137652958\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137639316-solution\">37<\/a><span class=\"os-divider\">. <\/span>If possible, find all values of\u00a0[latex] a [\/latex] such that there are no [latex] y- [\/latex]intercepts for [latex] f(x)=2|x+1|+a\/ [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135172151\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1165137641899\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137641901\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span><span class=\"TextRun SCXW89553349 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW89553349 BCX2\">Golden, Colorado and <\/span><span class=\"NormalTextRun ContextualSpellingAndGrammarErrorV2Themed SCXW89553349 BCX2\">Aurora,<\/span><span class=\"NormalTextRun SCXW89553349 BCX2\"> Colorado are on the same east-west line of the Denver metro area. Assume that Golden is <\/span><span class=\"NormalTextRun SCXW89553349 BCX2\">located<\/span><span class=\"NormalTextRun SCXW89553349 BCX2\"> at <\/span><span class=\"NormalTextRun ContextualSpellingAndGrammarErrorV2Themed SCXW89553349 BCX2\">the<\/span><span class=\"NormalTextRun SCXW89553349 BCX2\"> origin. The distance from Golden to <\/span><span class=\"NormalTextRun SCXW89553349 BCX2\">Aurora is about 34 miles. If x <\/span><\/span><span class=\"TextRun SCXW89553349 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW89553349 BCX2\">represents<\/span><span class=\"NormalTextRun SCXW89553349 BCX2\"> the distance from Aurora to Golden, express the relation between x<\/span><\/span> and 34 <span class=\"TextRun SCXW89553349 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW89553349 BCX2\">using absolute value notation.<\/span><\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137812302\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137812304\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137812302-solution\">39<\/a><span class=\"os-divider\">. <\/span>The true proportion\u00a0[latex] p [\/latex] of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137598001\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137562568\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable\u00a0 for the score.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137758810\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137758812\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137758810-solution\">41<\/a><span class=\"os-divider\">. <\/span>A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using\u00a0 as the diameter of the bearing, write this statement using absolute value notation.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137732323\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137732325\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is\u00a0 inches, express the tolerance using absolute value notation.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_2e387575-c04f-40e1-8895-195affae8fdb\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Graph an absolute value function.<\/li>\n<li>Solve an absolute value equation.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<figure id=\"attachment_616\" aria-describedby=\"caption-attachment-616\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-616 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.4-300x200.jpg\" alt=\"\" width=\"300\" height=\"200\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.4-300x200.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.4-65x43.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.4-225x150.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.4-350x233.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.4.jpg 721w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-616\" class=\"wp-caption-text\">Figure 1. &#8220;Colorado National Monument Entrance&#8221; by Danial Schwen is licensed under CC BY-SA 4.0<\/figcaption><\/figure>\n<div>\n<div>\n<p>Colorado is famous for its diverse landscape and amazing views. Suppose you are traveling from Denver to Grand Junction to visit the beautiful Colorado National Monument. As a student who is taking College Algebra, you cannot resist doing some mathematics before your trip. You want to calculate how many miles you will be driving. Since we can measure the distance in both directions, it is useful to consider distance as an absolute value function. In this section, we will continue our investigation of absolute value functions.<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137426078\" data-depth=\"1\">\n<h2 data-type=\"title\">Understanding Absolute Value<\/h2>\n<p id=\"fs-id1165135449691\">Recall that in its basic form\u00a0[latex]f(x)=|x|,[\/latex] the absolute value function is one of our toolkit functions. The <span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">absolute value<\/span> function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Absolute Value Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The absolute value function can be defined as a piecewise function<\/p>\n<p style=\"text-align: center;\">[latex]f(x) = |x| = \\begin{cases}x & \\text{if} & x \\geq 0 \\\\-x & \\text{if} & x < 0\\end{cases}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Using Absolute Value to Determine Resistance<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Electrical parts <span class=\"TextRun SCXW60696911 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW60696911 BCX2\">of a cellphone<\/span><\/span> come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often\u00a0[latex]\\pm1\\%, \\pm5\\%[\/latex] or [latex]\\pm10\\%[\/latex]<\/p>\n<p>Suppose we have a resistor rated at 680 ohms,\u00a0[latex]\\pm5\\%.[\/latex] Use the absolute value function to express the range of possible values of the actual resistance.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We can find that 5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance\u00a0[latex]R[\/latex] in ohms,<\/p>\n<p style=\"text-align: center;\">[latex]|R-680|\\le34[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Student who score within 20 points of 80 will <span class=\"TextRun SCXW127968588 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW127968588 BCX2\">pass a test for a summer course at CC<\/span><\/span>A. Write this as a distance from 80 using absolute value notation.<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135186288\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing an Absolute Value Function<\/h2>\n<p id=\"fs-id1165135570012\">The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the <span id=\"term-00003\" class=\"no-emphasis\" data-type=\"term\">origin<\/span> in Figure 2.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_617\" aria-describedby=\"caption-attachment-617\" style=\"width: 290px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-617\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-2-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-2-290x300.jpeg 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-2-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-2-225x233.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-2-350x362.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-2.jpeg 360w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><figcaption id=\"caption-attachment-617\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<p id=\"fs-id1165135639350\">Figure 3 shows the graph of\u00a0[latex]y=2|x+3|+4.[\/latex] The graph of\u00a0[latex]y=|x|[\/latex] has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at\u00a0[latex](3, 4)[\/latex] for this transformed function.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_618\" aria-describedby=\"caption-attachment-618\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-618\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-3-300x223.jpeg\" alt=\"\" width=\"300\" height=\"223\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-3-300x223.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-3-65x48.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-3-225x167.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-3-350x260.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-3.jpeg 500w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-618\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Writing an Equation for an Absolute Value Function Given a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write an equation for the function graphed in Figure 4.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_619\" aria-describedby=\"caption-attachment-619\" style=\"width: 290px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-619\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-4-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-4-290x300.jpeg 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-4-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-4-225x233.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-4-350x362.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-4.jpeg 360w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><figcaption id=\"caption-attachment-619\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure 5.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_620\" aria-describedby=\"caption-attachment-620\" style=\"width: 290px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-620\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-5-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-5-290x300.jpeg 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-5-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-5-225x233.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-5-350x362.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-5.jpeg 360w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><figcaption id=\"caption-attachment-620\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<p>We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance as shown in Figure 6.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_621\" aria-describedby=\"caption-attachment-621\" style=\"width: 290px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-621\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-6-290x300.jpeg\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-6-290x300.jpeg 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-6-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-6-225x233.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-6-350x362.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-6.jpeg 360w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><figcaption id=\"caption-attachment-621\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<p>From this transformation we can write the equation<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcll}f(x) &=& 2|x-3|-2, & \\quad \\text{treating the stretch as } a \\, \\text{vertical stretch, or} \\\\f(x) &=& |2(x-3)|-2, & \\quad \\text{treating the stretch as } a \\, \\text{horizontal compression}.\\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Note that these equations are algebraically equivalent\u2014the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression. Note also that if the vertical stretch factor is negative, there is also a reflection about the x-axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q:<\/strong> <strong>If we couldn\u2019t observe the stretch of the function from the graphs, could we algebraically determine it?<\/strong><\/p>\n<p><em data-effect=\"italics\">A: Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for\u00a0[latex]x[\/latex] and [latex]f(x).[\/latex]<\/em><\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a|x-3|-2[\/latex]<\/p>\n<p><em data-effect=\"italics\">Now substituting in the point [latex](1, 2)[\/latex]<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl} 2 &=& a|1-3|-2 \\\\ 4 &=& 2a \\\\ a &=& 2 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically reflected, and vertically shifted up 3 units.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q:<\/strong> <strong>Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?<br \/>\n<\/strong><\/p>\n<p id=\"fs-id1165137581861\"><em data-effect=\"italics\">A: Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.<br \/>\n<\/em><\/p>\n<p id=\"fs-id1165137444543\"><em data-effect=\"italics\">No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure 7).<\/em><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_622\" aria-describedby=\"caption-attachment-622\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-622\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-7-300x95.jpeg\" alt=\"\" width=\"300\" height=\"95\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-7-300x95.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-7-1024x325.jpeg 1024w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-7-768x244.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-7-65x21.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-7-225x71.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-7-350x111.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-7.jpeg 1138w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-622\" class=\"wp-caption-text\">Figure 7. (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165133257286\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving an Absolute Value Equation<\/h2>\n<p id=\"fs-id1165137401775\">In Other Type of Equations, we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as\u00a0[latex]8=|2x-6|,[\/latex] we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl rcl}2x - 6 &=& 8 & \\quad \\text{or} \\quad & 2x - 6 &=& -8 \\\\2x &=& 14 & & 2x &=& -2 \\\\x &=& 7 & & x &=& -1\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137641126\">Knowing how to solve problems involving <span id=\"term-00004\" class=\"no-emphasis\" data-type=\"term\">absolute value functions<\/span> is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\n<p id=\"fs-id1165137937577\">An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcll} |x| &=& 4, \\\\ |2x-1| &=& 3 & \\text{or} \\\\ |5x+2| &=& -4 &=& 9\\end{array}[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Solutions to Absolute Value Equations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For real numbers\u00a0[latex]A[\/latex] and\u00a0[latex]B[\/latex] an equation of the form\u00a0[latex]|A|=B,[\/latex] with\u00a0[latex]B\\ge0,[\/latex] will have solutions when\u00a0[latex]A=B[\/latex] or\u00a0[latex]A=-B.[\/latex] If\u00a0[latex]B<0,[\/latex] the equation\u00a0[latex]|A|=B[\/latex] has no solution.\n\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given the formula for an absolute value function, find the horizontal intercepts of its graph<\/strong>.<\/p>\n<ol>\n<li>Isolate the absolute value term.<\/li>\n<li>Use\u00a0[latex]|A|=B[\/latex] to write\u00a0[latex]A=B[\/latex] or\u00a0[latex]-A=B[\/latex] assuming [latex]B>0.[\/latex]<\/li>\n<li>Solve for [latex]x.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Finding the Zeros of an Absolute Value Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the function\u00a0[latex]f(x)=|4x+1|-7,[\/latex] find the values of\u00a0[latex]x[\/latex] such that [latex]f(x)=0.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcllllll}0 &=& |4x+1|-7 &&&&&\\text{Substitute 0 for } f(x). \\\\7 &=& |4x+1| &&&&&\\text{Isolate the absolute value on one side of the equation}. \\\\ \\\\ \\\\ \\\\ 7 &=& 4x+1 &\\text{or}& \\quad -7 &=& 4x+1 &\\text{Break into two separate equations and solve}. \\\\6 &=& 4x && -8 &=& 4x & \\\\ \\\\ \\\\ x &=& \\frac{6}{4} = 1.5 && x &=& \\frac{-8}{4} = -2 &\\end{array}[\/latex]<\/p>\n<p>The function outputs 0 when\u00a0[latex]x=\\frac{3}{2}[\/latex] or\u00a0[latex]x=-2.[\/latex] See Figure 8.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_623\" aria-describedby=\"caption-attachment-623\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-623\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-8-300x259.jpeg\" alt=\"\" width=\"300\" height=\"259\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-8-300x259.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-8-65x56.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-8-225x194.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-8-350x302.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.6-fig-8.jpeg 428w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-623\" class=\"wp-caption-text\">Figure 8<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the function\u00a0[latex]f(x)=|2x-1|-3,[\/latex] find the values of\u00a0[latex]x[\/latex] such that [latex]f(x)=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Should we always expect two answers when solving<\/strong> [latex]|A|=B?[\/latex]<\/p>\n<p>A:\u00a0<em data-effect=\"italics\">No. We may find one, two, or even no answers. For example, there is no solution to<\/em> [latex]2+|3x-5|=1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135571678\" data-depth=\"1\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access these online resources for additional instruction and practice with absolute value.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=X_gqB9bVOVE&amp;feature=youtu.be\/\">Graphing Absolute Value Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=zmlay7PlM-E&amp;feature=youtu.be\/\">Graphing Absolute Value Functions 2<\/a><\/li>\n<\/ul>\n<\/div>\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\">3.6 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165135255406\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165137406985\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165137734873\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135510060\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137734873-solution\">1<\/a><span class=\"os-divider\">. <\/span>How do you solve an absolute value equation?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137593210\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131968049\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>How can you tell whether an absolute value function has two <em data-effect=\"italics\">x<\/em>-intercepts without graphing the function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133103957\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133103959\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133103957-solution\">3<\/a><span class=\"os-divider\">. <\/span>When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135264708\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135149122\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>How can you use the graph of an absolute value function to determine the <em data-effect=\"italics\">x<\/em>-values for which the function values are negative?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134273549\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<div id=\"fs-id1165137841613\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137841615\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137841613-solution\">5<\/a><span class=\"os-divider\">. <\/span>Describe all numbers\u00a0[latex]x[\/latex] that are at a distance of 4 from the number 8. Express this set of numbers using absolute value notation.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135445894\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135445896\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span>Describe all numbers\u00a0[latex]x[\/latex] that are at a distance of\u00a0[latex]\\frac{1}{2}[\/latex] from the number\u00a0[latex]-4.[\/latex] Express this set of numbers using absolute value notation.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137542576\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137648320\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137542576-solution\">7<\/a><span class=\"os-divider\">. <\/span>Describe the situation in which the distance that point\u00a0[latex]x[\/latex] is from 10 is at least 15 units. Express this set of numbers using absolute value notation.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134057540\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137464076\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span>Find all function values\u00a0[latex]f(x)[\/latex] such that the distance from\u00a0[latex]f(x)[\/latex] to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137589731\">For the following exercises, find the \u00a0and intercepts of the graphs of each function.<\/p>\n<div id=\"fs-id1165134401702\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135362510\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134401702-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=4|x-3|+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137824535\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137824537\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-3|x-2|-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134371172\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134371174\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134371172-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-2|x+1|+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137656160\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137656163\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-5|x+2|+15[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"eip-613\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"eip-946\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"eip-613-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2|x-1|-6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"eip-240\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"eip-147\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=|-2x+1|-13[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"eip-452\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"eip-163\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"eip-452-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-|x-9|+16[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137602379\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165133047532\">For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.<\/p>\n<div id=\"fs-id1165137891404\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137817696\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]y=|x-1|[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137679099\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137679101\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137679099-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]y=|x+1|[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135422938\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135422940\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]y=|x|+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137406944\">For the following exercises, graph the given functions by hand.<\/p>\n<div id=\"fs-id1165135332726\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135332729\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135332726-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]y=|x|-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137601710\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137601713\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]y=-|x|[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137735265\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137431347\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137735265-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]y=-|x|-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137731590\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137603675\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]y=-|x-3|-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137394585\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137394587\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137394585-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-|x-1|-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135335986\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137651575\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-|x+3|+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137705796\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137705798\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137705796-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2|x+3|+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137427199\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137619904\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=3|x-2|+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137715460\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137715462\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137715460-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=|2x-4|-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135258293\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135258295\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=|3x+9|+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137464095\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137464097\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137464095-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=-|x-1|-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137443657\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137911316\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-|x+4|-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137803326\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137803328\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137803326-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{1}{2}|x+4|-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137897208\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<div id=\"fs-id1165137749758\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137749760\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Use a graphing utility to graph\u00a0[latex]f(x)=10|x-2|[\/latex] on the viewing window\u00a0[latex][0, 4].[\/latex] Identify the corresponding range. Show the graph.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137413783\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137434783\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137413783-solution\">33<\/a><span class=\"os-divider\">. <\/span>Use a graphing utility to graph\u00a0[latex]f(x)=-100|x|+100[\/latex] on the viewing window\u00a0[latex][-5, 5].[\/latex] Identify the corresponding range. Show the graph.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137831208\">For the following exercises, graph each function using a graphing utility. Specify the viewing window.<\/p>\n<div id=\"fs-id1165137762283\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135464843\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-0.1|0.1(0.2-x)|+0.3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"eip-id1165134039354\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"eip-id1165134039356\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"eip-id1165134039354-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=4\\cdot 10^9|x-(5\\cdot 10^9)|+2\\cdot 10^9[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137419467\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1165137901338\">For the following exercises, solve the inequality.<\/p>\n<div id=\"fs-id1165137434569\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137434570\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>If possible, find all values of\u00a0[latex]a[\/latex] such that there are no [latex]x-[\/latex]intercepts for [latex]f(x)=2|x+1|+a.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137639316\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137652958\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137639316-solution\">37<\/a><span class=\"os-divider\">. <\/span>If possible, find all values of\u00a0[latex]a[\/latex] such that there are no [latex]y-[\/latex]intercepts for [latex]f(x)=2|x+1|+a\/[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135172151\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1165137641899\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137641901\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span><span class=\"TextRun SCXW89553349 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW89553349 BCX2\">Golden, Colorado and <\/span><span class=\"NormalTextRun ContextualSpellingAndGrammarErrorV2Themed SCXW89553349 BCX2\">Aurora,<\/span><span class=\"NormalTextRun SCXW89553349 BCX2\"> Colorado are on the same east-west line of the Denver metro area. Assume that Golden is <\/span><span class=\"NormalTextRun SCXW89553349 BCX2\">located<\/span><span class=\"NormalTextRun SCXW89553349 BCX2\"> at <\/span><span class=\"NormalTextRun ContextualSpellingAndGrammarErrorV2Themed SCXW89553349 BCX2\">the<\/span><span class=\"NormalTextRun SCXW89553349 BCX2\"> origin. The distance from Golden to <\/span><span class=\"NormalTextRun SCXW89553349 BCX2\">Aurora is about 34 miles. If x <\/span><\/span><span class=\"TextRun SCXW89553349 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW89553349 BCX2\">represents<\/span><span class=\"NormalTextRun SCXW89553349 BCX2\"> the distance from Aurora to Golden, express the relation between x<\/span><\/span> and 34 <span class=\"TextRun SCXW89553349 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW89553349 BCX2\">using absolute value notation.<\/span><\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137812302\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137812304\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137812302-solution\">39<\/a><span class=\"os-divider\">. <\/span>The true proportion\u00a0[latex]p[\/latex] of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137598001\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137562568\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable\u00a0 for the score.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137758810\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137758812\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137758810-solution\">41<\/a><span class=\"os-divider\">. <\/span>A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using\u00a0 as the diameter of the bearing, write this statement using absolute value notation.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137732323\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137732325\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is\u00a0 inches, express the tolerance using absolute value notation.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n","protected":false},"author":158,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-137","chapter","type-chapter","status-publish","hentry"],"part":105,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/137","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":13,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/137\/revisions"}],"predecessor-version":[{"id":1503,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/137\/revisions\/1503"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/137\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=137"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=137"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=137"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}