{"id":81,"date":"2025-04-09T17:03:55","date_gmt":"2025-04-09T17:03:55","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/1-1-real-numbers-algebra-essentials-college-algebra-2e-openstax\/"},"modified":"2025-09-22T16:28:37","modified_gmt":"2025-09-22T16:28:37","slug":"1-1-real-numbers-algebra-essentials","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/1-1-real-numbers-algebra-essentials\/","title":{"raw":"1.1 Real Numbers: Algebra Essentials","rendered":"1.1 Real Numbers: Algebra Essentials"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_d0a86917-6447-4b49-bed1-efb5c352cc77\" class=\"chapter-content-module\" style=\"text-align: center;\" 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\" style=\"text-align: left;\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\" style=\"text-align: left;\">\r\n<p style=\"text-align: left;\">In this section you will:<\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: left;\">Classify a real number as a natural, whole, integer, rational, or irrational number.<\/li>\r\n \t<li style=\"text-align: left;\">Perform calculations using order of operations.<\/li>\r\n \t<li style=\"text-align: left;\">Use the following properties of numbers: commutative, associative, distributive, inverse, and identity.<\/li>\r\n \t<li style=\"text-align: left;\">Evaluate algebraic expressions.<\/li>\r\n \t<li style=\"text-align: left;\">Simplify algebraic expressions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id2610870\" style=\"text-align: left;\">It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity\u2014a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.<\/p>\r\n<p id=\"fs-id2597266\" style=\"text-align: left;\">Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.<\/p>\r\n<p id=\"fs-id2811451\" style=\"text-align: left;\">But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a \u201cbase state\u201d while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.<\/p>\r\n<p id=\"fs-id2558199\" style=\"text-align: left;\">Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.<\/p>\r\n<p id=\"fs-id3044208\" style=\"text-align: left;\">Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.<\/p>\r\n\r\n<section id=\"fs-id2374993\" data-depth=\"1\">\r\n<h2 style=\"text-align: left;\" data-type=\"title\">Classifying a Real Number<\/h2>\r\n<p id=\"fs-id2172910\" style=\"text-align: left;\">The numbers we use for counting, or enumerating items, are the <span id=\"term-00001\" data-type=\"term\">natural numbers<\/span>: 1, 2, 3, 4, 5, and so on. We describe them in set notation as [latex] \\{1, 2, 3, \\ldots\\} [\/latex] where the ellipsis (\u2026) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the <em data-effect=\"italics\">counting numbers<\/em>. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of <strong><span id=\"term-00002\" data-type=\"term\">whole numbers<\/span><\/strong> is the set of natural numbers plus zero: [latex] \\{0, 1, 2, 3, \\ldots\\} [\/latex]<\/p>\r\n<p style=\"text-align: left;\">The set of <strong><span id=\"term-00003\" data-type=\"term\">integers<\/span> <\/strong>adds the opposites of the natural numbers to the set of whole numbers: [latex] \\{\\ldots, -3, -2, -1, 0, 1, 2, 3, \\ldots\\} [\/latex] It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.<\/p>\r\n[latex] \\overset{\\text{negative integers}}{\\ldots, -3, -2, -1,} \\hspace{3em} \\overset{zero}{0,} \\hspace{3em} \\overset{\\text{positive integers}}{1, 2, 3, \\ldots} [\/latex]\r\n<p id=\"fs-id3036458\" style=\"text-align: left;\">The set of <strong><span id=\"term-00004\" data-type=\"term\">rational numbers<\/span><\/strong> is written as [latex] \\frac{m}{n} \\ \\text{m and n are integers and} \\ n \\not = 0 \\frac{15}{8} = 1.875 \\frac{4}{11} = 0.36363636\\ldots= 0.\\overline{36}. [\/latex] Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.<\/p>\r\n<p id=\"fs-id1841571\" style=\"text-align: left;\">Because they are fractions, any rational number can also be expressed as a terminating or repeating decimal. Any rational number can be represented as either:<\/p>\r\n<p style=\"text-align: left;\"><span class=\"token\">a) <\/span>a terminating decimal: [latex] \\frac{15}{8} = 1.875, [\/latex] or<\/p>\r\n<p style=\"text-align: left;\">b) a repeating decimal: [latex] \\frac{4}{11} = 0.36363636\\ldots= 0.\\overline{36} [\/latex]<\/p>\r\n<p id=\"fs-id1334141\" style=\"text-align: left;\">We use a line drawn over the repeating block of numbers instead of writing the group multiple times.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\" style=\"text-align: left;\">Example 1: Writing Integers as Rational Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\" style=\"text-align: left;\">\r\n<p style=\"text-align: left;\">Write each of the following as a rational number.<\/p>\r\n<p style=\"text-align: left;\">(a) 7<\/p>\r\n<p style=\"text-align: left;\">(b) 0<\/p>\r\n<p style=\"text-align: left;\">(c) -8<\/p>\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<p style=\"text-align: left;\">Write a fraction with the integer in the numerator and 1 in the denominator.<\/p>\r\n<p style=\"text-align: left;\">(a) [latex] 7 = \\frac{7}{1} [\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left;\">(b) [latex] 0 = \\frac{0}{1} [\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left;\">(c) [latex] -8 = -\\frac{8}{1} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><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\nWrite each of the following as a rational number.\r\n\r\n(a) 11\r\n\r\n(b) 3\r\n\r\n(c) -4\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Identifying Rational Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite each of the following rational numbers as either a terminating or repeating decimal.\r\n\r\n(a) [latex] -\\frac{5}{7} [\/latex]\r\n\r\n(b) [latex] \\frac{15}{5} [\/latex]\r\n\r\n(c) [latex] \\frac{13}{25} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Write each fraction as a decimal by dividing the numerator by the denominator.\r\n\r\n(a) [latex] -\\frac{5}{27} = -0.\\overline{714285}, [\/latex] a repeating decimal\r\n\r\n&nbsp;\r\n\r\n(b) [latex] \\frac{15}{3} = 3\\hspace{0.25em} \\text{(or 3.0)}, [\/latex] a terminating decimal\r\n\r\n&nbsp;\r\n\r\n(c) [latex] \\frac{13}{25} = 0.52,[\/latex] a terminating decimal\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><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 each of the following rational numbers as either a terminating or repeating decimal.\r\n\r\n(a) [latex] \\frac{68}{17} [\/latex]\r\n\r\n(b) [latex] \\frac{8}{13} [\/latex]\r\n\r\n(c) [latex] -\\frac{17}{20} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id2040230\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Irrational Numbers<\/h3>\r\n<p id=\"fs-id2393783\">At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even [latex] \\frac{3}{2} [\/latex] but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be <em data-effect=\"italics\">irrational<\/em> because they cannot be written as fractions. These numbers make up the set of <strong><span id=\"term-00005\" data-type=\"term\">irrational numbers<\/span><\/strong>. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\{h|h \\ \\text{is not a rational number}\\} [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Differentiating Rational and Irrational Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDetermine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.\r\n\r\n(a) [latex] \\sqrt{25} [\/latex]\r\n\r\n(b) [latex] \\frac{33}{9} [\/latex]\r\n\r\n(c) [latex] \\sqrt{11} [\/latex]\r\n\r\n(d) [latex] \\frac{17}{34} [\/latex]\r\n\r\n(e) [latex] 0.3033033303333\\ldots [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) [latex] \\sqrt{25}: \\hspace{0.5em} [\/latex] This can be simplified as [latex] \\sqrt{25} = 5. [\/latex] Therefore, [latex] \\sqrt{25} [\/latex] is rational.\r\n\r\n&nbsp;\r\n\r\n(b) [latex] \\frac{33}{9}: [\/latex] Because it is a fraction of integers, [latex] \\frac{33}{9} [\/latex] is a rational number. Next, simplify and divide.\r\n<p style=\"text-align: center;\">[latex] \\frac{33}{9} = \\frac{\\overset{11}{\\cancel{33}}}{\\underset{3}{\\cancel{9}}} = \\frac{11}{3} = 3.\\overline{6} [\/latex]<\/p>\r\nSo, [latex] \\frac{33}{9} [\/latex] is rational and a repeating decimal.\r\n\r\n&nbsp;\r\n\r\n(c) [latex] \\sqrt{11}: \\sqrt{11} [\/latex] is irrational because 11 is not a perfect square and [latex] \\sqrt{11} [\/latex] cannot be expressed as a fraction.\r\n\r\n&nbsp;\r\n\r\n(d) [latex] \\frac{17}{34}: [\/latex] Because it is a fraction of integers, [latex] \\frac{17}{34} [\/latex] is a rational number. Simplify and divide.\r\n<p style=\"text-align: center;\">[latex]\\frac{17}{34} = \\frac{\\overset{1}{\\cancel{17}}}{\\underset{2}{\\cancel{34}}} = \\frac{1}{2} = 0.5 [\/latex]<\/p>\r\nSo,[latex] \\frac{17}{34} [\/latex] is rational and a terminating decimal.\r\n\r\n&nbsp;\r\n\r\n(e) [latex] 0.3033033303333\\ldots [\/latex] is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1213813\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div class=\"os-solution-container\">\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\nDetermine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.\r\n\r\n(a) [latex] \\frac{7}{77} [\/latex]\r\n\r\n(b) [latex] \\sqrt{81} [\/latex]\r\n\r\n(c) [latex] 4.27027002700027\\ldots [\/latex]\r\n\r\n(d) [latex] \\frac{91}{13} [\/latex]\r\n\r\n(e) [latex] \\sqrt{39} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/section><section id=\"fs-id1598961\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real Numbers<\/h3>\r\n<p id=\"fs-id1493774\">Given any number <em data-effect=\"italics\">n<\/em>, we know that <em data-effect=\"italics\">n<\/em> is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of<strong> <span id=\"term-00006\" data-type=\"term\">real numbers<\/span><\/strong>. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Negative and positive real numbers include fractions, decimals, and irrational numbers according to their algebraic sign (+ or \u2013). Zero is considered neither positive nor negative.<\/p>\r\n<p id=\"fs-id2057634\">The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the <span id=\"term-00007\" data-type=\"term\">real number line<\/span> as shown in Figure 1<strong>.<\/strong><\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1096\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1096\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-1-300x34.webp\" alt=\"\" width=\"300\" height=\"34\" \/> Figure 1. The real number line[\/caption]\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Classifying Real Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nClassify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?\r\n\r\n(a) [latex] -\\frac{10}{3} [\/latex]\r\n\r\n(b) [latex] \\sqrt{5} [\/latex]\r\n\r\n(c) [latex] -\\sqrt{289} [\/latex]\r\n\r\n(d) [latex] -6\\pi [\/latex]\r\n\r\n(e) [latex] 0.615384615384\\ldots [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) [latex] -\\frac{10}{3} [\/latex] is negative and rational. It lies to the left of 0 on the number line.\r\n\r\n&nbsp;\r\n\r\n(b) [latex] \\sqrt{5} [\/latex] is positive and irrational. It lies to the right of 0.\r\n\r\n&nbsp;\r\n\r\n(c) [latex] -\\sqrt{289} = -\\sqrt{17^2} = -17 [\/latex] is negative and rational. It lies to the left of 0.\r\n\r\n&nbsp;\r\n\r\n(d) [latex] -6\\pi [\/latex] is negative and irrational. It lies to the left of 0.\r\n\r\n&nbsp;\r\n\r\n(e) [latex] 0.615384615384\\ldots [\/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div id=\"Example_01_01_04\" class=\"ui-has-child-title\" data-type=\"example\"><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1391400\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div class=\"os-solution-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nClassify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?\r\n\r\n(a) [latex] \\sqrt{73} [\/latex]\r\n\r\n(b) [latex] -11.411411411\\ldots [\/latex]\r\n\r\n(c) [latex] \\frac{47}{19} [\/latex]\r\n\r\n(d) [latex] -\\frac{\\sqrt{5}}{2} [\/latex]\r\n\r\n(e) [latex] 6.210735 [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1687126\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Sets of Numbers as Subsets<\/h3>\r\n<p id=\"fs-id2447911\">Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1097\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1097\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-2-300x144.webp\" alt=\"\" width=\"300\" height=\"144\" \/> Figure 2. Sets of numbers<br \/>N: the set of natural numbers<br \/>W: the set of whole numbers<br \/>I: the set of integers<br \/>Q: the set of rational numbers<br \/>Q\u00b4: the set of irrational numbers[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Sets of Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe set of <strong><span id=\"term-00008\" data-type=\"term\">natural numbers<\/span><\/strong> includes the numbers used for counting: [latex] \\left\\{1, 2, 3, \\ldots \\right\\} [\/latex]\r\n\r\nThe set of <strong><span id=\"term-00009\" data-type=\"term\">whole numbers<\/span><\/strong> is the set of natural numbers plus zero: [latex] \\left\\{0, 1, 2, 3, \\ldots \\right\\} [\/latex]\r\n\r\nThe set of <strong><span id=\"term-00010\" data-type=\"term\">integers<\/span> <\/strong>adds the negative natural numbers to the set of whole numbers: [latex] \\left\\{\\ldots, -3, -2, -1, 0, 1, 2, 3, \\ldots \\right\\} [\/latex]\r\n\r\nThe set of <strong><span id=\"term-00011\" data-type=\"term\">rational numbers<\/span><\/strong> includes fractions written as [latex] \\{\\frac{m}{n} | \\text{m and n are integers and} \\ n \\not = 0\\} [\/latex]\r\n\r\nThe set of <strong><span id=\"term-00012\" data-type=\"term\">irrational numbers<\/span><\/strong> is the set of numbers that are not rational, are nonrepeating, and are nonterminating: [latex] \\{h|h \\ \\text{is not a rational number}\\} [\/latex]\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 5: Differentiating the Sets of Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nClassify each number as being a natural number (<em data-effect=\"italics\">N<\/em>), whole number (<em data-effect=\"italics\">W<\/em>), integer (<em data-effect=\"italics\">I<\/em>), rational number (<em data-effect=\"italics\">Q<\/em>), and\/or irrational number (<em data-effect=\"italics\">Q\u2032<\/em>).\r\n\r\n(a) [latex] \\sqrt{36} [\/latex]\r\n\r\n(b) [latex] \\frac{8}{3} [\/latex]\r\n\r\n(c) [latex] \\sqrt{73} [\/latex]\r\n\r\n(d) [latex] -6 [\/latex]\r\n\r\n(e) [latex] 3.2121121112\\ldots [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">N<\/em><\/strong><\/td>\r\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">W<\/em><\/strong><\/td>\r\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">I<\/em><\/strong><\/td>\r\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">Q<\/em><\/strong><\/td>\r\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">Q\u2032<\/em><\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">(a) [latex] \\sqrt{36} = 6 [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">X<\/td>\r\n<td style=\"width: 16.6667%;\">X<\/td>\r\n<td style=\"width: 16.6667%;\">X<\/td>\r\n<td style=\"width: 16.6667%;\">X<\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">(b) [latex] \\frac{8}{3} = 2.\\overline{6} [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\">X<\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">(c) [latex] \\sqrt{73} [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\">X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">(d) [latex] -6 [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\">X<\/td>\r\n<td style=\"width: 16.6667%;\">X<\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">(e) [latex] 3.2121121112\\ldots [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\"><\/td>\r\n<td style=\"width: 16.6667%;\">X<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nClassify each number as being a natural number (<em data-effect=\"italics\">N<\/em>), whole number (<em data-effect=\"italics\">W<\/em>), integer (<em data-effect=\"italics\">I<\/em>), rational number (<em data-effect=\"italics\">Q<\/em>), and\/or irrational number (<em data-effect=\"italics\">Q\u2032<\/em>).\r\n\r\n(a) [latex] -\\frac{35}{7} [\/latex]\r\n\r\n(b) [latex] 0 [\/latex]\r\n\r\n(c) [latex] \\sqrt{169} [\/latex]\r\n\r\n(d) [latex] \\sqrt{24} [\/latex]\r\n\r\n(e) [latex] 4.763763763\\ldots [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id3574670\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Performing Calculations Using the Order of Operations<\/h2>\r\n<p id=\"fs-id1580945\">When we multiply a number by itself, we square it or raise it to a power of 2. For example, [latex] 4^2 = 4\\cdot 4 = 16. [\/latex] We can raise any number to any power. In general, the <strong><span id=\"term-00013\" data-type=\"term\">exponential notation<\/span><\/strong> [latex] a^n [\/latex] means that the number or variable [latex] a [\/latex] is used as a factor [latex] n [\/latex] times.<\/p>\r\n<p style=\"text-align: center;\">[latex] a^n = a\\cdot\u00a0 a\\cdot \\overset{\\text{n factors}}a\\cdot \\ldots \\cdot a [\/latex]<\/p>\r\n<p id=\"fs-id1469836\">In this notation, [latex] a^n [\/latex] is read as the <em data-effect=\"italics\">n<\/em>th power of [latex] a [\/latex] or [latex] a [\/latex] to the [latex] n [\/latex] where [latex] a [\/latex] is called the <strong><span id=\"term-00014\" data-type=\"term\">base<\/span> <\/strong>and [latex] n [\/latex] is called the <strong><span id=\"term-00015\" data-type=\"term\">exponent<\/span>.<\/strong> A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, [latex] 24 + 6\\cdot \\frac{2}3 - 4^2 [\/latex] is a mathematical expression.<\/p>\r\n<p id=\"fs-id2448785\">To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the <strong><span id=\"term-00016\" data-type=\"term\">order of operations<\/span><\/strong>. This is a sequence of rules for evaluating such expressions.<\/p>\r\n<p id=\"fs-id1634868\">Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.<\/p>\r\n<p id=\"fs-id2424391\">The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.<\/p>\r\n<p id=\"fs-id2558584\">Let\u2019s take a look at the expression provided.<\/p>\r\n<p style=\"text-align: center;\">[latex] 24 + 6\\cdot \\frac{2}3 - 4^2 [\/latex]<\/p>\r\n<p id=\"fs-id1345509\">There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify [latex] 4^2 [\/latex] as 16.<\/p>\r\n<p style=\"text-align: center;\">[latex] 24 + 6\\cdot \\frac{2}3 - 4^2 [\/latex]\r\n[latex] 24 + 6\\cdot \\frac{2}3 - 16 [\/latex]<\/p>\r\n<p id=\"fs-id894495\">Next, perform multiplication or division, left to right.<\/p>\r\n<p style=\"text-align: center;\">[latex] 24 + 6\\cdot \\frac{2}3 - 16 [\/latex]\r\n[latex] 24 + 4 - 16 [\/latex]<\/p>\r\n<p id=\"fs-id1315208\">Lastly, perform addition or subtraction, left to right.<\/p>\r\n<p style=\"text-align: center;\">[latex] 24 + 4 - 16 [\/latex]\r\n[latex] 28 - 16 [\/latex]\r\n[latex] 12 [\/latex]<\/p>\r\n<p id=\"fs-id894663\">Therefore, [latex] 24 + 6\\cdot \\frac{2}3 - 4^2 = 12 [\/latex]<\/p>\r\n<p id=\"fs-id3032363\">For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Oder of Operations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1324343\">Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym <strong>PEMDAS<\/strong>:<\/p>\r\n<p id=\"fs-id1331012\"><strong>P<\/strong>(arentheses)<span data-type=\"newline\">\r\n<\/span><strong>E<\/strong>(xponents)<span data-type=\"newline\">\r\n<\/span><strong>M<\/strong>(ultiplication) and <strong>D<\/strong>(ivision)<span data-type=\"newline\">\r\n<\/span><strong>A<\/strong>(ddition) and <strong>S<\/strong>(ubtraction)<\/p>\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 a mathematical expression, simplify it using the order of operations.<\/strong>\r\n<ol>\r\n \t<li><span class=\"os-stepwise-content\">Simplify any expressions within grouping symbols.<\/span><\/li>\r\n \t<li><span class=\"os-stepwise-content\">Simplify any expressions containing exponents or radicals.<\/span><\/li>\r\n \t<li><span class=\"os-stepwise-content\">Perform any multiplication and division in order, from left to right.<\/span><\/li>\r\n \t<li><span class=\"os-stepwise-content\">Perform any addition and subtraction in order, from left to right.<\/span><\/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 6: Using the Order of Operations<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the order of operations to evaluate each of the following expressions.\r\n\r\n(a) [latex] (3 \\cdot 2)^2 - 4(6 + 2)[\/latex]\r\n\r\n(b) [latex] \\frac{5^2-4}{7} - \\sqrt{11-2} [\/latex]\r\n\r\n(c) [latex] 6 - |5 - 8| + 3(4-1) [\/latex]\r\n\r\n(d) [latex] \\frac{14 - 3 \\cdot 2}{2 \\cdot 5 - 3^2} [\/latex]\r\n\r\n(e) [latex] 7(5 \\cdot 3) - 2[(6 - 3) - 4^2] + 1 [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a)\r\n\r\n[latex] \\begin{array}{rcll}(3 \\cdot 2)^2 - 4(6 + 2) &amp;=&amp; (6)^2 - 4(8) &amp; \\quad \\text{Simplify parentheses.} \\\\&amp;=&amp; 36 - 4(8) &amp; \\quad \\text{Simplify exponent.} \\\\&amp;=&amp; 36 - 32 &amp; \\quad \\text{Simplify multiplication.} \\\\&amp;=&amp; 4 &amp; \\quad \\text{Simplify subtraction.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(b)\r\n\r\n[latex] \\begin{array}{rcll}\\frac{5^2-4}{7} - \\sqrt{11-2} &amp;=&amp; \\frac{5^2-4}{7} - \\sqrt{9} &amp; \\quad \\text{Simplify grouping symbols (radical).} \\\\&amp;=&amp; \\frac{5^2-4}{7} - 3 &amp; \\quad \\text{Simplify radical.} \\\\&amp;=&amp; \\frac{25-4}{7} - 3 &amp; \\quad \\text{Simplify exponent.} \\\\&amp;=&amp; \\frac{21}{7} - 3 &amp; \\quad \\text{Simplify subtraction in numerator.} \\\\&amp;=&amp; 3 - 3 &amp; \\quad \\text{Simplify division.} \\\\&amp;=&amp; 0 &amp; \\quad \\text{Simplify subtraction.}\\end{array} [\/latex]\r\n\r\nNote that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.\r\n\r\n&nbsp;\r\n\r\n(c)\r\n\r\n[latex] \\begin{array}{rcll}6 - |5 - 8| + 3|4 - 1| &amp;=&amp; 6 - |-3| + 3(3) &amp; \\quad \\text{Simplify inside grouping symbols.} \\\\&amp;=&amp; 6 - (3) + 3(3) &amp; \\quad \\text{Simplify absolute value.} \\\\&amp;=&amp; 6 - 3 + 9 &amp; \\quad \\text{Simplify multiplication.} \\\\&amp;=&amp; 12 &amp; \\quad \\text{Simplify addition.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(d)\r\n\r\n[latex] \\begin{array}{rcll}\\frac{14 - 3 \\cdot 2}{2 \\cdot 5 - 3^2} &amp;=&amp; \\frac{14 - 3 \\cdot 2}{2 \\cdot 5 - 9} &amp; \\quad \\text{Simplify exponent.} \\\\&amp;=&amp; \\frac{14 - 6}{10 - 9} &amp; \\quad \\text{Simplify products.} \\\\&amp;=&amp; \\frac{8}{1} &amp; \\quad \\text{Simplify differences.} \\\\&amp;=&amp; 8 &amp; \\quad \\text{Simplify quotient.}\\end{array} [\/latex]\r\n\r\nIn this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.\r\n\r\n&nbsp;\r\n\r\n(e)\r\n\r\n[latex] \\begin{array}{rcll}7(5 \\cdot 3) - 2[(6 - 3) - 4^2] + 1 &amp;=&amp; 7(15) - 2[(3) - 4^2] + 1 &amp; \\quad \\text{Simplify inside parentheses.} \\\\&amp;=&amp; 7(15) - 2(3 - 16) + 1 &amp; \\quad \\text{Simplify exponent.} \\\\&amp;=&amp; 7(15) - 2(-13) + 1 &amp; \\quad \\text{Subtract.} \\\\&amp;=&amp; 105 + 26 + 1 &amp; \\quad \\text{Multiply.} \\\\&amp;=&amp; 132 &amp; \\quad \\text{Add.}\\end{array} [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the order of operations to evaluate each of the following expressions.\r\n\r\n(a) [latex] \\sqrt{5^2-4^2} + 7(5-4)^2 [\/latex]\r\n\r\n(b) [latex] 1+ \\frac{7\\cdot 5-8\\cdot 4}{9-6} [\/latex]\r\n\r\n(c) [latex] |1.8 - 4.3| + 0.4\\sqrt{15+10} [\/latex]\r\n\r\n(d) [latex] \\frac{1}{2}[5\\cdot 3^2-7^2]+\\frac{1}{3}\\cdot 9^2 [\/latex]\r\n\r\n(e) [latex] [(3-8)^2-4]-(3-8) [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id2783853\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Properties of Real Numbers<\/h2>\r\n<p id=\"fs-id1979200\">For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.<\/p>\r\n\r\n<section id=\"fs-id3578887\" data-depth=\"2\"><section id=\"fs-id3578889\" data-depth=\"3\">\r\n<h4 data-type=\"title\">Commutative Properties<\/h4>\r\n<p id=\"fs-id2523326\">The <strong><span id=\"term-00017\" data-type=\"term\">commutative property of addition<\/span><\/strong> states that numbers may be added in any order without affecting the sum.<\/p>\r\n<p style=\"text-align: center;\">[latex] a+b = b+a [\/latex]<\/p>\r\n<p id=\"fs-id3573981\">We can better see this relationship when using real numbers.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lclcl}(-2) + 7 = 5 &amp; &amp; \\text{and} &amp; &amp; 7 + (-2) = 5\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1432671\">Similarly, the <strong><span id=\"term-00018\" data-type=\"term\">commutative property of multiplication<\/span><\/strong> states that numbers may be multiplied in any order without affecting the product.<\/p>\r\n<p style=\"text-align: center;\">[latex] a\\cdot b=b\\cdot a [\/latex]<\/p>\r\n<p id=\"fs-id3260777\">Again, consider an example with real numbers.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ccccc}(-11) \\cdot (-4) = 44 &amp; &amp; \\text{and} &amp; &amp; (-4) \\cdot (-11) = 44\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id3033661\">It is important to note that neither subtraction nor division is commutative. For example, [latex] 17-5 [\/latex] is not the same as [latex] 5-17. [\/latex] Similarly, [latex] 20\\div{5}\\not=5\\div{20}. [\/latex]<\/p>\r\n\r\n<\/section><section id=\"fs-id1618548\" data-depth=\"3\">\r\n<h4 data-type=\"title\">Associative Properties<\/h4>\r\n<p id=\"fs-id2643647\">The <strong><span id=\"term-00019\" data-type=\"term\">associative property of multiplication<\/span><\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.<\/p>\r\n<p style=\"text-align: center;\">[latex] a(bc)=(ab)c [\/latex]<\/p>\r\n<p id=\"fs-id1314913\">Consider this example.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ccccc}(3 \\cdot 4) \\cdot 5 = 60 &amp; &amp; \\text{and} &amp; &amp; 3 \\cdot (4 \\cdot 5) = 60\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1718282\">The <strong><span id=\"term-00020\" data-type=\"term\">associative property of addition<\/span><\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/p>\r\n<p style=\"text-align: center;\">[latex] a+(b+c)=(a+b)+c [\/latex]<\/p>\r\n<p id=\"fs-id1734746\">This property can be especially helpful when dealing with negative integers. Consider this example.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ccccc}[15 + (-9)] + 23 = 29 &amp; &amp; \\text{and} &amp; &amp; 15 + [(-9) + 23] = 29\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id2213874\">Are subtraction and division associative? Review these examples.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rclrcl}8 - (3 - 15) &amp; \\stackrel{?}{=} &amp; (8 - 3) - 15 &amp; 64 \\div (8 \\div 4) &amp; \\stackrel{?}{=} &amp; (64 \\div 8) \\div 4 \\\\8 - (-12) &amp; = &amp; 5 - 15 &amp; 64 \\div 2 &amp; \\stackrel{?}{=} &amp; 8 \\div 4 \\\\20 &amp; \\neq &amp; -10 &amp; 32 &amp; \\neq &amp; 2\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1591297\">As we can see, neither subtraction nor division is associative.<\/p>\r\n\r\n<\/section><section id=\"fs-id1705288\" data-depth=\"3\">\r\n<h4 data-type=\"title\">Distributive Property<\/h4>\r\n<p id=\"fs-id2627113\">The <strong><span id=\"term-00021\" data-type=\"term\">distributive property<\/span><\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.<\/p>\r\n<p style=\"text-align: center;\">[latex] a\\cdot (b+c)=a\\cdot b+a\\cdot c [\/latex]<\/p>\r\n<p id=\"fs-id3594964\">This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.<\/p>\r\n<span id=\"fs-id1567314\" data-type=\"media\" data-alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.\"><img class=\"size-medium wp-image-1110 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-distributive-300x60.webp\" alt=\"\" width=\"300\" height=\"60\" \/><\/span>\r\n<p id=\"fs-id1719796\">Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by \u20137, and adding the products.<\/p>\r\n<p id=\"fs-id2186994\">To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}6 + (3 \\cdot 5) &amp; \\stackrel{?}{=} &amp; (6 + 3) \\cdot (6 + 5) \\\\6 + (15) &amp; \\stackrel{?}{=} &amp; (9) \\cdot (11) \\\\21 &amp; \\neq &amp; 99\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id2750433\">A special case of the distributive property occurs when a sum of terms is subtracted.<\/p>\r\n<p style=\"text-align: center;\">[latex] a-b=a+(-b) [\/latex]<\/p>\r\n<p id=\"fs-id2265668\">For example, consider the difference [latex] 12-(5+3) [\/latex]. We can rewrite the difference of the two terms 12 and [latex] (5+3) [\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex] (5+3) [\/latex] we add the opposite.<\/p>\r\n<p style=\"text-align: center;\">[latex] 12+(-1)\\cdot (5+3) [\/latex]<\/p>\r\n<p id=\"fs-id3586677\">Now, distribute [latex] -1 [\/latex] and simplify the result.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}12 - (5 + 3) &amp;=&amp; 12 + (-1) \\cdot (5 + 3) \\\\&amp;=&amp; 12 + [(\u22121) \\cdot 5 + (\u22121) \\cdot 3] \\\\&amp;=&amp; 12 + (-8) \\\\&amp;=&amp; 4\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1392553\">This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}12 - (5 + 3) &amp;=&amp; 12 + (-5 - 3) \\\\&amp;=&amp; 12 + (-8) \\\\&amp;=&amp; 4\\end{array} [\/latex]<\/p>\r\n\r\n<\/section><section id=\"fs-id2165198\" data-depth=\"3\">\r\n<h4 data-type=\"title\">Identity Properties<\/h4>\r\n<p id=\"fs-id3238896\">The <strong><span id=\"term-00022\" data-type=\"term\">identity property of addition<\/span><\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.<\/p>\r\n<p style=\"text-align: center;\">[latex] a+0=a [\/latex]<\/p>\r\n<p id=\"fs-id2440675\">The <strong><span id=\"term-00023\" data-type=\"term\">identity property of multiplication<\/span><\/strong> states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.<\/p>\r\n<p style=\"text-align: center;\">[latex] a\\cdot 1=a [\/latex]<\/p>\r\n\r\n<math display=\"block\"><\/math>\r\n<p id=\"fs-id1474435\">For example, we have [latex] (-6)+0=-6 [\/latex] and [latex] 23\\cdot 1=23. [\/latex] There are no exceptions for these properties; they work for every real number, including 0 and 1.<\/p>\r\n\r\n<\/section><section id=\"fs-id2591971\" data-depth=\"3\">\r\n<h4 data-type=\"title\">Inverse Properties<\/h4>\r\n<p id=\"fs-id2794277\">The <strong><span id=\"term-00024\" data-type=\"term\">inverse property of addition<\/span><\/strong> states that, for every real number <em data-effect=\"italics\">a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted by (\u2212<em data-effect=\"italics\">a<\/em>), that, when added to the original number, results in the additive identity, 0.<\/p>\r\n<p style=\"text-align: center;\">[latex] a+(-a)=0 [\/latex]<\/p>\r\n<p id=\"fs-id1678541\">For example, if [latex] a=-8 [\/latex] the additive inverse is 8, since [latex] (-8)+8=0 [\/latex]<\/p>\r\n<p id=\"fs-id2998343\">The <strong><span id=\"term-00025\" data-type=\"term\">inverse property of multiplication<\/span><\/strong> holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number <em data-effect=\"italics\">a<\/em>, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex] \\frac{1}{a} [\/latex] that, when multiplied by the original number, results in the multiplicative identity, 1.<\/p>\r\n<p style=\"text-align: center;\">[latex] a\\cdot \\frac{1}{a}=1 [\/latex]<\/p>\r\n<p id=\"fs-id1694769\">For example, if [latex] a=-\\frac{2}{3} [\/latex] the reciprocal, denoted [latex] \\frac{1}{a} [\/latex] is [latex] -\\frac{3}{2} [\/latex] because<\/p>\r\n<p style=\"text-align: center;\">[latex] a \\cdot \\frac{1}{a} = \\left(-\\frac{2}{3}\\right) \\cdot \\left(-\\frac{3}{2}\\right) = 1 [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Properties of Real Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe following properties hold for real numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Addition<\/strong><\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Multiplication<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><strong>Commutative Property<\/strong><\/td>\r\n<td style=\"width: 33.3333%;\">[latex] a+b=b+a [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex] a\\cdot b=b\\cdot a [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><strong>Associative Property<\/strong><\/td>\r\n<td style=\"width: 33.3333%;\">[latex] a+(b+c)=(a+b)+c [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex] a(bc)=(ab)c [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><strong>Distributive Property<\/strong><\/td>\r\n<td style=\"width: 33.3333%;\">[latex] a\\cdot (b+c)=a\\cdot b+a\\cdot c [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><strong>Identity Property<\/strong><\/td>\r\n<td style=\"width: 33.3333%;\">There exists a unique real number called the additive identity, 0, such that, for any real number <em data-effect=\"italics\">a<\/em>\r\n\r\n[latex] a+0=a [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em data-effect=\"italics\">a<\/em>\r\n\r\n[latex] a\\cdot 1=a [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><strong>Inverse Property<\/strong><\/td>\r\n<td style=\"width: 33.3333%;\">Every real number a has an additive inverse, or opposite, denoted <em data-effect=\"italics\">\u2013a<\/em>, such that\r\n\r\n[latex] a+(-a)=0 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">Every nonzero real number <em data-effect=\"italics\">a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex] \\frac{1}{a} [\/latex] such that\r\n\r\n[latex] a\\cdot (\\frac{1}{a})=1 [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Using Properties of Real Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n\r\n(a) [latex] 3 \\cdot 6 + 3 \\cdot 4 [\/latex]\r\n\r\n(b) [latex] (5+8)+(-8) [\/latex]\r\n\r\n(c) [latex] 6-(15+9) [\/latex]\r\n\r\n(d) [latex] \\frac{4}{7} \\cdot \\left(\\frac{2}{3} \\cdot \\frac{7}{4}\\right) [\/latex]\r\n\r\n(e) [latex] 100\\cdot [0.75+(-2.38)] [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a)\r\n\r\n[latex] \\begin{array}{rcll}3 \\cdot 6 + 3 \\cdot 4 &amp;=&amp; 3 \\cdot (6 + 4) &amp; \\quad \\text{Distributive property.} \\\\&amp;=&amp; 3 \\cdot 10 &amp; \\quad \\text{Simplify.} \\\\&amp;=&amp; 30 &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(b)\r\n\r\n[latex] \\begin{array}{rcll}(5 + 8) + (-8) &amp;=&amp; 5 + [8 + (-8)] &amp; \\quad \\text{Associative property of addition.} \\\\&amp;=&amp; 5 + 0 &amp; \\quad \\text{Inverse property of addition.} \\\\&amp;=&amp; 5 &amp; \\quad \\text{Identity property of addition.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(c)\r\n\r\n[latex] \\begin{array}{rcll}6 - (15 + 9) &amp;=&amp; 6 + [(-15) + (-9)] &amp; \\quad \\text{Distributive property.} \\\\&amp;=&amp; 6 + (-24) &amp; \\quad \\text{Simplify.} \\\\&amp;=&amp; -18 &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(d)\r\n\r\n[latex] \\begin{array}{rcll}\\frac{4}{7} \\cdot \\left(\\frac{2}{3} \\cdot \\frac{7}{4}\\right) &amp;=&amp; \\frac{4}{7} \\cdot \\left(\\frac{7}{4} \\cdot \\frac{2}{3}\\right) &amp; \\quad \\text{Commutative property of multiplication.} \\\\&amp;=&amp; \\left(\\frac{4}{7} \\cdot \\frac{7}{4}\\right) \\cdot \\frac{2}{3} &amp; \\quad \\text{Associative property of multiplication.} \\\\&amp;=&amp; 1 \\cdot \\frac{2}{3} &amp; \\quad \\text{Inverse property of multiplication.} \\\\&amp;=&amp; \\frac{2}{3} &amp; \\quad \\text{Identity property of multiplication.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(e)\r\n\r\n[latex] \\begin{array}{rcll}100 \\cdot [0.75 + (-2.38)] &amp;=&amp; 100 \\cdot 0.75 + 100 \\cdot (-2.38) &amp; \\quad \\text{Distributive property.} \\\\&amp;=&amp; 75 + (-238) &amp; \\quad \\text{Simplify.} \\\\&amp;=&amp; -163 &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n\r\n(a) [latex] \\left(-\\frac{23}{5}\\right)\\cdot [11\\cdot \\left(-\\frac{5}{23}\\right)] [\/latex]\r\n\r\n(b) [latex] 5\\cdot (6.2+0.4) [\/latex]\r\n\r\n(c) [latex] 18-(7-15) [\/latex]\r\n\r\n(d) [latex] \\frac{17}{18}+[\\frac{4}{9}+\\left(-\\frac{17}{18}\\right)] [\/latex]\r\n\r\n(e) [latex] 6\\cdot (-3)+6\\cdot 3 [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><section id=\"fs-id2146838\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Evaluating Algebraic Expressions<\/h3>\r\n<p id=\"fs-id2146844\">So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as [latex] x+5, \\frac{4}{3}\\pi r^2, [\/latex] or [latex] \\sqrt{2m^3n^2}. [\/latex] In the expression [latex] x+5, [\/latex] 5 is called a <strong><span id=\"term-00026\" data-type=\"term\">constant<\/span> <\/strong>because it does not vary and <em data-effect=\"italics\">x<\/em> is called a <strong><span id=\"term-00027\" data-type=\"term\">variable<\/span> <\/strong>because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An <strong><span id=\"term-00028\" data-type=\"term\">algebraic expression<\/span><\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.<\/p>\r\n<p id=\"fs-id2755741\">We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl rcl}(-3)^5 &amp;=&amp; (-3) \\cdot (-3) \\cdot (-3) \\cdot (-3) \\cdot (-3) &amp; x^5 &amp;=&amp; x \\cdot x \\cdot x \\cdot x \\cdot x \\\\(2 \\cdot 7)^3 &amp;=&amp; (2 \\cdot 7) \\cdot (2 \\cdot 7) \\cdot (2 \\cdot 7) &amp; (yz)^3 &amp;=&amp; (yz) \\cdot (yz) \\cdot (yz)\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1542482\">In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.<\/p>\r\n<p id=\"fs-id2464703\">Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Describing Algebraic Expressions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nList the constants and variables for each algebraic expression.\r\n\r\n(a) [latex] x+5 [\/latex]\r\n\r\n(b) [latex] \\frac{4}{3}\\pi r^3 [\/latex]\r\n\r\n(c) [latex] \\sqrt{2m^3n^2} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Constants<\/strong><\/td>\r\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Variables<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">(a) [latex] x+5 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex] 5 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex] x [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">(b) [latex] \\frac{4}{3}\\pi r^3 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex] \\frac{4}{3}, \\pi [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex] r [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">(c) [latex] \\sqrt{2m^3n^2} [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex] 2 [\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex] m, n [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nList the constants and variables for each algebraic expression.\r\n\r\n(a) [latex] 2\\pi r(r+h) [\/latex]\r\n\r\n(b) [latex] 2(L+W) [\/latex]\r\n\r\n(c) [latex] 4y^3+y [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Evaluating an Algebraic Expression at Different Values<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate the expression [latex] 2x-7 [\/latex] for each value for <em data-effect=\"italics\">x.<\/em>\r\n\r\n(a) [latex] x=0 [\/latex]\r\n\r\n(b) [latex] x=1 [\/latex]\r\n\r\n(c) [latex] x=\\frac{1}{2} [\/latex]\r\n\r\n(d) [latex] x=-4 [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) Substitute 0 for <em data-effect=\"italics\">x.<\/em>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}2x - 7 &amp;=&amp; 2(0) - 7 \\\\&amp;=&amp; 0 - 7 \\\\&amp;=&amp; -7\\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n(b) Substitute 1 for <em data-effect=\"italics\">x.<\/em>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}2x - 7 &amp;=&amp; 2(1) - 7 \\\\&amp;=&amp; 2 - 7 \\\\&amp;=&amp; -5\\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n(c) Substitute\u00a0[latex] \\frac{1}{2} [\/latex] for <em data-effect=\"italics\">x.<\/em>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}2x - 7 &amp;=&amp; 2\\left(\\frac{1}{2}\\right) - 7 \\\\&amp;=&amp; 1 - 7 \\\\&amp;=&amp; -6\\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n(d) Substitute -4 for <em data-effect=\"italics\">x.<\/em>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}2x - 7 &amp;=&amp; 2(-4) - 7 \\\\&amp;=&amp; -8 - 7 \\\\&amp;=&amp; -15\\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate the expression [latex] 11-3y [\/latex] for each value for <em data-effect=\"italics\">y.<\/em>\r\n\r\n(a) [latex] y=2 [\/latex]\r\n\r\n(b) [latex] y=0 [\/latex]\r\n\r\n(c) [latex] y=\\frac{2}{3} [\/latex]\r\n\r\n(d) [latex] y=-5 [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 10: Evaluating Algebraic Expressions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate each expression for the given values.\r\n\r\n(a) [latex] x+5 [\/latex] for [latex] x=-5 [\/latex]\r\n\r\n(b) [latex] \\frac{t}{2t-1} [\/latex] for [latex] t=10 [\/latex]\r\n\r\n(c) [latex] \\frac{4}{3}\\pi r^3 [\/latex] for [latex] r=5 [\/latex]\r\n\r\n(d) [latex] a+ab+b [\/latex] for [latex] a=11, b=-8 [\/latex]\r\n\r\n(e) [latex] \\sqrt{2m^3n^2} [\/latex] for [latex] m=2, n=3 [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) Substitute [latex] -5 [\/latex] for [latex] x. [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}x+5 &amp;=&amp; (-5)+5 \\\\ &amp;=&amp; 0\\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n(b) Substitute [latex] 10 [\/latex] for [latex] t. [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\frac{t}{2t-1} &amp;=&amp; \\frac{(10}{2(10)-1} \\\\ &amp;=&amp; \\frac{10}{20-1} \\\\ &amp;=&amp; \\frac{10}{19}\\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n(c) Substitute [latex] 5 [\/latex] for [latex] r. [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\frac{4}{3}\\pi r^3 &amp;=&amp; \\frac{4}{3}\\pi(5)^3 \\\\ &amp;=&amp; \\frac{4}{3}\\pi(125) \\\\ &amp;=&amp; \\frac{500}{3}\\pi\\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n(d) Substitute [latex] 11 [\/latex] for [latex] a [\/latex] and [latex] -8 [\/latex] for [latex] b. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}a+ab+b &amp;=&amp; (11)+(11)(-8)+(-8) \\\\ &amp;=&amp; 11-88-8 \\\\ &amp;=&amp; -85\\end{array} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n(e) Substitute [latex] 2 [\/latex] for [latex] m [\/latex] and [latex] 3 [\/latex] for [latex] n. [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\sqrt{2m^3n^2} &amp;=&amp; \\sqrt{2(2)^3(3)^2} \\\\ &amp;=&amp; \\sqrt{2(8)(9)} \\\\ &amp;=&amp; \\sqrt{144} \\\\ &amp;=&amp; 12\\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate each expression for the given values.\r\n\r\n(a) [latex] \\frac{y+3}{y-3} [\/latex] for [latex] y=5 [\/latex]\r\n\r\n(b) [latex] 7-2t [\/latex] for [latex] t=-2 [\/latex]\r\n\r\n(c) [latex] \\frac{1}{3}\\pi r^2 [\/latex] for [latex] r=11 [\/latex]\r\n\r\n(d) [latex] (p^2q)^3 [\/latex] for [latex] p=-2, q=3 [\/latex]\r\n\r\n(e) [latex] 4(m-n)-5(n-m) [\/latex] for [latex] m=\\frac{2}{3}, n=\\frac{1}{3} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id3660280\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Formulas<\/h3>\r\n<p id=\"fs-id2714749\">An <strong><span id=\"term-00029\" data-type=\"term\">equation<\/span> <\/strong>is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex] 2x+1=7 [\/latex] has the solution of 3 because when we substitute 3 for [latex] x [\/latex] in the equation, we obtain the true statement [latex] 2(3)+1=7. [\/latex]<\/p>\r\n<p id=\"fs-id1631296\">A <strong><span id=\"term-00030\" data-type=\"term\">formula<\/span> <\/strong>is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex] A [\/latex] of a circle in terms of the radius [latex] r [\/latex] of the circle: [latex] A=\\pi r^2 [\/latex] For any value of [latex] r, [\/latex] the area [latex] A [\/latex] can be found by evaluating the expression [latex] \\pi r^2. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 11: Using a Formula<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA right circular cylinder with radius [latex] r [\/latex] and height [latex] h [\/latex] has the surface area [latex] S [\/latex] (in square units) given by the formula [latex] S=2\\pi r(r+h). [\/latex] See Figure 3. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of [latex] \\pi . [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1112\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1112\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-3-300x256.webp\" alt=\"\" width=\"300\" height=\"256\" \/> Figure 3. Right circular cylinder[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Evaluate the expression [latex] 2\\pi r(r+h) [\/latex] for [latex] r=6 [\/latex] and [latex] h=9. [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}S &amp;=&amp; 2\\pi r(r+h) \\\\ &amp;=&amp; 2\\pi(6)[(6)+(9)] \\\\ &amp;=&amp; 2\\pi(6)(15) \\\\ &amp;=&amp; 180\\pi\\end{array} [\/latex]<\/p>\r\nThe surface area is [latex] 180\\pi [\/latex] square inches.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA photograph with length <em data-effect=\"italics\">L<\/em> and width <em data-effect=\"italics\">W<\/em> is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm<sup>2<\/sup>) is found to be [latex] A=(L+16)(W+16)-L\\cdot W [\/latex] See Figure 4. Find the area of a mat for a photograph with length 32 cm and width 24 cm.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_1113\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1113\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-4-300x251.webp\" alt=\"\" width=\"300\" height=\"251\" \/> Figure 4[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id2193236\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Simplifying Algebraic Expressions<\/h3>\r\n<p id=\"fs-id3229215\">Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 12: Simplifying Algebraic Expressions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify each algebraic expression.\r\n\r\n(a) [latex] 3x-2y+x-3y-7 [\/latex]\r\n\r\n(b) [latex] 2r-5(3-r)+4 [\/latex]\r\n\r\n(c) [latex] \\left(4t-\\frac{5}{4}s\\right)-\\left(\\frac{2}{3}t+2s\\right) [\/latex]\r\n\r\n(d) [latex] 2mn-5m+3mn+n [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a)\r\n\r\n[latex] \\begin{array}{rcll}3x - 2y + x - 3y - 7 &amp;=&amp; 3x + x - 2y - 3y - 7 &amp; \\quad \\text{Commutative property of addition.} \\\\&amp;=&amp; 4x - 5y - 7 &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(b)\r\n\r\n[latex] \\begin{array}{rcll}2r-5(3-r)+4 &amp;=&amp; 2r-15+5r+4 &amp; \\quad \\text{Distributive property.} \\\\&amp;=&amp; 2r+5r-15+4 &amp; \\quad \\text{Commutative property of addition.} \\\\&amp;=&amp; 7r-11 &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(c)\r\n\r\n[latex] \\begin{array}{rcll}\\left(4t-\\frac{5}{4}s\\right)-\\left(\\frac{2}{3}t+2s\\right) &amp;=&amp; 4t-\\frac{5}{4}s-\\frac{2}{3}t-2s &amp; \\quad \\text{Distributive property.} \\\\&amp;=&amp; 4t-\\frac{2}{3}t-\\frac{5}{4}s-2s &amp; \\quad \\text{Commutative property of addition.} \\\\&amp;=&amp; \\frac{10}{3}t-\\frac{13}{4}s &amp; \\quad \\text{Simplify.}\\end{array} [\/latex]\r\n\r\n&nbsp;\r\n\r\n(d)\r\n\r\n[latex] \\begin{array}{rcll}2mn-5m+3mn+n &amp;=&amp; 2mn+3mn-5m+n &amp; \\quad \\text{Commutative property of addition.} \\\\&amp;=&amp; 5mn-5m+n &amp; \\quad \\text{Simplify.} \\end{array} [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSimplify each algebraic expression.\r\n\r\n(a) [latex] \\frac{2}{3}y-2\\left(\\frac{4}{3}y+z\\right) [\/latex]\r\n\r\n(b) [latex] \\frac{5}{t}-2-\\frac{3}{t}+1 [\/latex]\r\n\r\n(c) [latex] 4p(q-1)+q(1-p) [\/latex]\r\n\r\n(d) [latex] 9r-(s+2r)+(6-s) [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 13: Simplifying a Formula<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA rectangle with length [latex] L [\/latex] and width [latex] W [\/latex] has a perimeter [latex] P [\/latex] given by [latex] P=L+W+L+W. [\/latex] Simplify this expression.\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}{rcll}P &amp;=&amp; L + W + L + W &amp; \\\\P &amp;=&amp; L + L + W + W &amp; \\quad \\text{Commutative property of addition} \\\\P &amp;=&amp; 2L + 2W &amp; \\quad \\text{Simplify} \\\\P &amp;=&amp; 2(L + W) &amp; \\quad \\text{Distributive property}\\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #13<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIf the amount [latex] P [\/latex] is deposited into an account paying simple interest [latex] r [\/latex] for time [latex] t, [\/latex] the total value of the deposit [latex] A [\/latex] is given by [latex] A=P+Prt. [\/latex] Simplify the expression. (This formula will be explored in more detail later in the course.)\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\" style=\"text-align: left;\"><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 real numbers.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=RJ7uU9HbdqA\">Simplify an Expression<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=RJ7uU9HbdqA\">Evaluate an Expression 1<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=8b-rf2AW3Ac\">Evaluate an Expression 2<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-eos os-section-exercises-container\" style=\"text-align: left;\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">1.1 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id3300486\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id2542028\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id2542033\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2542034\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2542033-solution\">1<\/a><span class=\"os-divider\">. <\/span>Is [latex] \\sqrt{2} [\/latex] an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1433666\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1433667\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2510664\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2510665\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2510664-solution\">3<\/a><span class=\"os-divider\">. <\/span>What do the Associative Properties allow us to do when following the order of operations? Explain your answer.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1571756\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id2521060\">For the following exercises, simplify the given expression.<\/p>\r\n\r\n<div id=\"fs-id2521063\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2521064\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span> [latex] 10+2\\cdot (5-3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2448113\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2448114\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2448113-solution\">5<\/a><span class=\"os-divider\">. <\/span> [latex] 6\\div2-(81\\div3^2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3665280\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3665281\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] 18+(6-8)^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2054506\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2054507\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2054506-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] -2\\cdot \\left[16\\div(8-4)^2\\right]^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1607809\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1460495\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] 4-6+2\\cdot 7 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2444193\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2444194\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2444193-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] 3(5-8) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3235427\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3235428\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] 4+6-10\\div2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1601248\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1601249\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1601248-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] 12\\div(36\\div9)+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2555791\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2555792\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] (4+5)^2\\div3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1500556\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1500557\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1500556-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] 3-12\\cdot 2+19 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1791590\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1791591\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] 2+8\\cdot 7\\div4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2396900\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2396901\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2396900-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] 5+(6+4)-11 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1450507\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1450508\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] 9-18\\div3^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2520962\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2520963\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2520962-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] 14 \\cdot3\\div7-6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1368488\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1368489\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] 9-(3+11)\\cdot 2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1619048\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1619049\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1619048-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] 6+2\\cdot 2-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1620842\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1620843\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] 64\\div(8+4\\cdot 2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2995850\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2995851\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2995850-solution\">21<\/a><span class=\"os-divider\">.<\/span> [latex] 9+4(2^2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1859164\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1859165\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] (12\\div3\\cdot 3)^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1433630\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1433631\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1433630-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] 25\\div5^2-7 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2158696\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2158697\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] (15-7)\\cdot (3-7) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1673686\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1673687\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1673686-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] 2\\cdot 4-9(-1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2555535\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2555536\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] 4^2-25\\cdot \\frac{1}{5} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2708694\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2708695\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2708694-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] 12(3-1)\\div6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1373736\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1373741\">For the following exercises, evaluate the expression using the given value of the variable.<\/p>\r\n\r\n<div id=\"fs-id3349803\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3349804\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] 8(x+3)-64 [\/latex] for [latex] x=2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id2248088\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2248090\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2248088-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] 4y+8-2y [\/latex] for [latex] y=3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1634685\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1634686\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] (11a+3)-18a [\/latex] for [latex] a=-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1610297\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1610298\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1610297-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] 4z-2z(1+4)-36 [\/latex] for [latex] z=5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2183439\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2183440\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] 4y(7-2)^2+200 [\/latex] for [latex] y=-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1442438\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1442439\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1442438-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] -(2x)^2+1+3 [\/latex] for [latex] x=2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2570388\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2570389\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] 8(2+4)-15b+b [\/latex] for [latex] b=-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2714429\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2714430\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2714429-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] 2(11c-4)-36 [\/latex] for [latex] c=0 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1453902\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1453903\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] 4(3-1)x-4 [\/latex] for [latex] x=10 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1485040\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1485041\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1485040-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] \\frac{1}{4}(8w-4^2) [\/latex] for [latex] w=1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1734699\" style=\"text-align: left;\">For the following exercises, simplify the expression.<\/p>\r\n\r\n<div id=\"fs-id1734702\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1734703\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] 4x+x(13-7) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2113191\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2113192\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2113191-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] 2y-(4)^2y-11 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3185006\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3185007\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] \\frac{a}{2^3}(64)-12a\\div6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2107133\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2107134\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2107133-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] 8b-4b(3)+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2444084\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3260784\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] 5l\\div3l \\cdot (9-6) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2194512\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2194513\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2194512-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] 7z-3+z \\cdot 6^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2619567\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2619568\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] 4\\cdot 3+18x\\div9-12 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1458150\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1458151\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1458150-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] 9(y+8)-27 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1443223\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1443224\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] (\\frac{9}{6}t-4)2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1443990\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1443991\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1443990-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] 6+12b-3\\cdot6b [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2771916\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2771917\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] 18y-2(1+7y) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3283284\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1917673\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id3283284-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] (\\frac{4}{9})^2\\cdot 27x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2980532\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2980533\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] 8(3-m)+1(-8) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2193704\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2193705\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2193704-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] 9x+4x(2+3)-4(2x+3x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2825015\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2825016\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] 5^2-4(3x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id2098225\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<p id=\"fs-id1485257\">For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor\u2019s dog.<\/p>\r\n\r\n<div id=\"fs-id1485264\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2404926\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1485264-solution\">53<\/a><span class=\"os-divider\">. <\/span>Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"eip-id1167336272625\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"eip-id1167336272627\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>How much money does Fred keep?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"eip-52\">For the following exercises, solve the given problem.<\/p>\r\n\r\n<div id=\"fs-id2658142\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1540914\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2658142-solution\">55<\/a><span class=\"os-divider\">. <\/span>According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by [latex] \\pi . [\/latex] Is the circumference of a quarter a whole number, a rational number, or an irrational number?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2483553\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2483554\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id3615404\">For the following exercises, consider this scenario: There is a mound of [latex] g [\/latex] pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.<\/p>\r\n\r\n<div id=\"fs-id1842419\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1842420\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1842419-solution\">57<\/a><span class=\"os-divider\">. <\/span>Write the equation that describes the situation.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1596547\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1596548\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Solve for <em data-effect=\"italics\">g<\/em>.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"eip-373\">For the following exercise, solve the given problem.<\/p>\r\n\r\n<div id=\"fs-id1276001\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1276002\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1276001-solution\">59<\/a><span class=\"os-divider\">. <\/span>Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that [latex] 2,500,000-x=0. [\/latex] What property of addition tells us what the value of <em data-effect=\"italics\">x<\/em> must be?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2178453\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id2460118\">For the following exercises, use a graphing calculator to solve for <em data-effect=\"italics\">x<\/em>. Round the answers to the nearest hundredth.<\/p>\r\n\r\n<div id=\"fs-id1703228\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1703229\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex] 0.5(12.3)^2-48x=\\frac{3}{5} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1315170\" class=\"material-set-2 os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1315171\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1315170-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex] (0.25-0.75)^2x-7.2=9.9 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1720808\" data-depth=\"2\">\r\n<h3 style=\"text-align: left;\" data-type=\"title\">Extensions<\/h3>\r\n<div id=\"fs-id1523458\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1523459\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>If a whole number is not a natural number, what must the number be?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1454051\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1454052\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1454051-solution\">63<\/a><span class=\"os-divider\">. <\/span>Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2806511\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2806512\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2538076\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2538077\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2538076-solution\">65<\/a><span class=\"os-divider\">. <\/span>Determine whether the simplified expression is rational or irrational: [latex] \\sqrt{-18-4(5)(-1)}. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1440424\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1440425\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. Determine whether the simplified expression is rational or irrational: <\/span>[latex] \\sqrt{-16+4(5)+5}. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1979146\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2608456\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1979146-solution\">67<\/a><span class=\"os-divider\">. <\/span>The division of two natural numbers will always result in what type of number?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2053806\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2053807\" data-type=\"problem\">\r\n<p style=\"text-align: left;\"><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>What property of real numbers would simplify the following expression: [latex] 4+7(x-1)? [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_d0a86917-6447-4b49-bed1-efb5c352cc77\" class=\"chapter-content-module\" style=\"text-align: center;\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\" style=\"text-align: left;\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\" style=\"text-align: left;\">\n<p style=\"text-align: left;\">In this section you will:<\/p>\n<ul>\n<li style=\"text-align: left;\">Classify a real number as a natural, whole, integer, rational, or irrational number.<\/li>\n<li style=\"text-align: left;\">Perform calculations using order of operations.<\/li>\n<li style=\"text-align: left;\">Use the following properties of numbers: commutative, associative, distributive, inverse, and identity.<\/li>\n<li style=\"text-align: left;\">Evaluate algebraic expressions.<\/li>\n<li style=\"text-align: left;\">Simplify algebraic expressions.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p id=\"fs-id2610870\" style=\"text-align: left;\">It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity\u2014a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.<\/p>\n<p id=\"fs-id2597266\" style=\"text-align: left;\">Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.<\/p>\n<p id=\"fs-id2811451\" style=\"text-align: left;\">But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a \u201cbase state\u201d while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.<\/p>\n<p id=\"fs-id2558199\" style=\"text-align: left;\">Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.<\/p>\n<p id=\"fs-id3044208\" style=\"text-align: left;\">Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.<\/p>\n<section id=\"fs-id2374993\" data-depth=\"1\">\n<h2 style=\"text-align: left;\" data-type=\"title\">Classifying a Real Number<\/h2>\n<p id=\"fs-id2172910\" style=\"text-align: left;\">The numbers we use for counting, or enumerating items, are the <span id=\"term-00001\" data-type=\"term\">natural numbers<\/span>: 1, 2, 3, 4, 5, and so on. We describe them in set notation as [latex]\\{1, 2, 3, \\ldots\\}[\/latex] where the ellipsis (\u2026) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the <em data-effect=\"italics\">counting numbers<\/em>. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of <strong><span id=\"term-00002\" data-type=\"term\">whole numbers<\/span><\/strong> is the set of natural numbers plus zero: [latex]\\{0, 1, 2, 3, \\ldots\\}[\/latex]<\/p>\n<p style=\"text-align: left;\">The set of <strong><span id=\"term-00003\" data-type=\"term\">integers<\/span> <\/strong>adds the opposites of the natural numbers to the set of whole numbers: [latex]\\{\\ldots, -3, -2, -1, 0, 1, 2, 3, \\ldots\\}[\/latex] It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.<\/p>\n<p>[latex]\\overset{\\text{negative integers}}{\\ldots, -3, -2, -1,} \\hspace{3em} \\overset{zero}{0,} \\hspace{3em} \\overset{\\text{positive integers}}{1, 2, 3, \\ldots}[\/latex]<\/p>\n<p id=\"fs-id3036458\" style=\"text-align: left;\">The set of <strong><span id=\"term-00004\" data-type=\"term\">rational numbers<\/span><\/strong> is written as [latex]\\frac{m}{n} \\ \\text{m and n are integers and} \\ n \\not = 0 \\frac{15}{8} = 1.875 \\frac{4}{11} = 0.36363636\\ldots= 0.\\overline{36}.[\/latex] Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.<\/p>\n<p id=\"fs-id1841571\" style=\"text-align: left;\">Because they are fractions, any rational number can also be expressed as a terminating or repeating decimal. Any rational number can be represented as either:<\/p>\n<p style=\"text-align: left;\"><span class=\"token\">a) <\/span>a terminating decimal: [latex]\\frac{15}{8} = 1.875,[\/latex] or<\/p>\n<p style=\"text-align: left;\">b) a repeating decimal: [latex]\\frac{4}{11} = 0.36363636\\ldots= 0.\\overline{36}[\/latex]<\/p>\n<p id=\"fs-id1334141\" style=\"text-align: left;\">We use a line drawn over the repeating block of numbers instead of writing the group multiple times.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\" style=\"text-align: left;\">Example 1: Writing Integers as Rational Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\" style=\"text-align: left;\">\n<p style=\"text-align: left;\">Write each of the following as a rational number.<\/p>\n<p style=\"text-align: left;\">(a) 7<\/p>\n<p style=\"text-align: left;\">(b) 0<\/p>\n<p style=\"text-align: left;\">(c) -8<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p style=\"text-align: left;\">Write a fraction with the integer in the numerator and 1 in the denominator.<\/p>\n<p style=\"text-align: left;\">(a) [latex]7 = \\frac{7}{1}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">(b) [latex]0 = \\frac{0}{1}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">(c) [latex]-8 = -\\frac{8}{1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write each of the following as a rational number.<\/p>\n<p>(a) 11<\/p>\n<p>(b) 3<\/p>\n<p>(c) -4<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Identifying Rational Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write each of the following rational numbers as either a terminating or repeating decimal.<\/p>\n<p>(a) [latex]-\\frac{5}{7}[\/latex]<\/p>\n<p>(b) [latex]\\frac{15}{5}[\/latex]<\/p>\n<p>(c) [latex]\\frac{13}{25}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Write each fraction as a decimal by dividing the numerator by the denominator.<\/p>\n<p>(a) [latex]-\\frac{5}{27} = -0.\\overline{714285},[\/latex] a repeating decimal<\/p>\n<p>&nbsp;<\/p>\n<p>(b) [latex]\\frac{15}{3} = 3\\hspace{0.25em} \\text{(or 3.0)},[\/latex] a terminating decimal<\/p>\n<p>&nbsp;<\/p>\n<p>(c) [latex]\\frac{13}{25} = 0.52,[\/latex] a terminating decimal<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write each of the following rational numbers as either a terminating or repeating decimal.<\/p>\n<p>(a) [latex]\\frac{68}{17}[\/latex]<\/p>\n<p>(b) [latex]\\frac{8}{13}[\/latex]<\/p>\n<p>(c) [latex]-\\frac{17}{20}[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id2040230\" data-depth=\"2\">\n<h3 data-type=\"title\">Irrational Numbers<\/h3>\n<p id=\"fs-id2393783\">At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even [latex]\\frac{3}{2}[\/latex] but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be <em data-effect=\"italics\">irrational<\/em> because they cannot be written as fractions. These numbers make up the set of <strong><span id=\"term-00005\" data-type=\"term\">irrational numbers<\/span><\/strong>. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.<\/p>\n<p style=\"text-align: center;\">[latex]\\{h|h \\ \\text{is not a rational number}\\}[\/latex]<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Differentiating Rational and Irrational Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.<\/p>\n<p>(a) [latex]\\sqrt{25}[\/latex]<\/p>\n<p>(b) [latex]\\frac{33}{9}[\/latex]<\/p>\n<p>(c) [latex]\\sqrt{11}[\/latex]<\/p>\n<p>(d) [latex]\\frac{17}{34}[\/latex]<\/p>\n<p>(e) [latex]0.3033033303333\\ldots[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) [latex]\\sqrt{25}: \\hspace{0.5em}[\/latex] This can be simplified as [latex]\\sqrt{25} = 5.[\/latex] Therefore, [latex]\\sqrt{25}[\/latex] is rational.<\/p>\n<p>&nbsp;<\/p>\n<p>(b) [latex]\\frac{33}{9}:[\/latex] Because it is a fraction of integers, [latex]\\frac{33}{9}[\/latex] is a rational number. Next, simplify and divide.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{33}{9} = \\frac{\\overset{11}{\\cancel{33}}}{\\underset{3}{\\cancel{9}}} = \\frac{11}{3} = 3.\\overline{6}[\/latex]<\/p>\n<p>So, [latex]\\frac{33}{9}[\/latex] is rational and a repeating decimal.<\/p>\n<p>&nbsp;<\/p>\n<p>(c) [latex]\\sqrt{11}: \\sqrt{11}[\/latex] is irrational because 11 is not a perfect square and [latex]\\sqrt{11}[\/latex] cannot be expressed as a fraction.<\/p>\n<p>&nbsp;<\/p>\n<p>(d) [latex]\\frac{17}{34}:[\/latex] Because it is a fraction of integers, [latex]\\frac{17}{34}[\/latex] is a rational number. Simplify and divide.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{17}{34} = \\frac{\\overset{1}{\\cancel{17}}}{\\underset{2}{\\cancel{34}}} = \\frac{1}{2} = 0.5[\/latex]<\/p>\n<p>So,[latex]\\frac{17}{34}[\/latex] is rational and a terminating decimal.<\/p>\n<p>&nbsp;<\/p>\n<p>(e) [latex]0.3033033303333\\ldots[\/latex] is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1213813\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div class=\"os-solution-container\">\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>Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.<\/p>\n<p>(a) [latex]\\frac{7}{77}[\/latex]<\/p>\n<p>(b) [latex]\\sqrt{81}[\/latex]<\/p>\n<p>(c) [latex]4.27027002700027\\ldots[\/latex]<\/p>\n<p>(d) [latex]\\frac{91}{13}[\/latex]<\/p>\n<p>(e) [latex]\\sqrt{39}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id1598961\" data-depth=\"2\">\n<h3 data-type=\"title\">Real Numbers<\/h3>\n<p id=\"fs-id1493774\">Given any number <em data-effect=\"italics\">n<\/em>, we know that <em data-effect=\"italics\">n<\/em> is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of<strong> <span id=\"term-00006\" data-type=\"term\">real numbers<\/span><\/strong>. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Negative and positive real numbers include fractions, decimals, and irrational numbers according to their algebraic sign (+ or \u2013). Zero is considered neither positive nor negative.<\/p>\n<p id=\"fs-id2057634\">The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the <span id=\"term-00007\" data-type=\"term\">real number line<\/span> as shown in Figure 1<strong>.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1096\" aria-describedby=\"caption-attachment-1096\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1096\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-1-300x34.webp\" alt=\"\" width=\"300\" height=\"34\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-1-300x34.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-1-65x7.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-1-225x25.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-1-350x39.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-1.webp 428w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1096\" class=\"wp-caption-text\">Figure 1. The real number line<\/figcaption><\/figure>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Classifying Real Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?<\/p>\n<p>(a) [latex]-\\frac{10}{3}[\/latex]<\/p>\n<p>(b) [latex]\\sqrt{5}[\/latex]<\/p>\n<p>(c) [latex]-\\sqrt{289}[\/latex]<\/p>\n<p>(d) [latex]-6\\pi[\/latex]<\/p>\n<p>(e) [latex]0.615384615384\\ldots[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) [latex]-\\frac{10}{3}[\/latex] is negative and rational. It lies to the left of 0 on the number line.<\/p>\n<p>&nbsp;<\/p>\n<p>(b) [latex]\\sqrt{5}[\/latex] is positive and irrational. It lies to the right of 0.<\/p>\n<p>&nbsp;<\/p>\n<p>(c) [latex]-\\sqrt{289} = -\\sqrt{17^2} = -17[\/latex] is negative and rational. It lies to the left of 0.<\/p>\n<p>&nbsp;<\/p>\n<p>(d) [latex]-6\\pi[\/latex] is negative and irrational. It lies to the left of 0.<\/p>\n<p>&nbsp;<\/p>\n<p>(e) [latex]0.615384615384\\ldots[\/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"Example_01_01_04\" class=\"ui-has-child-title\" data-type=\"example\">\n<section>\n<div class=\"body\">\n<div id=\"fs-id1391400\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div class=\"os-solution-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?<\/p>\n<p>(a) [latex]\\sqrt{73}[\/latex]<\/p>\n<p>(b) [latex]-11.411411411\\ldots[\/latex]<\/p>\n<p>(c) [latex]\\frac{47}{19}[\/latex]<\/p>\n<p>(d) [latex]-\\frac{\\sqrt{5}}{2}[\/latex]<\/p>\n<p>(e) [latex]6.210735[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1687126\" data-depth=\"2\">\n<h3 data-type=\"title\">Sets of Numbers as Subsets<\/h3>\n<p id=\"fs-id2447911\">Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1097\" aria-describedby=\"caption-attachment-1097\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1097\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-2-300x144.webp\" alt=\"\" width=\"300\" height=\"144\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-2-300x144.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-2-65x31.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-2-225x108.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-2-350x169.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-2.webp 731w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1097\" class=\"wp-caption-text\">Figure 2. Sets of numbers<br \/>N: the set of natural numbers<br \/>W: the set of whole numbers<br \/>I: the set of integers<br \/>Q: the set of rational numbers<br \/>Q\u00b4: the set of irrational numbers<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Sets of Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The set of <strong><span id=\"term-00008\" data-type=\"term\">natural numbers<\/span><\/strong> includes the numbers used for counting: [latex]\\left\\{1, 2, 3, \\ldots \\right\\}[\/latex]<\/p>\n<p>The set of <strong><span id=\"term-00009\" data-type=\"term\">whole numbers<\/span><\/strong> is the set of natural numbers plus zero: [latex]\\left\\{0, 1, 2, 3, \\ldots \\right\\}[\/latex]<\/p>\n<p>The set of <strong><span id=\"term-00010\" data-type=\"term\">integers<\/span> <\/strong>adds the negative natural numbers to the set of whole numbers: [latex]\\left\\{\\ldots, -3, -2, -1, 0, 1, 2, 3, \\ldots \\right\\}[\/latex]<\/p>\n<p>The set of <strong><span id=\"term-00011\" data-type=\"term\">rational numbers<\/span><\/strong> includes fractions written as [latex]\\{\\frac{m}{n} | \\text{m and n are integers and} \\ n \\not = 0\\}[\/latex]<\/p>\n<p>The set of <strong><span id=\"term-00012\" data-type=\"term\">irrational numbers<\/span><\/strong> is the set of numbers that are not rational, are nonrepeating, and are nonterminating: [latex]\\{h|h \\ \\text{is not a rational number}\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Differentiating the Sets of Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Classify each number as being a natural number (<em data-effect=\"italics\">N<\/em>), whole number (<em data-effect=\"italics\">W<\/em>), integer (<em data-effect=\"italics\">I<\/em>), rational number (<em data-effect=\"italics\">Q<\/em>), and\/or irrational number (<em data-effect=\"italics\">Q\u2032<\/em>).<\/p>\n<p>(a) [latex]\\sqrt{36}[\/latex]<\/p>\n<p>(b) [latex]\\frac{8}{3}[\/latex]<\/p>\n<p>(c) [latex]\\sqrt{73}[\/latex]<\/p>\n<p>(d) [latex]-6[\/latex]<\/p>\n<p>(e) [latex]3.2121121112\\ldots[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">N<\/em><\/strong><\/td>\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">W<\/em><\/strong><\/td>\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">I<\/em><\/strong><\/td>\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">Q<\/em><\/strong><\/td>\n<td style=\"width: 16.6667%;\"><strong><em data-effect=\"italics\">Q\u2032<\/em><\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">(a) [latex]\\sqrt{36} = 6[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">X<\/td>\n<td style=\"width: 16.6667%;\">X<\/td>\n<td style=\"width: 16.6667%;\">X<\/td>\n<td style=\"width: 16.6667%;\">X<\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">(b) [latex]\\frac{8}{3} = 2.\\overline{6}[\/latex]<\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\">X<\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">(c) [latex]\\sqrt{73}[\/latex]<\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\">X<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">(d) [latex]-6[\/latex]<\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\">X<\/td>\n<td style=\"width: 16.6667%;\">X<\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">(e) [latex]3.2121121112\\ldots[\/latex]<\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\"><\/td>\n<td style=\"width: 16.6667%;\">X<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Classify each number as being a natural number (<em data-effect=\"italics\">N<\/em>), whole number (<em data-effect=\"italics\">W<\/em>), integer (<em data-effect=\"italics\">I<\/em>), rational number (<em data-effect=\"italics\">Q<\/em>), and\/or irrational number (<em data-effect=\"italics\">Q\u2032<\/em>).<\/p>\n<p>(a) [latex]-\\frac{35}{7}[\/latex]<\/p>\n<p>(b) [latex]0[\/latex]<\/p>\n<p>(c) [latex]\\sqrt{169}[\/latex]<\/p>\n<p>(d) [latex]\\sqrt{24}[\/latex]<\/p>\n<p>(e) [latex]4.763763763\\ldots[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id3574670\" data-depth=\"1\">\n<h2 data-type=\"title\">Performing Calculations Using the Order of Operations<\/h2>\n<p id=\"fs-id1580945\">When we multiply a number by itself, we square it or raise it to a power of 2. For example, [latex]4^2 = 4\\cdot 4 = 16.[\/latex] We can raise any number to any power. In general, the <strong><span id=\"term-00013\" data-type=\"term\">exponential notation<\/span><\/strong> [latex]a^n[\/latex] means that the number or variable [latex]a[\/latex] is used as a factor [latex]n[\/latex] times.<\/p>\n<p style=\"text-align: center;\">[latex]a^n = a\\cdot\u00a0 a\\cdot \\overset{\\text{n factors}}a\\cdot \\ldots \\cdot a[\/latex]<\/p>\n<p id=\"fs-id1469836\">In this notation, [latex]a^n[\/latex] is read as the <em data-effect=\"italics\">n<\/em>th power of [latex]a[\/latex] or [latex]a[\/latex] to the [latex]n[\/latex] where [latex]a[\/latex] is called the <strong><span id=\"term-00014\" data-type=\"term\">base<\/span> <\/strong>and [latex]n[\/latex] is called the <strong><span id=\"term-00015\" data-type=\"term\">exponent<\/span>.<\/strong> A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, [latex]24 + 6\\cdot \\frac{2}3 - 4^2[\/latex] is a mathematical expression.<\/p>\n<p id=\"fs-id2448785\">To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the <strong><span id=\"term-00016\" data-type=\"term\">order of operations<\/span><\/strong>. This is a sequence of rules for evaluating such expressions.<\/p>\n<p id=\"fs-id1634868\">Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.<\/p>\n<p id=\"fs-id2424391\">The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.<\/p>\n<p id=\"fs-id2558584\">Let\u2019s take a look at the expression provided.<\/p>\n<p style=\"text-align: center;\">[latex]24 + 6\\cdot \\frac{2}3 - 4^2[\/latex]<\/p>\n<p id=\"fs-id1345509\">There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify [latex]4^2[\/latex] as 16.<\/p>\n<p style=\"text-align: center;\">[latex]24 + 6\\cdot \\frac{2}3 - 4^2[\/latex]<br \/>\n[latex]24 + 6\\cdot \\frac{2}3 - 16[\/latex]<\/p>\n<p id=\"fs-id894495\">Next, perform multiplication or division, left to right.<\/p>\n<p style=\"text-align: center;\">[latex]24 + 6\\cdot \\frac{2}3 - 16[\/latex]<br \/>\n[latex]24 + 4 - 16[\/latex]<\/p>\n<p id=\"fs-id1315208\">Lastly, perform addition or subtraction, left to right.<\/p>\n<p style=\"text-align: center;\">[latex]24 + 4 - 16[\/latex]<br \/>\n[latex]28 - 16[\/latex]<br \/>\n[latex]12[\/latex]<\/p>\n<p id=\"fs-id894663\">Therefore, [latex]24 + 6\\cdot \\frac{2}3 - 4^2 = 12[\/latex]<\/p>\n<p id=\"fs-id3032363\">For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Oder of Operations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1324343\">Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym <strong>PEMDAS<\/strong>:<\/p>\n<p id=\"fs-id1331012\"><strong>P<\/strong>(arentheses)<span data-type=\"newline\"><br \/>\n<\/span><strong>E<\/strong>(xponents)<span data-type=\"newline\"><br \/>\n<\/span><strong>M<\/strong>(ultiplication) and <strong>D<\/strong>(ivision)<span data-type=\"newline\"><br \/>\n<\/span><strong>A<\/strong>(ddition) and <strong>S<\/strong>(ubtraction)<\/p>\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 a mathematical expression, simplify it using the order of operations.<\/strong><\/p>\n<ol>\n<li><span class=\"os-stepwise-content\">Simplify any expressions within grouping symbols.<\/span><\/li>\n<li><span class=\"os-stepwise-content\">Simplify any expressions containing exponents or radicals.<\/span><\/li>\n<li><span class=\"os-stepwise-content\">Perform any multiplication and division in order, from left to right.<\/span><\/li>\n<li><span class=\"os-stepwise-content\">Perform any addition and subtraction in order, from left to right.<\/span><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Using the Order of Operations<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the order of operations to evaluate each of the following expressions.<\/p>\n<p>(a) [latex](3 \\cdot 2)^2 - 4(6 + 2)[\/latex]<\/p>\n<p>(b) [latex]\\frac{5^2-4}{7} - \\sqrt{11-2}[\/latex]<\/p>\n<p>(c) [latex]6 - |5 - 8| + 3(4-1)[\/latex]<\/p>\n<p>(d) [latex]\\frac{14 - 3 \\cdot 2}{2 \\cdot 5 - 3^2}[\/latex]<\/p>\n<p>(e) [latex]7(5 \\cdot 3) - 2[(6 - 3) - 4^2] + 1[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a)<\/p>\n<p>[latex]\\begin{array}{rcll}(3 \\cdot 2)^2 - 4(6 + 2) &=& (6)^2 - 4(8) & \\quad \\text{Simplify parentheses.} \\\\&=& 36 - 4(8) & \\quad \\text{Simplify exponent.} \\\\&=& 36 - 32 & \\quad \\text{Simplify multiplication.} \\\\&=& 4 & \\quad \\text{Simplify subtraction.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(b)<\/p>\n<p>[latex]\\begin{array}{rcll}\\frac{5^2-4}{7} - \\sqrt{11-2} &=& \\frac{5^2-4}{7} - \\sqrt{9} & \\quad \\text{Simplify grouping symbols (radical).} \\\\&=& \\frac{5^2-4}{7} - 3 & \\quad \\text{Simplify radical.} \\\\&=& \\frac{25-4}{7} - 3 & \\quad \\text{Simplify exponent.} \\\\&=& \\frac{21}{7} - 3 & \\quad \\text{Simplify subtraction in numerator.} \\\\&=& 3 - 3 & \\quad \\text{Simplify division.} \\\\&=& 0 & \\quad \\text{Simplify subtraction.}\\end{array}[\/latex]<\/p>\n<p>Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.<\/p>\n<p>&nbsp;<\/p>\n<p>(c)<\/p>\n<p>[latex]\\begin{array}{rcll}6 - |5 - 8| + 3|4 - 1| &=& 6 - |-3| + 3(3) & \\quad \\text{Simplify inside grouping symbols.} \\\\&=& 6 - (3) + 3(3) & \\quad \\text{Simplify absolute value.} \\\\&=& 6 - 3 + 9 & \\quad \\text{Simplify multiplication.} \\\\&=& 12 & \\quad \\text{Simplify addition.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(d)<\/p>\n<p>[latex]\\begin{array}{rcll}\\frac{14 - 3 \\cdot 2}{2 \\cdot 5 - 3^2} &=& \\frac{14 - 3 \\cdot 2}{2 \\cdot 5 - 9} & \\quad \\text{Simplify exponent.} \\\\&=& \\frac{14 - 6}{10 - 9} & \\quad \\text{Simplify products.} \\\\&=& \\frac{8}{1} & \\quad \\text{Simplify differences.} \\\\&=& 8 & \\quad \\text{Simplify quotient.}\\end{array}[\/latex]<\/p>\n<p>In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.<\/p>\n<p>&nbsp;<\/p>\n<p>(e)<\/p>\n<p>[latex]\\begin{array}{rcll}7(5 \\cdot 3) - 2[(6 - 3) - 4^2] + 1 &=& 7(15) - 2[(3) - 4^2] + 1 & \\quad \\text{Simplify inside parentheses.} \\\\&=& 7(15) - 2(3 - 16) + 1 & \\quad \\text{Simplify exponent.} \\\\&=& 7(15) - 2(-13) + 1 & \\quad \\text{Subtract.} \\\\&=& 105 + 26 + 1 & \\quad \\text{Multiply.} \\\\&=& 132 & \\quad \\text{Add.}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the order of operations to evaluate each of the following expressions.<\/p>\n<p>(a) [latex]\\sqrt{5^2-4^2} + 7(5-4)^2[\/latex]<\/p>\n<p>(b) [latex]1+ \\frac{7\\cdot 5-8\\cdot 4}{9-6}[\/latex]<\/p>\n<p>(c) [latex]|1.8 - 4.3| + 0.4\\sqrt{15+10}[\/latex]<\/p>\n<p>(d) [latex]\\frac{1}{2}[5\\cdot 3^2-7^2]+\\frac{1}{3}\\cdot 9^2[\/latex]<\/p>\n<p>(e) [latex][(3-8)^2-4]-(3-8)[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id2783853\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Properties of Real Numbers<\/h2>\n<p id=\"fs-id1979200\">For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.<\/p>\n<section id=\"fs-id3578887\" data-depth=\"2\">\n<section id=\"fs-id3578889\" data-depth=\"3\">\n<h4 data-type=\"title\">Commutative Properties<\/h4>\n<p id=\"fs-id2523326\">The <strong><span id=\"term-00017\" data-type=\"term\">commutative property of addition<\/span><\/strong> states that numbers may be added in any order without affecting the sum.<\/p>\n<p style=\"text-align: center;\">[latex]a+b = b+a[\/latex]<\/p>\n<p id=\"fs-id3573981\">We can better see this relationship when using real numbers.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lclcl}(-2) + 7 = 5 & & \\text{and} & & 7 + (-2) = 5\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1432671\">Similarly, the <strong><span id=\"term-00018\" data-type=\"term\">commutative property of multiplication<\/span><\/strong> states that numbers may be multiplied in any order without affecting the product.<\/p>\n<p style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/p>\n<p id=\"fs-id3260777\">Again, consider an example with real numbers.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}(-11) \\cdot (-4) = 44 & & \\text{and} & & (-4) \\cdot (-11) = 44\\end{array}[\/latex]<\/p>\n<p id=\"fs-id3033661\">It is important to note that neither subtraction nor division is commutative. For example, [latex]17-5[\/latex] is not the same as [latex]5-17.[\/latex] Similarly, [latex]20\\div{5}\\not=5\\div{20}.[\/latex]<\/p>\n<\/section>\n<section id=\"fs-id1618548\" data-depth=\"3\">\n<h4 data-type=\"title\">Associative Properties<\/h4>\n<p id=\"fs-id2643647\">The <strong><span id=\"term-00019\" data-type=\"term\">associative property of multiplication<\/span><\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.<\/p>\n<p style=\"text-align: center;\">[latex]a(bc)=(ab)c[\/latex]<\/p>\n<p id=\"fs-id1314913\">Consider this example.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}(3 \\cdot 4) \\cdot 5 = 60 & & \\text{and} & & 3 \\cdot (4 \\cdot 5) = 60\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1718282\">The <strong><span id=\"term-00020\" data-type=\"term\">associative property of addition<\/span><\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/p>\n<p style=\"text-align: center;\">[latex]a+(b+c)=(a+b)+c[\/latex]<\/p>\n<p id=\"fs-id1734746\">This property can be especially helpful when dealing with negative integers. Consider this example.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}[15 + (-9)] + 23 = 29 & & \\text{and} & & 15 + [(-9) + 23] = 29\\end{array}[\/latex]<\/p>\n<p id=\"fs-id2213874\">Are subtraction and division associative? Review these examples.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rclrcl}8 - (3 - 15) & \\stackrel{?}{=} & (8 - 3) - 15 & 64 \\div (8 \\div 4) & \\stackrel{?}{=} & (64 \\div 8) \\div 4 \\\\8 - (-12) & = & 5 - 15 & 64 \\div 2 & \\stackrel{?}{=} & 8 \\div 4 \\\\20 & \\neq & -10 & 32 & \\neq & 2\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1591297\">As we can see, neither subtraction nor division is associative.<\/p>\n<\/section>\n<section id=\"fs-id1705288\" data-depth=\"3\">\n<h4 data-type=\"title\">Distributive Property<\/h4>\n<p id=\"fs-id2627113\">The <strong><span id=\"term-00021\" data-type=\"term\">distributive property<\/span><\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.<\/p>\n<p style=\"text-align: center;\">[latex]a\\cdot (b+c)=a\\cdot b+a\\cdot c[\/latex]<\/p>\n<p id=\"fs-id3594964\">This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.<\/p>\n<p><span id=\"fs-id1567314\" data-type=\"media\" data-alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1110 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-distributive-300x60.webp\" alt=\"\" width=\"300\" height=\"60\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-distributive-300x60.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-distributive-65x13.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-distributive-225x45.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-distributive-350x70.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-distributive.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/span><\/p>\n<p id=\"fs-id1719796\">Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by \u20137, and adding the products.<\/p>\n<p id=\"fs-id2186994\">To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}6 + (3 \\cdot 5) & \\stackrel{?}{=} & (6 + 3) \\cdot (6 + 5) \\\\6 + (15) & \\stackrel{?}{=} & (9) \\cdot (11) \\\\21 & \\neq & 99\\end{array}[\/latex]<\/p>\n<p id=\"fs-id2750433\">A special case of the distributive property occurs when a sum of terms is subtracted.<\/p>\n<p style=\"text-align: center;\">[latex]a-b=a+(-b)[\/latex]<\/p>\n<p id=\"fs-id2265668\">For example, consider the difference [latex]12-(5+3)[\/latex]. We can rewrite the difference of the two terms 12 and [latex](5+3)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex](5+3)[\/latex] we add the opposite.<\/p>\n<p style=\"text-align: center;\">[latex]12+(-1)\\cdot (5+3)[\/latex]<\/p>\n<p id=\"fs-id3586677\">Now, distribute [latex]-1[\/latex] and simplify the result.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}12 - (5 + 3) &=& 12 + (-1) \\cdot (5 + 3) \\\\&=& 12 + [(\u22121) \\cdot 5 + (\u22121) \\cdot 3] \\\\&=& 12 + (-8) \\\\&=& 4\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1392553\">This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}12 - (5 + 3) &=& 12 + (-5 - 3) \\\\&=& 12 + (-8) \\\\&=& 4\\end{array}[\/latex]<\/p>\n<\/section>\n<section id=\"fs-id2165198\" data-depth=\"3\">\n<h4 data-type=\"title\">Identity Properties<\/h4>\n<p id=\"fs-id3238896\">The <strong><span id=\"term-00022\" data-type=\"term\">identity property of addition<\/span><\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.<\/p>\n<p style=\"text-align: center;\">[latex]a+0=a[\/latex]<\/p>\n<p id=\"fs-id2440675\">The <strong><span id=\"term-00023\" data-type=\"term\">identity property of multiplication<\/span><\/strong> states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.<\/p>\n<p style=\"text-align: center;\">[latex]a\\cdot 1=a[\/latex]<\/p>\n<p><math display=\"block\"><\/math><\/p>\n<p id=\"fs-id1474435\">For example, we have [latex](-6)+0=-6[\/latex] and [latex]23\\cdot 1=23.[\/latex] There are no exceptions for these properties; they work for every real number, including 0 and 1.<\/p>\n<\/section>\n<section id=\"fs-id2591971\" data-depth=\"3\">\n<h4 data-type=\"title\">Inverse Properties<\/h4>\n<p id=\"fs-id2794277\">The <strong><span id=\"term-00024\" data-type=\"term\">inverse property of addition<\/span><\/strong> states that, for every real number <em data-effect=\"italics\">a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted by (\u2212<em data-effect=\"italics\">a<\/em>), that, when added to the original number, results in the additive identity, 0.<\/p>\n<p style=\"text-align: center;\">[latex]a+(-a)=0[\/latex]<\/p>\n<p id=\"fs-id1678541\">For example, if [latex]a=-8[\/latex] the additive inverse is 8, since [latex](-8)+8=0[\/latex]<\/p>\n<p id=\"fs-id2998343\">The <strong><span id=\"term-00025\" data-type=\"term\">inverse property of multiplication<\/span><\/strong> holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number <em data-effect=\"italics\">a<\/em>, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\frac{1}{a}[\/latex] that, when multiplied by the original number, results in the multiplicative identity, 1.<\/p>\n<p style=\"text-align: center;\">[latex]a\\cdot \\frac{1}{a}=1[\/latex]<\/p>\n<p id=\"fs-id1694769\">For example, if [latex]a=-\\frac{2}{3}[\/latex] the reciprocal, denoted [latex]\\frac{1}{a}[\/latex] is [latex]-\\frac{3}{2}[\/latex] because<\/p>\n<p style=\"text-align: center;\">[latex]a \\cdot \\frac{1}{a} = \\left(-\\frac{2}{3}\\right) \\cdot \\left(-\\frac{3}{2}\\right) = 1[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Properties of Real Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The following properties hold for real numbers <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\"><\/td>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Addition<\/strong><\/td>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Multiplication<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><strong>Commutative Property<\/strong><\/td>\n<td style=\"width: 33.3333%;\">[latex]a+b=b+a[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><strong>Associative Property<\/strong><\/td>\n<td style=\"width: 33.3333%;\">[latex]a+(b+c)=(a+b)+c[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]a(bc)=(ab)c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><strong>Distributive Property<\/strong><\/td>\n<td style=\"width: 33.3333%;\">[latex]a\\cdot (b+c)=a\\cdot b+a\\cdot c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><strong>Identity Property<\/strong><\/td>\n<td style=\"width: 33.3333%;\">There exists a unique real number called the additive identity, 0, such that, for any real number <em data-effect=\"italics\">a<\/em><\/p>\n<p>[latex]a+0=a[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em data-effect=\"italics\">a<\/em><\/p>\n<p>[latex]a\\cdot 1=a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><strong>Inverse Property<\/strong><\/td>\n<td style=\"width: 33.3333%;\">Every real number a has an additive inverse, or opposite, denoted <em data-effect=\"italics\">\u2013a<\/em>, such that<\/p>\n<p>[latex]a+(-a)=0[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">Every nonzero real number <em data-effect=\"italics\">a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\frac{1}{a}[\/latex] such that<\/p>\n<p>[latex]a\\cdot (\\frac{1}{a})=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Using Properties of Real Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<p>(a) [latex]3 \\cdot 6 + 3 \\cdot 4[\/latex]<\/p>\n<p>(b) [latex](5+8)+(-8)[\/latex]<\/p>\n<p>(c) [latex]6-(15+9)[\/latex]<\/p>\n<p>(d) [latex]\\frac{4}{7} \\cdot \\left(\\frac{2}{3} \\cdot \\frac{7}{4}\\right)[\/latex]<\/p>\n<p>(e) [latex]100\\cdot [0.75+(-2.38)][\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a)<\/p>\n<p>[latex]\\begin{array}{rcll}3 \\cdot 6 + 3 \\cdot 4 &=& 3 \\cdot (6 + 4) & \\quad \\text{Distributive property.} \\\\&=& 3 \\cdot 10 & \\quad \\text{Simplify.} \\\\&=& 30 & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(b)<\/p>\n<p>[latex]\\begin{array}{rcll}(5 + 8) + (-8) &=& 5 + [8 + (-8)] & \\quad \\text{Associative property of addition.} \\\\&=& 5 + 0 & \\quad \\text{Inverse property of addition.} \\\\&=& 5 & \\quad \\text{Identity property of addition.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(c)<\/p>\n<p>[latex]\\begin{array}{rcll}6 - (15 + 9) &=& 6 + [(-15) + (-9)] & \\quad \\text{Distributive property.} \\\\&=& 6 + (-24) & \\quad \\text{Simplify.} \\\\&=& -18 & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(d)<\/p>\n<p>[latex]\\begin{array}{rcll}\\frac{4}{7} \\cdot \\left(\\frac{2}{3} \\cdot \\frac{7}{4}\\right) &=& \\frac{4}{7} \\cdot \\left(\\frac{7}{4} \\cdot \\frac{2}{3}\\right) & \\quad \\text{Commutative property of multiplication.} \\\\&=& \\left(\\frac{4}{7} \\cdot \\frac{7}{4}\\right) \\cdot \\frac{2}{3} & \\quad \\text{Associative property of multiplication.} \\\\&=& 1 \\cdot \\frac{2}{3} & \\quad \\text{Inverse property of multiplication.} \\\\&=& \\frac{2}{3} & \\quad \\text{Identity property of multiplication.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(e)<\/p>\n<p>[latex]\\begin{array}{rcll}100 \\cdot [0.75 + (-2.38)] &=& 100 \\cdot 0.75 + 100 \\cdot (-2.38) & \\quad \\text{Distributive property.} \\\\&=& 75 + (-238) & \\quad \\text{Simplify.} \\\\&=& -163 & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<p>(a) [latex]\\left(-\\frac{23}{5}\\right)\\cdot [11\\cdot \\left(-\\frac{5}{23}\\right)][\/latex]<\/p>\n<p>(b) [latex]5\\cdot (6.2+0.4)[\/latex]<\/p>\n<p>(c) [latex]18-(7-15)[\/latex]<\/p>\n<p>(d) [latex]\\frac{17}{18}+[\\frac{4}{9}+\\left(-\\frac{17}{18}\\right)][\/latex]<\/p>\n<p>(e) [latex]6\\cdot (-3)+6\\cdot 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id2146838\" data-depth=\"2\">\n<h3 data-type=\"title\">Evaluating Algebraic Expressions<\/h3>\n<p id=\"fs-id2146844\">So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as [latex]x+5, \\frac{4}{3}\\pi r^2,[\/latex] or [latex]\\sqrt{2m^3n^2}.[\/latex] In the expression [latex]x+5,[\/latex] 5 is called a <strong><span id=\"term-00026\" data-type=\"term\">constant<\/span> <\/strong>because it does not vary and <em data-effect=\"italics\">x<\/em> is called a <strong><span id=\"term-00027\" data-type=\"term\">variable<\/span> <\/strong>because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An <strong><span id=\"term-00028\" data-type=\"term\">algebraic expression<\/span><\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.<\/p>\n<p id=\"fs-id2755741\">We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl rcl}(-3)^5 &=& (-3) \\cdot (-3) \\cdot (-3) \\cdot (-3) \\cdot (-3) & x^5 &=& x \\cdot x \\cdot x \\cdot x \\cdot x \\\\(2 \\cdot 7)^3 &=& (2 \\cdot 7) \\cdot (2 \\cdot 7) \\cdot (2 \\cdot 7) & (yz)^3 &=& (yz) \\cdot (yz) \\cdot (yz)\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1542482\">In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.<\/p>\n<p id=\"fs-id2464703\">Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Describing Algebraic Expressions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>List the constants and variables for each algebraic expression.<\/p>\n<p>(a) [latex]x+5[\/latex]<\/p>\n<p>(b) [latex]\\frac{4}{3}\\pi r^3[\/latex]<\/p>\n<p>(c) [latex]\\sqrt{2m^3n^2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\"><\/td>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Constants<\/strong><\/td>\n<td style=\"width: 33.3333%; text-align: center;\"><strong>Variables<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">(a) [latex]x+5[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">(b) [latex]\\frac{4}{3}\\pi r^3[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]\\frac{4}{3}, \\pi[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]r[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">(c) [latex]\\sqrt{2m^3n^2}[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]m, n[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>List the constants and variables for each algebraic expression.<\/p>\n<p>(a) [latex]2\\pi r(r+h)[\/latex]<\/p>\n<p>(b) [latex]2(L+W)[\/latex]<\/p>\n<p>(c) [latex]4y^3+y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Evaluating an Algebraic Expression at Different Values<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate the expression [latex]2x-7[\/latex] for each value for <em data-effect=\"italics\">x.<\/em><\/p>\n<p>(a) [latex]x=0[\/latex]<\/p>\n<p>(b) [latex]x=1[\/latex]<\/p>\n<p>(c) [latex]x=\\frac{1}{2}[\/latex]<\/p>\n<p>(d) [latex]x=-4[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) Substitute 0 for <em data-effect=\"italics\">x.<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}2x - 7 &=& 2(0) - 7 \\\\&=& 0 - 7 \\\\&=& -7\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(b) Substitute 1 for <em data-effect=\"italics\">x.<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}2x - 7 &=& 2(1) - 7 \\\\&=& 2 - 7 \\\\&=& -5\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(c) Substitute\u00a0[latex]\\frac{1}{2}[\/latex] for <em data-effect=\"italics\">x.<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}2x - 7 &=& 2\\left(\\frac{1}{2}\\right) - 7 \\\\&=& 1 - 7 \\\\&=& -6\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(d) Substitute -4 for <em data-effect=\"italics\">x.<\/em><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}2x - 7 &=& 2(-4) - 7 \\\\&=& -8 - 7 \\\\&=& -15\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate the expression [latex]11-3y[\/latex] for each value for <em data-effect=\"italics\">y.<\/em><\/p>\n<p>(a) [latex]y=2[\/latex]<\/p>\n<p>(b) [latex]y=0[\/latex]<\/p>\n<p>(c) [latex]y=\\frac{2}{3}[\/latex]<\/p>\n<p>(d) [latex]y=-5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 10: Evaluating Algebraic Expressions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate each expression for the given values.<\/p>\n<p>(a) [latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/p>\n<p>(b) [latex]\\frac{t}{2t-1}[\/latex] for [latex]t=10[\/latex]<\/p>\n<p>(c) [latex]\\frac{4}{3}\\pi r^3[\/latex] for [latex]r=5[\/latex]<\/p>\n<p>(d) [latex]a+ab+b[\/latex] for [latex]a=11, b=-8[\/latex]<\/p>\n<p>(e) [latex]\\sqrt{2m^3n^2}[\/latex] for [latex]m=2, n=3[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) Substitute [latex]-5[\/latex] for [latex]x.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}x+5 &=& (-5)+5 \\\\ &=& 0\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(b) Substitute [latex]10[\/latex] for [latex]t.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\frac{t}{2t-1} &=& \\frac{(10}{2(10)-1} \\\\ &=& \\frac{10}{20-1} \\\\ &=& \\frac{10}{19}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(c) Substitute [latex]5[\/latex] for [latex]r.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\frac{4}{3}\\pi r^3 &=& \\frac{4}{3}\\pi(5)^3 \\\\ &=& \\frac{4}{3}\\pi(125) \\\\ &=& \\frac{500}{3}\\pi\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(d) Substitute [latex]11[\/latex] for [latex]a[\/latex] and [latex]-8[\/latex] for [latex]b.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}a+ab+b &=& (11)+(11)(-8)+(-8) \\\\ &=& 11-88-8 \\\\ &=& -85\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(e) Substitute [latex]2[\/latex] for [latex]m[\/latex] and [latex]3[\/latex] for [latex]n.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\sqrt{2m^3n^2} &=& \\sqrt{2(2)^3(3)^2} \\\\ &=& \\sqrt{2(8)(9)} \\\\ &=& \\sqrt{144} \\\\ &=& 12\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate each expression for the given values.<\/p>\n<p>(a) [latex]\\frac{y+3}{y-3}[\/latex] for [latex]y=5[\/latex]<\/p>\n<p>(b) [latex]7-2t[\/latex] for [latex]t=-2[\/latex]<\/p>\n<p>(c) [latex]\\frac{1}{3}\\pi r^2[\/latex] for [latex]r=11[\/latex]<\/p>\n<p>(d) [latex](p^2q)^3[\/latex] for [latex]p=-2, q=3[\/latex]<\/p>\n<p>(e) [latex]4(m-n)-5(n-m)[\/latex] for [latex]m=\\frac{2}{3}, n=\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id3660280\" data-depth=\"2\">\n<h3 data-type=\"title\">Formulas<\/h3>\n<p id=\"fs-id2714749\">An <strong><span id=\"term-00029\" data-type=\"term\">equation<\/span> <\/strong>is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the solution of 3 because when we substitute 3 for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2(3)+1=7.[\/latex]<\/p>\n<p id=\"fs-id1631296\">A <strong><span id=\"term-00030\" data-type=\"term\">formula<\/span> <\/strong>is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi r^2[\/latex] For any value of [latex]r,[\/latex] the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi r^2.[\/latex]<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 11: Using a Formula<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r(r+h).[\/latex] See Figure 3. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of [latex]\\pi .[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1112\" aria-describedby=\"caption-attachment-1112\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1112\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-3-300x256.webp\" alt=\"\" width=\"300\" height=\"256\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-3-300x256.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-3-65x55.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-3-225x192.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-3.webp 328w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1112\" class=\"wp-caption-text\">Figure 3. Right circular cylinder<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Evaluate the expression [latex]2\\pi r(r+h)[\/latex] for [latex]r=6[\/latex] and [latex]h=9.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}S &=& 2\\pi r(r+h) \\\\ &=& 2\\pi(6)[(6)+(9)] \\\\ &=& 2\\pi(6)(15) \\\\ &=& 180\\pi\\end{array}[\/latex]<\/p>\n<p>The surface area is [latex]180\\pi[\/latex] square inches.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A photograph with length <em data-effect=\"italics\">L<\/em> and width <em data-effect=\"italics\">W<\/em> is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm<sup>2<\/sup>) is found to be [latex]A=(L+16)(W+16)-L\\cdot W[\/latex] See Figure 4. Find the area of a mat for a photograph with length 32 cm and width 24 cm.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1113\" aria-describedby=\"caption-attachment-1113\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1113\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-4-300x251.webp\" alt=\"\" width=\"300\" height=\"251\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-4-300x251.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-4-65x54.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-4-225x188.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-4-350x292.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/1.1-fig-4.webp 449w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-1113\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<section id=\"fs-id2193236\" data-depth=\"2\">\n<h3 data-type=\"title\">Simplifying Algebraic Expressions<\/h3>\n<p id=\"fs-id3229215\">Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 12: Simplifying Algebraic Expressions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify each algebraic expression.<\/p>\n<p>(a) [latex]3x-2y+x-3y-7[\/latex]<\/p>\n<p>(b) [latex]2r-5(3-r)+4[\/latex]<\/p>\n<p>(c) [latex]\\left(4t-\\frac{5}{4}s\\right)-\\left(\\frac{2}{3}t+2s\\right)[\/latex]<\/p>\n<p>(d) [latex]2mn-5m+3mn+n[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a)<\/p>\n<p>[latex]\\begin{array}{rcll}3x - 2y + x - 3y - 7 &=& 3x + x - 2y - 3y - 7 & \\quad \\text{Commutative property of addition.} \\\\&=& 4x - 5y - 7 & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(b)<\/p>\n<p>[latex]\\begin{array}{rcll}2r-5(3-r)+4 &=& 2r-15+5r+4 & \\quad \\text{Distributive property.} \\\\&=& 2r+5r-15+4 & \\quad \\text{Commutative property of addition.} \\\\&=& 7r-11 & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(c)<\/p>\n<p>[latex]\\begin{array}{rcll}\\left(4t-\\frac{5}{4}s\\right)-\\left(\\frac{2}{3}t+2s\\right) &=& 4t-\\frac{5}{4}s-\\frac{2}{3}t-2s & \\quad \\text{Distributive property.} \\\\&=& 4t-\\frac{2}{3}t-\\frac{5}{4}s-2s & \\quad \\text{Commutative property of addition.} \\\\&=& \\frac{10}{3}t-\\frac{13}{4}s & \\quad \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>(d)<\/p>\n<p>[latex]\\begin{array}{rcll}2mn-5m+3mn+n &=& 2mn+3mn-5m+n & \\quad \\text{Commutative property of addition.} \\\\&=& 5mn-5m+n & \\quad \\text{Simplify.} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Simplify each algebraic expression.<\/p>\n<p>(a) [latex]\\frac{2}{3}y-2\\left(\\frac{4}{3}y+z\\right)[\/latex]<\/p>\n<p>(b) [latex]\\frac{5}{t}-2-\\frac{3}{t}+1[\/latex]<\/p>\n<p>(c) [latex]4p(q-1)+q(1-p)[\/latex]<\/p>\n<p>(d) [latex]9r-(s+2r)+(6-s)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 13: Simplifying a Formula<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A rectangle with length [latex]L[\/latex] and width [latex]W[\/latex] has a perimeter [latex]P[\/latex] given by [latex]P=L+W+L+W.[\/latex] Simplify this expression.<\/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}{rcll}P &=& L + W + L + W & \\\\P &=& L + L + W + W & \\quad \\text{Commutative property of addition} \\\\P &=& 2L + 2W & \\quad \\text{Simplify} \\\\P &=& 2(L + W) & \\quad \\text{Distributive property}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\" style=\"text-align: left;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If the amount [latex]P[\/latex] is deposited into an account paying simple interest [latex]r[\/latex] for time [latex]t,[\/latex] the total value of the deposit [latex]A[\/latex] is given by [latex]A=P+Prt.[\/latex] Simplify the expression. (This formula will be explored in more detail later in the course.)<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\" style=\"text-align: left;\">\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 real numbers.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=RJ7uU9HbdqA\">Simplify an Expression<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=RJ7uU9HbdqA\">Evaluate an Expression 1<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=8b-rf2AW3Ac\">Evaluate an Expression 2<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"os-eos os-section-exercises-container\" style=\"text-align: left;\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">1.1 Section Exercises<\/span><\/h2>\n<section id=\"fs-id3300486\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id2542028\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id2542033\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2542034\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2542033-solution\">1<\/a><span class=\"os-divider\">. <\/span>Is [latex]\\sqrt{2}[\/latex] an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1433666\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1433667\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2510664\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2510665\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2510664-solution\">3<\/a><span class=\"os-divider\">. <\/span>What do the Associative Properties allow us to do when following the order of operations? Explain your answer.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1571756\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id2521060\">For the following exercises, simplify the given expression.<\/p>\n<div id=\"fs-id2521063\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2521064\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span> [latex]10+2\\cdot (5-3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2448113\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2448114\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2448113-solution\">5<\/a><span class=\"os-divider\">. <\/span> [latex]6\\div2-(81\\div3^2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3665280\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3665281\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]18+(6-8)^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2054506\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2054507\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2054506-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]-2\\cdot \\left[16\\div(8-4)^2\\right]^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1607809\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1460495\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]4-6+2\\cdot 7[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2444193\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2444194\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2444193-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]3(5-8)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3235427\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3235428\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]4+6-10\\div2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1601248\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1601249\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1601248-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]12\\div(36\\div9)+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2555791\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2555792\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex](4+5)^2\\div3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1500556\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1500557\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1500556-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]3-12\\cdot 2+19[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1791590\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1791591\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]2+8\\cdot 7\\div4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2396900\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2396901\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2396900-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]5+(6+4)-11[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1450507\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1450508\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]9-18\\div3^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2520962\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2520963\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2520962-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]14 \\cdot3\\div7-6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1368488\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1368489\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]9-(3+11)\\cdot 2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1619048\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1619049\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1619048-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]6+2\\cdot 2-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1620842\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1620843\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]64\\div(8+4\\cdot 2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2995850\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2995851\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2995850-solution\">21<\/a><span class=\"os-divider\">.<\/span> [latex]9+4(2^2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1859164\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1859165\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex](12\\div3\\cdot 3)^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1433630\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1433631\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1433630-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]25\\div5^2-7[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2158696\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2158697\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex](15-7)\\cdot (3-7)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1673686\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1673687\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1673686-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]2\\cdot 4-9(-1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2555535\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2555536\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]4^2-25\\cdot \\frac{1}{5}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2708694\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2708695\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2708694-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]12(3-1)\\div6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1373736\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1373741\">For the following exercises, evaluate the expression using the given value of the variable.<\/p>\n<div id=\"fs-id3349803\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3349804\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]8(x+3)-64[\/latex] for [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id2248088\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2248090\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2248088-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]4y+8-2y[\/latex] for [latex]y=3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1634685\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1634686\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex](11a+3)-18a[\/latex] for [latex]a=-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1610297\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1610298\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1610297-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]4z-2z(1+4)-36[\/latex] for [latex]z=5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2183439\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2183440\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex]4y(7-2)^2+200[\/latex] for [latex]y=-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1442438\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1442439\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1442438-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex]-(2x)^2+1+3[\/latex] for [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2570388\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2570389\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]8(2+4)-15b+b[\/latex] for [latex]b=-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2714429\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2714430\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2714429-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]2(11c-4)-36[\/latex] for [latex]c=0[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1453902\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1453903\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex]4(3-1)x-4[\/latex] for [latex]x=10[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1485040\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1485041\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1485040-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex]\\frac{1}{4}(8w-4^2)[\/latex] for [latex]w=1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1734699\" style=\"text-align: left;\">For the following exercises, simplify the expression.<\/p>\n<div id=\"fs-id1734702\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1734703\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]4x+x(13-7)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2113191\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2113192\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2113191-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]2y-(4)^2y-11[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3185006\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3185007\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex]\\frac{a}{2^3}(64)-12a\\div6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2107133\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2107134\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2107133-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]8b-4b(3)+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2444084\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3260784\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]5l\\div3l \\cdot (9-6)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2194512\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2194513\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2194512-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]7z-3+z \\cdot 6^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2619567\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2619568\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]4\\cdot 3+18x\\div9-12[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1458150\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1458151\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1458150-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]9(y+8)-27[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1443223\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1443224\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex](\\frac{9}{6}t-4)2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1443990\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1443991\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1443990-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]6+12b-3\\cdot6b[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2771916\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2771917\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]18y-2(1+7y)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3283284\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1917673\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id3283284-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex](\\frac{4}{9})^2\\cdot 27x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2980532\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2980533\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]8(3-m)+1(-8)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2193704\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2193705\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2193704-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]9x+4x(2+3)-4(2x+3x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2825015\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2825016\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]5^2-4(3x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id2098225\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<p id=\"fs-id1485257\">For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor\u2019s dog.<\/p>\n<div id=\"fs-id1485264\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2404926\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1485264-solution\">53<\/a><span class=\"os-divider\">. <\/span>Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"eip-id1167336272625\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"eip-id1167336272627\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>How much money does Fred keep?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"eip-52\">For the following exercises, solve the given problem.<\/p>\n<div id=\"fs-id2658142\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1540914\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2658142-solution\">55<\/a><span class=\"os-divider\">. <\/span>According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by [latex]\\pi .[\/latex] Is the circumference of a quarter a whole number, a rational number, or an irrational number?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2483553\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2483554\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id3615404\">For the following exercises, consider this scenario: There is a mound of [latex]g[\/latex] pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.<\/p>\n<div id=\"fs-id1842419\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1842420\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1842419-solution\">57<\/a><span class=\"os-divider\">. <\/span>Write the equation that describes the situation.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1596547\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1596548\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Solve for <em data-effect=\"italics\">g<\/em>.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"eip-373\">For the following exercise, solve the given problem.<\/p>\n<div id=\"fs-id1276001\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1276002\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1276001-solution\">59<\/a><span class=\"os-divider\">. <\/span>Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that [latex]2,500,000-x=0.[\/latex] What property of addition tells us what the value of <em data-effect=\"italics\">x<\/em> must be?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2178453\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id2460118\">For the following exercises, use a graphing calculator to solve for <em data-effect=\"italics\">x<\/em>. Round the answers to the nearest hundredth.<\/p>\n<div id=\"fs-id1703228\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1703229\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex]0.5(12.3)^2-48x=\\frac{3}{5}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1315170\" class=\"material-set-2 os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1315171\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1315170-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex](0.25-0.75)^2x-7.2=9.9[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1720808\" data-depth=\"2\">\n<h3 style=\"text-align: left;\" data-type=\"title\">Extensions<\/h3>\n<div id=\"fs-id1523458\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1523459\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>If a whole number is not a natural number, what must the number be?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1454051\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1454052\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1454051-solution\">63<\/a><span class=\"os-divider\">. <\/span>Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2806511\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2806512\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2538076\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2538077\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id2538076-solution\">65<\/a><span class=\"os-divider\">. <\/span>Determine whether the simplified expression is rational or irrational: [latex]\\sqrt{-18-4(5)(-1)}.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1440424\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1440425\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. Determine whether the simplified expression is rational or irrational: <\/span>[latex]\\sqrt{-16+4(5)+5}.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1979146\" class=\"os-hasSolution\" style=\"text-align: left;\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2608456\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-1\" data-page-slug=\"chapter-1\" data-page-uuid=\"55bb85dc-ccb1-5df0-bb07-a4d1ea6890f8\" data-page-fragment=\"fs-id1979146-solution\">67<\/a><span class=\"os-divider\">. <\/span>The division of two natural numbers will always result in what type of number?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2053806\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2053807\" data-type=\"problem\">\n<p style=\"text-align: left;\"><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>What property of real numbers would simplify the following expression: [latex]4+7(x-1)?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n","protected":false},"author":158,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-81","chapter","type-chapter","status-publish","hentry"],"part":27,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/81","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":127,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/81\/revisions"}],"predecessor-version":[{"id":1799,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/81\/revisions\/1799"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/27"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/81\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=81"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=81"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=81"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=81"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}