{"id":1739,"date":"2023-10-12T00:32:03","date_gmt":"2023-10-12T00:32:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-to-real-numbers\/"},"modified":"2025-11-19T00:34:07","modified_gmt":"2025-11-19T00:34:07","slug":"introduction-to-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-to-real-numbers\/","title":{"raw":"Real Numbers","rendered":"Real Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li class=\"li2\"><span class=\"s1\">Classify a real number.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Perform calculations using order of operations.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Use the properties of real numbers.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Evaluate and simplify algebraic expressions.<\/span><\/li>\r\n \t<li>Express an interval.<\/li>\r\n<\/ul>\r\n<\/div>\r\nBecause of the evolution of the number system, we can now perform complex calculations using several\u00a0categories of real numbers. In this section we will explore sets of numbers, perform calculations with different kinds of numbers, and begin to learn about the use of numbers in algebraic expressions.\r\n<h2>Classify a Real Number<\/h2>\r\nThe numbers we use for counting, or enumerating items, are the <strong>natural numbers<\/strong>: 1, 2, 3, 4, 5, and so on. We describe them in set notation as {1, 2, 3, ...} where the ellipsis (\u2026) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the <em>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>whole numbers<\/strong> is the set of natural numbers plus zero: {0, 1, 2, 3,...}.\r\n\r\nThe set of <strong>integers<\/strong> adds the opposites of the natural numbers to the set of whole numbers: {...,-3, -2, -1, 0, 1, 2, 3,...}. 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.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{\\text{negative integers}} &amp;&amp; {\\text{zero}} &amp;&amp; {\\text{positive integers}}\\\\&amp;{\\dots ,-3,-2,-1,} &amp;&amp; {0,} &amp;&amp; {1,2,3,\\dots } \\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\nThe set of <strong>rational numbers<\/strong> is written as [latex]\\left\\{\\frac{m}{n}|m\\text{ and }{n}\\text{ are integers and }{n}\\ne{ 0 }\\right\\}[\/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.\r\n\r\nBecause they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:\r\n<ol>\r\n \t<li>a terminating decimal: [latex]\\frac{15}{8}=1.875[\/latex], or<\/li>\r\n \t<li>a repeating decimal: [latex]\\frac{4}{11}=0.36363636\\dots =0.\\overline{36}[\/latex]<\/li>\r\n<\/ol>\r\nWe use a line drawn over the repeating block of numbers instead of writing the group multiple times.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing Integers as Rational Numbers<\/h3>\r\nWrite each of the following as a rational number.\r\n<ol>\r\n \t<li>7<\/li>\r\n \t<li>0<\/li>\r\n \t<li>\u20138<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"534535\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"534535\"]\r\n\r\nWrite a fraction with the integer in the numerator and 1 in the denominator.\r\n<ol>\r\n \t<li>[latex]7=\\dfrac{7}{1}[\/latex]<\/li>\r\n \t<li>[latex]0=\\dfrac{0}{1}[\/latex]<\/li>\r\n \t<li>[latex]-8=-\\dfrac{8}{1}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite each of the following as a rational number.\r\n<ol>\r\n \t<li>11<\/li>\r\n \t<li>3<\/li>\r\n \t<li>\u20134<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"755048\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"755048\"]\r\n<ol>\r\n \t<li>[latex]\\dfrac{11}{1}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{1}[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{4}{1}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying Rational Numbers<\/h3>\r\nWrite each of the following rational numbers as either a terminating or repeating decimal.\r\n<ol>\r\n \t<li>[latex]-\\dfrac{5}{7}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{15}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{13}{25}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"549544\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"549544\"]\r\n\r\nWrite each fraction as a decimal by dividing the numerator by the denominator.\r\n<ol>\r\n \t<li>[latex]-\\dfrac{5}{7}=-0.\\overline{714285}[\/latex], a repeating decimal<\/li>\r\n \t<li>[latex]\\dfrac{15}{5}=3[\/latex] (or 3.0), a terminating decimal<\/li>\r\n \t<li>[latex]\\dfrac{13}{25}=0.52[\/latex], a terminating decimal<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<h2>Irrational Numbers<\/h2>\r\nAt 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>irrational<\/em> because they cannot be written as fractions. These numbers make up the set of <strong>irrational numbers<\/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.\r\n<div style=\"text-align: center;\">{h | h is not a rational number}<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Differentiating Rational and Irrational Numbers<\/h3>\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<ol>\r\n \t<li>[latex]\\sqrt{25}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{33}{9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{11}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{17}{34}[\/latex]<\/li>\r\n \t<li>[latex]0.3033033303333\\dots[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"502265\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"502265\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt{25}:[\/latex] This can be simplified as [latex]\\sqrt{25}=5[\/latex]. Therefore, [latex]\\sqrt{25}[\/latex] is rational.<\/li>\r\n \t<li>[latex]\\dfrac{33}{9}:[\/latex] Because it is a fraction, [latex]\\dfrac{33}{9}[\/latex] is a rational number. Next, simplify and divide.\r\n<div style=\"text-align: center;\">[latex]\\dfrac{33}{9}=\\dfrac{{11}\\cdot{3}}{{3}\\cdot{3}}=\\dfrac{11}{3}=3.\\overline{6}[\/latex]<\/div>\r\nSo, [latex]\\dfrac{33}{9}[\/latex] is rational and a repeating decimal.<\/li>\r\n \t<li>[latex]\\sqrt{11}:[\/latex] This cannot be simplified any further. Therefore, [latex]\\sqrt{11}[\/latex] is an irrational number.<\/li>\r\n \t<li>[latex]\\dfrac{17}{34}:[\/latex] Because it is a fraction, [latex]\\dfrac{17}{34}[\/latex] is a rational number. Simplify and divide.\r\n<div style=\"text-align: center;\">[latex]\\dfrac{17}{34}=\\dfrac{1\\cdot{17}}{2\\cdot{17}}=\\dfrac{1}{2}=0.5[\/latex]<\/div>\r\nSo, [latex]\\dfrac{17}{34}[\/latex] is rational and a terminating decimal.<\/li>\r\n \t<li>0.3033033303333... 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.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92383&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div>\r\n<h3>Real Numbers<\/h3>\r\nGiven any number <em>n<\/em>, we know that <em>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>real numbers<\/strong>. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or \u2013). Zero is considered neither positive nor negative.\r\n\r\nThe 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 <strong>real number line<\/strong>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223810\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" \/> The real number line[\/caption]\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Classifying Real Numbers<\/h3>\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<ol>\r\n \t<li>[latex]-\\dfrac{10}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{5}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt{289}[\/latex]<\/li>\r\n \t<li>[latex]-6\\pi[\/latex]<\/li>\r\n \t<li>[latex]0.616161\\dots[\/latex]<\/li>\r\n \t<li>[latex] 0.13 [\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"705558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"705558\"]\r\n<ol>\r\n \t<li>[latex]-\\dfrac{10}{3}[\/latex] is negative and rational. It lies to the left of 0 on the number line.<\/li>\r\n \t<li>[latex]\\sqrt{5}[\/latex] is positive and irrational. It lies to the right of 0.<\/li>\r\n \t<li>[latex]-\\sqrt{289}=-\\sqrt{{17}^{2}}=-17[\/latex] is negative and rational. It lies to the left of 0.<\/li>\r\n \t<li>[latex]-6\\pi [\/latex] is negative and irrational. It lies to the left of 0.<\/li>\r\n \t<li>[latex]0.616161\\dots [\/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.<\/li>\r\n \t<li>[latex] 0.13 [\/latex] is a finite decimal and may be written as 13\/100. \u00a0So it is rational and positive.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\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<ol>\r\n \t<li>[latex]\\sqrt{73}[\/latex]<\/li>\r\n \t<li>[latex]-11.411411411\\dots [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{47}{19}[\/latex]<\/li>\r\n \t<li>[latex]-\\dfrac{\\sqrt{5}}{2}[\/latex]<\/li>\r\n \t<li>[latex]6.210735[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"155954\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"155954\"]\r\n<ol>\r\n \t<li>positive, irrational; right<\/li>\r\n \t<li>negative, rational; left<\/li>\r\n \t<li>positive, rational; right<\/li>\r\n \t<li>negative, irrational; left<\/li>\r\n \t<li>positive, rational; right<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Sets of Numbers as Subsets<\/h3>\r\nBeginning 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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223813\/CNX_CAT_Figure_01_01_001.jpg\" alt=\"A large box labeled: Real Numbers encloses five circles. Four of these circles enclose each other and the other is separate from the rest. The innermost circle contains: 1, 2, 3\u2026 N. The circle enclosing that circle contains: 0 W. The circle enclosing that circle contains: \u2026, -3, -2, -1 I. The outermost circle contains: m\/n, n not equal to zero Q. The separate circle contains: pi, square root of two, etc Q\u00b4.\" width=\"731\" height=\"352\" \/> Sets of numbers. \u00a0 <em>N<\/em>: the set of natural numbers \u00a0 <em>W<\/em>: the set of whole numbers \u00a0 <em>I<\/em>: the set of integers \u00a0 <em>Q<\/em>: the set of rational numbers \u00a0 <em>Q\u00b4<\/em>: the set of irrational numbers[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sets of Numbers<\/h3>\r\nThe set of <strong>natural numbers<\/strong> includes the numbers used for counting: [latex]\\{1,2,3,\\dots\\}[\/latex].\r\n\r\nThe set of <strong>whole numbers<\/strong> is the set of natural numbers plus zero: [latex]\\{0,1,2,3,\\dots\\}[\/latex].\r\n\r\nThe set of <strong>integers<\/strong> adds the negative natural numbers to the set of whole numbers: [latex]\\{\\dots,-3,-2,-1,0,1,2,3,\\dots\\}[\/latex].\r\n\r\nThe set of <strong>rational numbers<\/strong> includes fractions written as [latex]\\{\\frac{m}{n}|m\\text{ and }n\\text{ are integers and }n\\ne 0\\}[\/latex].\r\n\r\nThe set of <strong>irrational numbers<\/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 class=\"textbox exercises\">\r\n<h3>Example: Differentiating the Sets of Numbers<\/h3>\r\nClassify each number as being a natural number, whole number, integer, rational number, and\/or irrational number.\r\n<ul>\r\n \t<li>[latex]\\sqrt{36}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{8}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{73}[\/latex]<\/li>\r\n \t<li>[latex]-6[\/latex]<\/li>\r\n \t<li>[latex]3.2121121112\\dots [\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"779749\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"779749\"]\r\n<table class=\"lines\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\"><\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">\u00a0natural number<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">\u00a0whole number<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">integer<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">rational number<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">irrational number<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14.7017px;\">\r\n<td style=\"height: 14.7017px; width: 266.364px;\">\u00a0[latex]\\sqrt{36}=6[\/latex]<\/td>\r\n<td style=\"height: 14.7017px; width: 104.545px;\">\u00a0yes<\/td>\r\n<td style=\"height: 14.7017px; width: 95.4545px;\">\u00a0yes<\/td>\r\n<td style=\"height: 14.7017px; width: 45.4545px;\">yes<\/td>\r\n<td style=\"height: 14.7017px; width: 104.545px;\">yes<\/td>\r\n<td style=\"height: 14.7017px; width: 114.545px;\">no<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]\\dfrac{8}{3}=2.\\overline{6}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">yes<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">no<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]\\sqrt{73}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">yes<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex] \u20136 [\/latex]<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">yes<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">\u00a0yes<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">\u00a0no<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]3.2121121112\\dots[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">yes<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nClassify each number as being a natural number (<em>N<\/em>), whole number (<em>W<\/em>), integer (<em>I<\/em>), rational number (<em>Q<\/em>), and\/or irrational number (<em>Q'<\/em>).\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-\\dfrac{35}{7}[\/latex]<\/li>\r\n \t<li>[latex]0[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{169}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{24}[\/latex]<\/li>\r\n \t<li>[latex]4.763763763\\dots [\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"266197\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"266197\"]\r\n<table class=\"lines\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\"><\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">\u00a0natural number<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">\u00a0whole number<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">integer<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">rational number<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">irrational number<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14.7017px;\">\r\n<td style=\"height: 14.7017px; width: 266.364px;\">[latex]-\\dfrac{35}{7}[\/latex]<\/td>\r\n<td style=\"height: 14.7017px; width: 104.545px;\">\u00a0yes<\/td>\r\n<td style=\"height: 14.7017px; width: 95.4545px;\">\u00a0yes<\/td>\r\n<td style=\"height: 14.7017px; width: 45.4545px;\">yes<\/td>\r\n<td style=\"height: 14.7017px; width: 104.545px;\">yes<\/td>\r\n<td style=\"height: 14.7017px; width: 114.545px;\">no<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex] 0 [\/latex]<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">yes<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">yes<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">yes<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">no<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]\\sqrt{169}[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">\u00a0yes<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">yes<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">yes<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">yes<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">no<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex] \\sqrt{24} [\/latex]<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">yes<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px; width: 266.364px;\">[latex]4.763763763\\dots[\/latex]<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\r\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\r\n<td style=\"height: 14px; width: 104.545px;\">yes<\/td>\r\n<td style=\"height: 14px; width: 114.545px;\">no<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=13740&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"100%\"><\/iframe>\r\n\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=13741&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"100%\"><\/iframe>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch this video for an overview of the sets of numbers, and how to identify which set a number belongs to.\r\nhttps:\/\/youtu.be\/htP2goe31MM\r\n<h2>Properties of Real Numbers<\/h2>\r\nWhen 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>exponential notation<\/strong> [latex]{a}^{n}[\/latex] means that the number or variable [latex]a[\/latex] is used as a factor [latex]n[\/latex] times.\r\n<div style=\"text-align: center;\">[latex]a^{n}=a\\cdot a\\cdot a\\cdot \\dots \\cdot a[\/latex]<\/div>\r\nIn this notation, [latex]{a}^{n}[\/latex] is read as the <em>n<\/em>th power of [latex]a[\/latex], where [latex]a[\/latex] is called the <strong>base<\/strong> and [latex]n[\/latex] is called the <strong>exponent<\/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.\r\n\r\nTo evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the <strong>order of operations<\/strong>. This is a sequence of rules for evaluating such expressions.\r\n\r\nRecall 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.\r\n\r\nThe 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.\r\n\r\nLet\u2019s take a look at the expression provided.\r\n<div style=\"text-align: center;\">[latex]24+6\\cdot \\dfrac{2}{3}-{4}^{2}[\/latex]<\/div>\r\nThere 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.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+6\\cdot\\frac{2}{3}-4^{2} \\\\ 24+6\\cdot\\frac{2}{3}-16\\end{gathered}[\/latex]<\/div>\r\nNext, perform multiplication or division, left to right.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+6\\cdot\\frac{2}{3}-16 \\\\ 24+4-16\\end{gathered}[\/latex]<\/div>\r\nLastly, perform addition or subtraction, left to right.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+4-16 \\\\ 28-16 \\\\ 12\\end{gathered}[\/latex]<\/div>\r\nTherefore, [latex]24+6\\cdot \\dfrac{2}{3}-{4}^{2}=12[\/latex].\r\n\r\nFor 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.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Order of Operations<\/h3>\r\nOperations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym <strong>PEMDAS<\/strong>:\r\n\r\n<strong>P<\/strong>(arentheses)\r\n\r\n<strong>E<\/strong>(xponents)\r\n\r\n<strong>M<\/strong>(ultiplication) and <strong>D<\/strong>(ivision)\r\n\r\n<strong>A<\/strong>(ddition) and <strong>S<\/strong>(ubtraction)\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a mathematical expression, simplify it using the order of operations.<\/h3>\r\n<ol>\r\n \t<li>Simplify any expressions within grouping symbols.<\/li>\r\n \t<li>Simplify any expressions containing exponents or radicals.<\/li>\r\n \t<li>Perform any multiplication and division in order, from left to right.<\/li>\r\n \t<li>Perform any addition and subtraction in order, from left to right.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Order of Operations<\/h3>\r\nUse the order of operations to evaluate each of the following expressions.\r\n<ol>\r\n \t<li>[latex]{\\left(3\\cdot 2\\right)}^{2}-4\\left(6+2\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{5}^{2}-4}{7}-\\sqrt{11 - 2}[\/latex]<\/li>\r\n \t<li>[latex]6-|5 - 8|+3\\left(4 - 1\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{14 - 3\\cdot 2}{2\\cdot 5-{3}^{2}}[\/latex]<\/li>\r\n \t<li>[latex]7\\left(5\\cdot 3\\right)-2\\left[\\left(6 - 3\\right)-{4}^{2}\\right]+1[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"371324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371324\"]\r\n\r\n1.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(3\\cdot 2\\right)^{2} &amp; =\\left(6\\right)^{2}-4\\left(8\\right) &amp;&amp; \\text{Simplify parentheses} \\\\ &amp; =36-4\\left(8\\right) &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =36-32 &amp;&amp; \\text{Simplify multiplication} \\\\ &amp; =4 &amp;&amp; \\text{Simplify subtraction}\\end{align}[\/latex]<\/p>\r\n2.\r\n<p class=\"p1\" style=\"text-align: center;\"><span class=\"s1\">[latex]\\begin{align}\\frac{5^{2}-4}{7}-\\sqrt{11-2} &amp; =\\frac{5^{2}-4}{7}-\\sqrt{9} &amp;&amp; \\text{Simplify grouping systems (radical)} \\\\ &amp; =\\frac{5^{2}-4}{7}-3 &amp;&amp; \\text{Simplify radical} \\\\ &amp; =\\frac{25-4}{7}-3 &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =\\frac{21}{7}-3 &amp;&amp; \\text{Simplify subtraction in numerator} \\\\ &amp; =3-3 &amp;&amp; \\text{Simplify division} \\\\ &amp; =0 &amp;&amp; \\text{Simplify subtraction}\\end{align}[\/latex]<\/span><\/p>\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\n3.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}6-|5-8|+3\\left(4-1\\right) &amp; =6-|-3|+3\\left(3\\right) &amp;&amp; \\text{Simplify inside grouping system} \\\\ &amp; =6-3+3\\left(3\\right) &amp;&amp; \\text{Simplify absolute value} \\\\ &amp; =6-3+9 &amp;&amp; \\text{Simplify multiplication} \\\\ &amp; =3+9 &amp;&amp; \\text{Simplify subtraction} \\\\ &amp; =12 &amp;&amp; \\text{Simplify addition}\\end{align}[\/latex]<\/p>\r\n4.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{14-3\\cdot2}{2\\cdot5-3^{2}} &amp; =\\frac{14-3\\cdot2}{2\\cdot5-9} &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =\\frac{14-6}{10-9} &amp;&amp; \\text{Simplify products} \\\\ &amp; =\\frac{8}{1} &amp;&amp; \\text{Simplify quotient} \\\\ &amp; =8 &amp;&amp; \\text{Simplify quotient}\\end{align}[\/latex]\r\nIn this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.<\/p>\r\n5.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}7\\left(5\\cdot3\\right)-2[\\left(6-3\\right)-4^{2}]+1 &amp; =7\\left(15\\right)-2[\\left(3\\right)-4^{2}]+1 &amp;&amp; \\text{Simplify inside parentheses} \\\\ &amp; 7\\left(15\\right)-2\\left(3-16\\right)+1 &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =7\\left(15\\right)-2\\left(-13\\right)+1 &amp;&amp; \\text{Subtract} \\\\ &amp; =105+26+1 &amp;&amp; \\text{Multiply} \\\\ &amp; =132 &amp;&amp; \\text{Add}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=259&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=99379&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nWatch the following video for more examples of using the order of operations to simplify an expression.\r\n\r\nhttps:\/\/youtu.be\/9suc63qB96o\r\n<h2>Using Properties of Real Numbers<\/h2>\r\nFor 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.\r\n<h3>Commutative Properties<\/h3>\r\nThe <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.\r\n<div style=\"text-align: center;\">[latex]a+b=b+a[\/latex]<\/div>\r\nWe can better see this relationship when using real numbers.\r\n<div style=\"text-align: center;\">[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/div>\r\nSimilarly, the <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.\r\n<div style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\r\nAgain, consider an example with real numbers.\r\n<div style=\"text-align: center;\">[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/div>\r\nIt 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\\ne 5\\div 20[\/latex].\r\n<h3>Associative Properties<\/h3>\r\nThe <strong>associative property of multiplication<\/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.\r\n<div style=\"text-align: center;\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\r\nConsider this example.\r\n<div style=\"text-align: center;\">[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/div>\r\nThe <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.\r\n<div style=\"text-align: center;\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\r\nThis property can be especially helpful when dealing with negative integers. Consider this example.\r\n<div style=\"text-align: center;\">[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/div>\r\nAre subtraction and division associative? Review these examples.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}8-\\left(3-15\\right) &amp; \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) &amp; \\stackrel{?}=5-15 \\\\ 20 &amp; \\neq 20-10 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}64\\div\\left(8\\div4\\right)&amp;\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 &amp; \\stackrel{?}{=}8\\div4 \\\\ 32 &amp; \\neq 2 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\nAs we can see, neither subtraction nor division is associative.\r\n<h3>Distributive Property<\/h3>\r\nThe <strong>distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\r\nThis property combines both addition and multiplication (and is the only property to do so). Let us consider an example.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223815\/CNX_CAT_Figure_01_01_003.jpg\" 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.\" \/>\r\n\r\nNote 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.\r\n\r\nTo 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.\r\n<div style=\"text-align: center;\">[latex]\\begin{align} 6+\\left(3\\cdot 5\\right)&amp; \\stackrel{?}{=} \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ 6+\\left(15\\right)&amp; \\stackrel{?}{=} \\left(9\\right)\\cdot \\left(11\\right) \\\\ 21&amp; \\ne 99 \\end{align}[\/latex]<\/div>\r\nMultiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.\r\n\r\nA special case of the distributive property occurs when a sum of terms is subtracted.\r\n<div style=\"text-align: center;\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\r\nFor example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms 12 and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.\r\n<div style=\"text-align: center;\">[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\r\nNow, distribute [latex]-1[\/latex] and simplify the result.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}12+\\left(-1\\right)\\cdot\\left(5+3\\right)&amp;=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\&amp;=12+(-5-3) \\\\&amp;=12+\\left(-8\\right) \\\\&amp;=4 \\end{align}[\/latex]<\/div>\r\nThis 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.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}12-\\left(5+3\\right) &amp;=12+\\left(-5-3\\right) \\\\ &amp;=12+\\left(-8\\right) \\\\ &amp;=4\\end{align}[\/latex]<\/div>\r\n<h3>Identity Properties<\/h3>\r\nThe <strong>identity property of addition<\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.\r\n<div style=\"text-align: center;\">\u00a0[latex]a+0=a[\/latex]<\/div>\r\n<div><\/div>\r\nThe <strong>identity property of multiplication<\/strong> states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.\r\n<div style=\"text-align: center;\">[latex]a\\cdot 1=a[\/latex]<\/div>\r\nFor example, we have [latex]\\left(-6\\right)+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.\r\n<h3>Inverse Properties<\/h3>\r\nThe <strong>inverse property of addition<\/strong> states that, for every real number <em>a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted\u2212<em>a<\/em>, that, when added to the original number, results in the additive identity, 0.\r\n<div style=\"text-align: center;\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\r\nFor example, if [latex]a=-8[\/latex], the additive inverse is 8, since [latex]\\left(-8\\right)+8=0[\/latex].\r\n\r\nThe <strong>inverse property of multiplication<\/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>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.\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=1[\/latex]<\/div>\r\nFor example, if [latex]a=-\\frac{2}{3}[\/latex], the reciprocal, denoted [latex]\\frac{1}{a}[\/latex], is [latex]-\\frac{3}{2}[\/latex]\u00a0because\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=\\left(-\\dfrac{2}{3}\\right)\\cdot \\left(-\\dfrac{3}{2}\\right)=1[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Properties of Real Numbers<\/h3>\r\nThe following properties hold for real numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>.\r\n<table summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Addition<\/th>\r\n<th>Multiplication<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong>Commutative Property<\/strong><\/td>\r\n<td>[latex]a+b=b+a[\/latex]<\/td>\r\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Associative Property<\/strong><\/td>\r\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\r\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Distributive Property<\/strong><\/td>\r\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Identity Property<\/strong><\/td>\r\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em>\r\n<div>[latex]a+0=a[\/latex]<\/div><\/td>\r\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em>\r\n<div>[latex]a\\cdot 1=a[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Inverse Property<\/strong><\/td>\r\n<td>Every real number a has an additive inverse, or opposite, denoted <em>\u2013a<\/em>, such that\r\n<div>[latex]a+\\left(-a\\right)=0[\/latex]<\/div><\/td>\r\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\frac{1}{a}[\/latex], such that\r\n<div>[latex]a\\cdot \\left(\\dfrac{1}{a}\\right)=1[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Properties of Real Numbers<\/h3>\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n<ol>\r\n \t<li>[latex]3\\cdot 6+3\\cdot 4[\/latex]<\/li>\r\n \t<li>[latex]\\left(5+8\\right)+\\left(-8\\right)[\/latex]<\/li>\r\n \t<li>[latex]6-\\left(15+9\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{7}\\cdot \\left(\\frac{2}{3}\\cdot \\dfrac{7}{4}\\right)[\/latex]<\/li>\r\n \t<li>[latex]100\\cdot \\left[0.75+\\left(-2.38\\right)\\right][\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"892710\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"892710\"]\r\n\r\n1.\r\n\r\n[latex]\\begin{align}3\\cdot6+3\\cdot4 &amp;=3\\cdot\\left(6+4\\right) &amp;&amp; \\text{Distributive property} \\\\ &amp;=3\\cdot10 &amp;&amp; \\text{Simplify} \\\\ &amp; =30 &amp;&amp; \\text{Simplify}\\end{align}[\/latex]\r\n\r\n2.\r\n\r\n[latex]\\begin{align}\\left(5+8\\right)+\\left(-8\\right) &amp;=5+\\left[8+\\left(-8\\right)\\right] &amp;&amp;\\text{Associative property of addition} \\\\ &amp;=5+0 &amp;&amp; \\text{Inverse property of addition} \\\\ &amp;=5 &amp;&amp;\\text{Identity property of addition}\\end{align}[\/latex]\r\n\r\n3.\r\n\r\n[latex]\\begin{align}6-\\left(15+9\\right) &amp; =6+(-15-9) &amp;&amp; \\text{Distributive property} \\\\ &amp; =6+\\left(-24\\right) &amp;&amp; \\text{Simplify} \\\\ &amp; =-18 &amp;&amp; \\text{Simplify}\\end{align}[\/latex]\r\n\r\n4.\r\n\r\n[latex]\\begin{align}\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) &amp; =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) &amp;&amp; \\text{Commutative property of multiplication} \\\\ &amp; =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3} &amp;&amp; \\text{Associative property of multiplication} \\\\ &amp; =1\\cdot\\frac{2}{3} &amp;&amp; \\text{Inverse property of multiplication} \\\\ &amp; =\\frac{2}{3} &amp;&amp; \\text{Identity property of multiplication}\\end{align}[\/latex]\r\n\r\n5.\r\n\r\n[latex]\\begin{align}100\\cdot[0.75+\\left(-2.38\\right)] &amp; =100\\cdot0.75+100\\cdot\\left(-2.38\\right) &amp;&amp; \\text{Distributive property} \\\\ &amp; =75+\\left(-238\\right) &amp;&amp; \\text{Simplify} \\\\ &amp; =-163 &amp;&amp; \\text{Simplify}\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n<ol>\r\n \t<li>[latex]\\left(-\\dfrac{23}{5}\\right)\\cdot \\left[11\\cdot \\left(-\\dfrac{5}{23}\\right)\\right][\/latex]<\/li>\r\n \t<li>[latex]5\\cdot \\left(6.2+0.4\\right)[\/latex]<\/li>\r\n \t<li>[latex]18-\\left(7 - 15\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{17}{18}+\\cdot \\left[\\dfrac{4}{9}+\\left(-\\dfrac{17}{18}\\right)\\right][\/latex]<\/li>\r\n \t<li>[latex]6\\cdot \\left(-3\\right)+6\\cdot 3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"881536\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"881536\"]\r\n<ol>\r\n \t<li>11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;<\/li>\r\n \t<li>33, distributive property;<\/li>\r\n \t<li>26, distributive property;<\/li>\r\n \t<li>[latex]\\dfrac{4}{9}[\/latex], commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;<\/li>\r\n \t<li>0, distributive property, inverse property of addition, identity property of addition<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92360&amp;theme=oea&amp;iframe_resize_id=mom115\" width=\"100%\" height=\"400\"><\/iframe>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92361&amp;theme=oea&amp;iframe_resize_id=mom120\" width=\"100%\" height=\"400\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Evaluate and Simplify Algebraic Expressions<\/h2>\r\nSo 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}^{3}[\/latex], or [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]. In the expression [latex]x+5, 5[\/latex] is called a <strong>constant<\/strong> because it does not vary and <em>x<\/em> is called a <strong>variable<\/strong> because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.\r\n\r\nWe 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.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;\\left(-3\\right)^{5}=\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right) &amp;&amp; x^{5}=x\\cdot x\\cdot x\\cdot x\\cdot x \\\\ &amp;\\left(2\\cdot7\\right)^{3}=\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right) &amp;&amp; \\left(yz\\right)^{3}=\\left(yz\\right)\\cdot\\left(yz\\right)\\cdot\\left(yz\\right)\\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\nIn each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.\r\n\r\nAny 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.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Describing Algebraic Expressions<\/h3>\r\nList the constants and variables for each algebraic expression.\r\n<ol>\r\n \t<li><em>x<\/em> + 5<\/li>\r\n \t<li>[latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"790423\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"790423\"]\r\n<table summary=\"A table with four rows and three columns. The first entry of the first row is empty, but the second entry reads: Constants, and the third reads: Variables. The first entry of the second row reads: x plus five. The second column entry reads: five. The third column entry reads: x. The first entry of the third row reads: four-thirds pi times r cubed. The second column entry reads: four-thirds, pi. The third column entry reads: r. The first entry of the fourth row reads: the square root of two times m cubed times n squared. The second column entry reads: two. The third column entry reads: m, n.\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Constants<\/th>\r\n<th>Variables<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1. <em>x<\/em> + 5<\/td>\r\n<td>5<\/td>\r\n<td><em>x<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. [latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/td>\r\n<td>[latex]\\frac{4}{3},\\pi [\/latex]<\/td>\r\n<td>[latex]r[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3. [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>[latex]m,n[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109667&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"300\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating an Algebraic Expression at Different Values<\/h3>\r\nEvaluate the expression [latex]2x - 7[\/latex] for each value for <em>x.<\/em>\r\n<ol>\r\n \t<li>[latex]x=0[\/latex]<\/li>\r\n \t<li>[latex]x=1[\/latex]<\/li>\r\n \t<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]x=-4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"421675\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"421675\"]\r\n<ol>\r\n \t<li>Substitute 0 for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(0\\right)-7 \\\\ &amp; =0-7 \\\\ &amp; =-7\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 1 for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(1\\right)-7 \\\\ &amp; =2-7 \\\\ &amp; =-5\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]\\dfrac{1}{2}[\/latex] for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(\\frac{1}{2}\\right)-7 \\\\ &amp; =1-7 \\\\ &amp; =-6\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(-4\\right)-7 \\\\ &amp; =-8-7 \\\\ &amp; =-15\\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div style=\"text-align: center;\"><\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1976&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"200\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Algebraic Expressions<\/h3>\r\nEvaluate each expression for the given values.\r\n<ol>\r\n \t<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\r\n \t<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\r\n \t<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"182854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"182854\"]\r\n<ol>\r\n \t<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &amp;=\\left(-5\\right)+5 \\\\ &amp;=0\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 10 for [latex]t[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} &amp; =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ &amp; =\\frac{10}{20-1} \\\\ &amp; =\\frac{10}{19}\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 5 for [latex]r[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} &amp; =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ &amp; =\\frac{4}{3}\\pi\\left(125\\right) \\\\ &amp; =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 11 for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b &amp; =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ &amp; =11-8-8 \\\\ &amp; =-85\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 2 for [latex]m[\/latex] and 3 for [latex]n[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} &amp; =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ &amp; =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ &amp; =\\sqrt{144} \\\\ &amp; =12\\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=483&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<iframe id=\"mom12\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92388&amp;theme=oea&amp;iframe_resize_id=mom12\" width=\"100%\" height=\"200\"><\/iframe>\r\n\r\n<iframe id=\"mom13\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109700&amp;theme=oea&amp;iframe_resize_id=mom13\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\nIn the following video we present more examples of how to evaluate an expression for a given value.\r\n\r\nhttps:\/\/youtu.be\/MkRdwV4n91g\r\n<h2>Formulas<\/h2>\r\nAn <strong>equation<\/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 unique solution [latex]x=3[\/latex] because when we substitute 3 for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].\r\n\r\nA <strong>formula<\/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].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Formula<\/h3>\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\\left(r+h\\right)[\/latex]. 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[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223817\/CNX_CAT_Figure_01_01_004.jpg\" alt=\"A right circular cylinder with an arrow extending from the center of the top circle outward to the edge, labeled: r. Another arrow beside the image going from top to bottom, labeled: h.\" width=\"487\" height=\"279\" \/> Right circular cylinder[\/caption]\r\n\r\n[reveal-answer q=\"257174\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"257174\"]\r\n\r\nEvaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}S&amp;=2\\pi r\\left(r+h\\right) \\\\ &amp; =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ &amp; =2\\pi\\left(6\\right)\\left(15\\right) \\\\ &amp; =180\\pi\\end{align}[\/latex]<\/div>\r\nThe surface area is [latex]180\\pi [\/latex] square inches.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223819\/CNX_CAT_Figure_01_01_005.jpg\" alt=\"\/ An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.\" width=\"487\" height=\"407\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\nA photograph with length <em>L<\/em> and width <em>W<\/em> is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm<sup>2<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a matte for a photograph with length 32 cm and width 24 cm.\r\n\r\n[reveal-answer q=\"846181\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"846181\"]\r\n\r\n1,152 cm<sup>2<\/sup>\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110263&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"300\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Simplify Algebraic Expressions<\/h2>\r\nSometimes 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.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Algebraic Expressions<\/h3>\r\nSimplify each algebraic expression.\r\n<ol>\r\n \t<li>[latex]3x - 2y+x - 3y - 7[\/latex]<\/li>\r\n \t<li>[latex]2r - 5\\left(3-r\\right)+4[\/latex]<\/li>\r\n \t<li>[latex]\\left(4t-\\dfrac{5}{4}s\\right)-\\left(\\dfrac{2}{3}t+2s\\right)[\/latex]<\/li>\r\n \t<li>[latex]2mn - 5m+3mn+n[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"286046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"286046\"]\r\n<ol>\r\n \t<li>[latex]\\begin{align}3x-2y+x-3y-7 &amp; =3x+x-2y-3y-7 &amp;&amp; \\text{Commutative property of addition} \\\\ &amp; =4x-5y-7 &amp;&amp; \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align}2r-5\\left(3-r\\right)+4 &amp; =2r-15+5r+4 &amp;&amp; \\text{Distributive property}\\\\&amp;=2r+5r-15+4 &amp;&amp; \\text{Commutative property of addition} \\\\ &amp; =7r-11 &amp;&amp; \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align} 4t-\\frac{5}{4}s -\\left(\\frac{2}{3}t+2s\\right) &amp;=4t-\\frac{5}{4}s-\\frac{2}{3}t-2s &amp;&amp;\\text{Distributive property}\\\\&amp;=4t-\\frac{2}{3}t-\\frac{5}{4}s-2s &amp;&amp; \\text{Commutative property of addition}\\\\&amp;=\\frac{12}{3}t-\\frac{2}{3}t-\\frac{5}{4}s-\\frac{8}{4}s &amp;&amp; \\text{Common Denominators}\\\\ &amp; =\\frac{10}{3}t-\\frac{13}{4}s &amp;&amp; \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align}mn-5m+3mn+n &amp; =2mn+3mn-5m+n &amp;&amp; \\text{Commutative property of addition} \\\\ &amp; =5mn-5m+n &amp;&amp; \\text{Simplify}\\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom30\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=50617&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"300\"><\/iframe>\r\n<iframe id=\"mom40\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1980&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"300\"><\/iframe>\r\n<iframe id=\"mom50\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3616&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"300\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying a Formula<\/h3>\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[reveal-answer q=\"921194\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"921194\"]\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;P=L+W+L+W \\\\ &amp;P=L+L+W+W &amp;&amp; \\text{Commutative property of addition} \\\\ &amp;P=2L+2W &amp;&amp; \\text{Simplify} \\\\ &amp;P=2\\left(L+W\\right) &amp;&amp; \\text{Distributive property}\\end{align}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\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[reveal-answer q=\"600440\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"600440\"]\r\n\r\n[latex]A=P\\left(1+rt\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Writing Inequalities<\/h2>\r\nIndicating an <strong>interval<\/strong>\u00a0such as [latex]x\\ge 4[\/latex] can be achieved in several ways.\r\n\r\nWe can use a number line as shown below.\u00a0The blue ray begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225859\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" \/>\r\n\r\nWe can use <strong>set-builder notation<\/strong>: [latex]\\{x|x\\ge 4\\}[\/latex], which translates to \"all real numbers <em>x <\/em>such that <em>x <\/em>is greater than or equal to 4.\" Notice that braces are used to indicate a set.\r\n\r\nThe third method is <strong>interval notation<\/strong>, where solution sets are indicated with parentheses or brackets. The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This is perhaps the most useful method as it applies to concepts studied later in this course and to other higher-level math courses.\r\n\r\nThe main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be \"equaled.\" A few examples of an <strong>interval<\/strong>, or a set of numbers in which a solution falls, are [latex]\\left[-2,6\\right)[\/latex], or all numbers between [latex]-2[\/latex] and [latex]6[\/latex], including [latex]-2[\/latex], but not including [latex]6[\/latex]; [latex]\\left(-1,0\\right)[\/latex], all real numbers between, but not including [latex]-1[\/latex] and [latex]0[\/latex]; and [latex]\\left(-\\infty ,1\\right][\/latex], all real numbers less than and including [latex]1[\/latex]. The table below\u00a0outlines the possibilities.\r\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\r\n<thead>\r\n<tr>\r\n<th>Set Indicated<\/th>\r\n<th>Set-Builder Notation<\/th>\r\n<th>Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, but not including <em>a <\/em>or <em>b<\/em><\/td>\r\n<td>[latex]{x|a&lt;x&lt;b}[\/latex]<\/td>\r\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\r\n<td>[latex]\\{x|x&gt;a\\}[\/latex]<\/td>\r\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\r\n<td>[latex]\\{x|x&lt;b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\{x|x\\ge a\\}[\/latex]<\/td>\r\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\{x|x\\le b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\{x|a\\le x&lt;b\\}[\/latex]<\/td>\r\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\{x|a&lt;x\\le b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\r\n<td>[latex]\\{x|a\\le x\\le b\\}[\/latex]<\/td>\r\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\r\n<td>[latex]\\{x|x&lt;a\\text{ and }x&gt;b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers<\/td>\r\n<td>[latex]\\{x|x\\text{ is all real numbers}\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Interval Notation to Express All Real Numbers Greater Than or Equal to a Number<\/h3>\r\nUse interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].\r\n\r\n[reveal-answer q=\"143041\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"143041\"]\r\n\r\nUse a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse interval notation to indicate all real numbers between and including [latex]-3[\/latex] and [latex]5[\/latex].\r\n\r\n[reveal-answer q=\"814810\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814810\"]\r\n\r\n[latex]\\left[-3,5\\right][\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=58&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92604&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Interval Notation to Express All Real Numbers Less Than or Equal to <em>a <\/em>or Greater Than or Equal to <em>b<\/em><\/h3>\r\nWrite the interval expressing all real numbers less than or equal to [latex]-1[\/latex] or greater than or equal to [latex]1[\/latex].\r\n\r\n[reveal-answer q=\"797079\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797079\"]\r\n\r\nWe have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at [latex]-\\infty [\/latex] and ends at [latex]-1[\/latex], which is written as [latex]\\left(-\\infty ,-1\\right][\/latex].\r\n\r\nThe second interval must show all real numbers greater than or equal to [latex]1[\/latex], which is written as [latex]\\left[1,\\infty \\right)[\/latex]. However, we want to combine these two sets. We accomplish this by inserting the union symbol, [latex]\\cup [\/latex], between the two intervals.\r\n<div style=\"text-align: center;\">[latex]\\left(-\\infty ,-1\\right]\\cup \\left[1,\\infty \\right)[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nExpress all real numbers less than [latex]-2[\/latex] or greater than or equal to 3 in interval notation.\r\n\r\n[reveal-answer q=\"729196\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"729196\"]\r\n\r\n[latex]\\left(-\\infty ,-2\\right)\\cup \\left[3,\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2748&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>Rational numbers may be written as fractions or terminating or repeating decimals.<\/li>\r\n \t<li>Determine whether a number is rational or irrational by writing it as a decimal.<\/li>\r\n \t<li>The rational numbers and irrational numbers make up the set of real numbers. A number can be classified as natural, whole, integer, rational, or irrational.<\/li>\r\n \t<li>The order of operations is used to evaluate expressions.<\/li>\r\n \t<li>The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties.<\/li>\r\n \t<li>Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. They take on a numerical value when evaluated by replacing variables with constants.<\/li>\r\n \t<li>Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>algebraic expression\u00a0<\/strong>constants and variables combined using addition, subtraction, multiplication, and division\r\n\r\n<strong>associative property of addition\u00a0<\/strong>the sum of three numbers may be grouped differently without affecting the result; in symbols, [latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]\r\n\r\n<strong>associative property of multiplication\u00a0<\/strong>the product of three numbers may be grouped differently without affecting the result; in symbols, [latex]a\\cdot \\left(b\\cdot c\\right)=\\left(a\\cdot b\\right)\\cdot c[\/latex]\r\n\r\n<strong>base\u00a0<\/strong>in exponential notation, the expression that is being multiplied\r\n\r\n<strong>commutative property of addition\u00a0<\/strong>two numbers may be added in either order without affecting the result; in symbols, [latex]a+b=b+a[\/latex]\r\n\r\n<strong>commutative property of multiplication\u00a0<\/strong>two numbers may be multiplied in any order without affecting the result; in symbols, [latex]a\\cdot b=b\\cdot a[\/latex]\r\n\r\n<strong>constant\u00a0<\/strong>a quantity that does not change value\r\n\r\n<strong>distributive property\u00a0<\/strong>the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, [latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]\r\n\r\n<strong>equation\u00a0<\/strong>a mathematical statement indicating that two expressions are equal\r\n\r\n<strong>exponent\u00a0<\/strong>in exponential notation, the raised number or variable that indicates how many times the base is being multiplied\r\n\r\n<strong>exponential notation\u00a0<\/strong>a shorthand method of writing products of the same factor\r\n\r\n<strong>formula\u00a0<\/strong>an equation expressing a relationship between constant and variable quantities\r\n\r\n<strong>identity property of addition\u00a0<\/strong>there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, [latex]a+0=a[\/latex]\r\n\r\n<strong>identity property of multiplication\u00a0<\/strong>there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, [latex]a\\cdot 1=a[\/latex]\r\n\r\n<strong>integers\u00a0<\/strong>the set consisting of the natural numbers, their opposites, and 0: [latex]\\{\\dots ,-3,-2,-1,0,1,2,3,\\dots \\}[\/latex]\r\n\r\n<strong>inverse property of addition\u00a0<\/strong>for every real number [latex]a[\/latex], there is a unique number, called the additive inverse (or opposite), denoted [latex]-a[\/latex], which, when added to the original number, results in the additive identity, 0; in symbols, [latex]a+\\left(-a\\right)=0[\/latex]\r\n\r\n<strong>inverse property of multiplication\u00a0<\/strong>for every non-zero real number [latex]a[\/latex], there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\dfrac{1}{a}[\/latex], which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, [latex]a\\cdot \\dfrac{1}{a}=1[\/latex]\r\n\r\n<strong>irrational numbers\u00a0<\/strong>the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers\r\n\r\n<strong>natural numbers\u00a0<\/strong>the set of counting numbers: [latex]\\{1,2,3,\\dots \\}[\/latex]\r\n\r\n<strong>order of operations\u00a0<\/strong>a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations\r\n\r\n<strong>rational numbers\u00a0<\/strong>the set of all numbers of the form [latex]\\dfrac{m}{n}[\/latex], where [latex]m[\/latex] and [latex]n[\/latex] are integers and [latex]n\\ne 0[\/latex]. Any rational number may be written as a fraction or a terminating or repeating decimal.\r\n\r\n<strong>real number line\u00a0<\/strong>a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.\r\n\r\n<strong>real numbers\u00a0<\/strong>the sets of rational numbers and irrational numbers taken together\r\n\r\n<strong>variable\u00a0<\/strong>a quantity that may change value\r\n\r\n<strong>whole numbers\u00a0<\/strong>the set consisting of 0 plus the natural numbers: [latex]\\{0,1,2,3,\\dots \\}[\/latex]","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li class=\"li2\"><span class=\"s1\">Classify a real number.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Perform calculations using order of operations.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use the properties of real numbers.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Evaluate and simplify algebraic expressions.<\/span><\/li>\n<li>Express an interval.<\/li>\n<\/ul>\n<\/div>\n<p>Because of the evolution of the number system, we can now perform complex calculations using several\u00a0categories of real numbers. In this section we will explore sets of numbers, perform calculations with different kinds of numbers, and begin to learn about the use of numbers in algebraic expressions.<\/p>\n<h2>Classify a Real Number<\/h2>\n<p>The numbers we use for counting, or enumerating items, are the <strong>natural numbers<\/strong>: 1, 2, 3, 4, 5, and so on. We describe them in set notation as {1, 2, 3, &#8230;} where the ellipsis (\u2026) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the <em>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>whole numbers<\/strong> is the set of natural numbers plus zero: {0, 1, 2, 3,&#8230;}.<\/p>\n<p>The set of <strong>integers<\/strong> adds the opposites of the natural numbers to the set of whole numbers: {&#8230;,-3, -2, -1, 0, 1, 2, 3,&#8230;}. 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<div style=\"text-align: center;\">[latex]\\begin{align}&{\\text{negative integers}} && {\\text{zero}} && {\\text{positive integers}}\\\\&{\\dots ,-3,-2,-1,} && {0,} && {1,2,3,\\dots } \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<p>The set of <strong>rational numbers<\/strong> is written as [latex]\\left\\{\\frac{m}{n}|m\\text{ and }{n}\\text{ are integers and }{n}\\ne{ 0 }\\right\\}[\/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>Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:<\/p>\n<ol>\n<li>a terminating decimal: [latex]\\frac{15}{8}=1.875[\/latex], or<\/li>\n<li>a repeating decimal: [latex]\\frac{4}{11}=0.36363636\\dots =0.\\overline{36}[\/latex]<\/li>\n<\/ol>\n<p>We use a line drawn over the repeating block of numbers instead of writing the group multiple times.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Integers as Rational Numbers<\/h3>\n<p>Write each of the following as a rational number.<\/p>\n<ol>\n<li>7<\/li>\n<li>0<\/li>\n<li>\u20138<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q534535\">Show Solution<\/span><\/p>\n<div id=\"q534535\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write a fraction with the integer in the numerator and 1 in the denominator.<\/p>\n<ol>\n<li>[latex]7=\\dfrac{7}{1}[\/latex]<\/li>\n<li>[latex]0=\\dfrac{0}{1}[\/latex]<\/li>\n<li>[latex]-8=-\\dfrac{8}{1}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following as a rational number.<\/p>\n<ol>\n<li>11<\/li>\n<li>3<\/li>\n<li>\u20134<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q755048\">Show Solution<\/span><\/p>\n<div id=\"q755048\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\dfrac{11}{1}[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{1}[\/latex]<\/li>\n<li>[latex]-\\dfrac{4}{1}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Rational Numbers<\/h3>\n<p>Write each of the following rational numbers as either a terminating or repeating decimal.<\/p>\n<ol>\n<li>[latex]-\\dfrac{5}{7}[\/latex]<\/li>\n<li>[latex]\\dfrac{15}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{13}{25}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q549544\">Show Solution<\/span><\/p>\n<div id=\"q549544\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write each fraction as a decimal by dividing the numerator by the denominator.<\/p>\n<ol>\n<li>[latex]-\\dfrac{5}{7}=-0.\\overline{714285}[\/latex], a repeating decimal<\/li>\n<li>[latex]\\dfrac{15}{5}=3[\/latex] (or 3.0), a terminating decimal<\/li>\n<li>[latex]\\dfrac{13}{25}=0.52[\/latex], a terminating decimal<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<h2>Irrational Numbers<\/h2>\n<p>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>irrational<\/em> because they cannot be written as fractions. These numbers make up the set of <strong>irrational numbers<\/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<div style=\"text-align: center;\">{h | h is not a rational number}<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Differentiating Rational and Irrational Numbers<\/h3>\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<ol>\n<li>[latex]\\sqrt{25}[\/latex]<\/li>\n<li>[latex]\\dfrac{33}{9}[\/latex]<\/li>\n<li>[latex]\\sqrt{11}[\/latex]<\/li>\n<li>[latex]\\dfrac{17}{34}[\/latex]<\/li>\n<li>[latex]0.3033033303333\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q502265\">Show Solution<\/span><\/p>\n<div id=\"q502265\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{25}:[\/latex] This can be simplified as [latex]\\sqrt{25}=5[\/latex]. Therefore, [latex]\\sqrt{25}[\/latex] is rational.<\/li>\n<li>[latex]\\dfrac{33}{9}:[\/latex] Because it is a fraction, [latex]\\dfrac{33}{9}[\/latex] is a rational number. Next, simplify and divide.\n<div style=\"text-align: center;\">[latex]\\dfrac{33}{9}=\\dfrac{{11}\\cdot{3}}{{3}\\cdot{3}}=\\dfrac{11}{3}=3.\\overline{6}[\/latex]<\/div>\n<p>So, [latex]\\dfrac{33}{9}[\/latex] is rational and a repeating decimal.<\/li>\n<li>[latex]\\sqrt{11}:[\/latex] This cannot be simplified any further. Therefore, [latex]\\sqrt{11}[\/latex] is an irrational number.<\/li>\n<li>[latex]\\dfrac{17}{34}:[\/latex] Because it is a fraction, [latex]\\dfrac{17}{34}[\/latex] is a rational number. Simplify and divide.\n<div style=\"text-align: center;\">[latex]\\dfrac{17}{34}=\\dfrac{1\\cdot{17}}{2\\cdot{17}}=\\dfrac{1}{2}=0.5[\/latex]<\/div>\n<p>So, [latex]\\dfrac{17}{34}[\/latex] is rational and a terminating decimal.<\/li>\n<li>0.3033033303333&#8230; 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.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92383&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div>\n<h3>Real Numbers<\/h3>\n<p>Given any number <em>n<\/em>, we know that <em>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>real numbers<\/strong>. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or \u2013). Zero is considered neither positive nor negative.<\/p>\n<p>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 <strong>real number line<\/strong>.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223810\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" \/><\/p>\n<p class=\"wp-caption-text\">The real number line<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Classifying Real Numbers<\/h3>\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<ol>\n<li>[latex]-\\dfrac{10}{3}[\/latex]<\/li>\n<li>[latex]\\sqrt{5}[\/latex]<\/li>\n<li>[latex]-\\sqrt{289}[\/latex]<\/li>\n<li>[latex]-6\\pi[\/latex]<\/li>\n<li>[latex]0.616161\\dots[\/latex]<\/li>\n<li>[latex]0.13[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q705558\">Show Solution<\/span><\/p>\n<div id=\"q705558\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]-\\dfrac{10}{3}[\/latex] is negative and rational. It lies to the left of 0 on the number line.<\/li>\n<li>[latex]\\sqrt{5}[\/latex] is positive and irrational. It lies to the right of 0.<\/li>\n<li>[latex]-\\sqrt{289}=-\\sqrt{{17}^{2}}=-17[\/latex] is negative and rational. It lies to the left of 0.<\/li>\n<li>[latex]-6\\pi[\/latex] is negative and irrational. It lies to the left of 0.<\/li>\n<li>[latex]0.616161\\dots[\/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.<\/li>\n<li>[latex]0.13[\/latex] is a finite decimal and may be written as 13\/100. \u00a0So it is rational and positive.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\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<ol>\n<li>[latex]\\sqrt{73}[\/latex]<\/li>\n<li>[latex]-11.411411411\\dots[\/latex]<\/li>\n<li>[latex]\\dfrac{47}{19}[\/latex]<\/li>\n<li>[latex]-\\dfrac{\\sqrt{5}}{2}[\/latex]<\/li>\n<li>[latex]6.210735[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q155954\">Show Solution<\/span><\/p>\n<div id=\"q155954\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>positive, irrational; right<\/li>\n<li>negative, rational; left<\/li>\n<li>positive, rational; right<\/li>\n<li>negative, irrational; left<\/li>\n<li>positive, rational; right<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h3>Sets of Numbers as Subsets<\/h3>\n<p>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.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223813\/CNX_CAT_Figure_01_01_001.jpg\" alt=\"A large box labeled: Real Numbers encloses five circles. Four of these circles enclose each other and the other is separate from the rest. The innermost circle contains: 1, 2, 3\u2026 N. The circle enclosing that circle contains: 0 W. The circle enclosing that circle contains: \u2026, -3, -2, -1 I. The outermost circle contains: m\/n, n not equal to zero Q. The separate circle contains: pi, square root of two, etc Q\u00b4.\" width=\"731\" height=\"352\" \/><\/p>\n<p class=\"wp-caption-text\">Sets of numbers. \u00a0 <em>N<\/em>: the set of natural numbers \u00a0 <em>W<\/em>: the set of whole numbers \u00a0 <em>I<\/em>: the set of integers \u00a0 <em>Q<\/em>: the set of rational numbers \u00a0 <em>Q\u00b4<\/em>: the set of irrational numbers<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Sets of Numbers<\/h3>\n<p>The set of <strong>natural numbers<\/strong> includes the numbers used for counting: [latex]\\{1,2,3,\\dots\\}[\/latex].<\/p>\n<p>The set of <strong>whole numbers<\/strong> is the set of natural numbers plus zero: [latex]\\{0,1,2,3,\\dots\\}[\/latex].<\/p>\n<p>The set of <strong>integers<\/strong> adds the negative natural numbers to the set of whole numbers: [latex]\\{\\dots,-3,-2,-1,0,1,2,3,\\dots\\}[\/latex].<\/p>\n<p>The set of <strong>rational numbers<\/strong> includes fractions written as [latex]\\{\\frac{m}{n}|m\\text{ and }n\\text{ are integers and }n\\ne 0\\}[\/latex].<\/p>\n<p>The set of <strong>irrational numbers<\/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 class=\"textbox exercises\">\n<h3>Example: Differentiating the Sets of Numbers<\/h3>\n<p>Classify each number as being a natural number, whole number, integer, rational number, and\/or irrational number.<\/p>\n<ul>\n<li>[latex]\\sqrt{36}[\/latex]<\/li>\n<li>[latex]\\dfrac{8}{3}[\/latex]<\/li>\n<li>[latex]\\sqrt{73}[\/latex]<\/li>\n<li>[latex]-6[\/latex]<\/li>\n<li>[latex]3.2121121112\\dots[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q779749\">Show Solution<\/span><\/p>\n<div id=\"q779749\" class=\"hidden-answer\" style=\"display: none\">\n<table class=\"lines\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\"><\/td>\n<td style=\"height: 14px; width: 104.545px;\">\u00a0natural number<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">\u00a0whole number<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">integer<\/td>\n<td style=\"height: 14px; width: 104.545px;\">rational number<\/td>\n<td style=\"height: 14px; width: 114.545px;\">irrational number<\/td>\n<\/tr>\n<tr style=\"height: 14.7017px;\">\n<td style=\"height: 14.7017px; width: 266.364px;\">\u00a0[latex]\\sqrt{36}=6[\/latex]<\/td>\n<td style=\"height: 14.7017px; width: 104.545px;\">\u00a0yes<\/td>\n<td style=\"height: 14.7017px; width: 95.4545px;\">\u00a0yes<\/td>\n<td style=\"height: 14.7017px; width: 45.4545px;\">yes<\/td>\n<td style=\"height: 14.7017px; width: 104.545px;\">yes<\/td>\n<td style=\"height: 14.7017px; width: 114.545px;\">no<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]\\dfrac{8}{3}=2.\\overline{6}[\/latex]<\/td>\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 104.545px;\">yes<\/td>\n<td style=\"height: 14px; width: 114.545px;\">no<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]\\sqrt{73}[\/latex]<\/td>\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\n<td style=\"height: 14px; width: 114.545px;\">yes<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]\u20136[\/latex]<\/td>\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">yes<\/td>\n<td style=\"height: 14px; width: 104.545px;\">\u00a0yes<\/td>\n<td style=\"height: 14px; width: 114.545px;\">\u00a0no<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]3.2121121112\\dots[\/latex]<\/td>\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\n<td style=\"height: 14px; width: 114.545px;\">yes<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Classify each number as being a natural number (<em>N<\/em>), whole number (<em>W<\/em>), integer (<em>I<\/em>), rational number (<em>Q<\/em>), and\/or irrational number (<em>Q&#8217;<\/em>).<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]-\\dfrac{35}{7}[\/latex]<\/li>\n<li>[latex]0[\/latex]<\/li>\n<li>[latex]\\sqrt{169}[\/latex]<\/li>\n<li>[latex]\\sqrt{24}[\/latex]<\/li>\n<li>[latex]4.763763763\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266197\">Show Solution<\/span><\/p>\n<div id=\"q266197\" class=\"hidden-answer\" style=\"display: none\">\n<table class=\"lines\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\"><\/td>\n<td style=\"height: 14px; width: 104.545px;\">\u00a0natural number<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">\u00a0whole number<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">integer<\/td>\n<td style=\"height: 14px; width: 104.545px;\">rational number<\/td>\n<td style=\"height: 14px; width: 114.545px;\">irrational number<\/td>\n<\/tr>\n<tr style=\"height: 14.7017px;\">\n<td style=\"height: 14.7017px; width: 266.364px;\">[latex]-\\dfrac{35}{7}[\/latex]<\/td>\n<td style=\"height: 14.7017px; width: 104.545px;\">\u00a0yes<\/td>\n<td style=\"height: 14.7017px; width: 95.4545px;\">\u00a0yes<\/td>\n<td style=\"height: 14.7017px; width: 45.4545px;\">yes<\/td>\n<td style=\"height: 14.7017px; width: 104.545px;\">yes<\/td>\n<td style=\"height: 14.7017px; width: 114.545px;\">no<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]0[\/latex]<\/td>\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">yes<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">yes<\/td>\n<td style=\"height: 14px; width: 104.545px;\">yes<\/td>\n<td style=\"height: 14px; width: 114.545px;\">no<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]\\sqrt{169}[\/latex]<\/td>\n<td style=\"height: 14px; width: 104.545px;\">\u00a0yes<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">yes<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">yes<\/td>\n<td style=\"height: 14px; width: 104.545px;\">yes<\/td>\n<td style=\"height: 14px; width: 114.545px;\">no<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\">\u00a0[latex]\\sqrt{24}[\/latex]<\/td>\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 104.545px;\">no<\/td>\n<td style=\"height: 14px; width: 114.545px;\">yes<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 266.364px;\">[latex]4.763763763\\dots[\/latex]<\/td>\n<td style=\"height: 14px; width: 104.545px;\">\u00a0no<\/td>\n<td style=\"height: 14px; width: 95.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 45.4545px;\">no<\/td>\n<td style=\"height: 14px; width: 104.545px;\">yes<\/td>\n<td style=\"height: 14px; width: 114.545px;\">no<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=13740&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"100%\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=13741&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"100%\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch this video for an overview of the sets of numbers, and how to identify which set a number belongs to.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Identifying Sets of Real Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/htP2goe31MM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Properties of Real Numbers<\/h2>\n<p>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>exponential notation<\/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<div style=\"text-align: center;\">[latex]a^{n}=a\\cdot a\\cdot a\\cdot \\dots \\cdot a[\/latex]<\/div>\n<p>In this notation, [latex]{a}^{n}[\/latex] is read as the <em>n<\/em>th power of [latex]a[\/latex], where [latex]a[\/latex] is called the <strong>base<\/strong> and [latex]n[\/latex] is called the <strong>exponent<\/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>To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the <strong>order of operations<\/strong>. This is a sequence of rules for evaluating such expressions.<\/p>\n<p>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>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>Let\u2019s take a look at the expression provided.<\/p>\n<div style=\"text-align: center;\">[latex]24+6\\cdot \\dfrac{2}{3}-{4}^{2}[\/latex]<\/div>\n<p>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<div style=\"text-align: center;\">[latex]\\begin{gathered}24+6\\cdot\\frac{2}{3}-4^{2} \\\\ 24+6\\cdot\\frac{2}{3}-16\\end{gathered}[\/latex]<\/div>\n<p>Next, perform multiplication or division, left to right.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+6\\cdot\\frac{2}{3}-16 \\\\ 24+4-16\\end{gathered}[\/latex]<\/div>\n<p>Lastly, perform addition or subtraction, left to right.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+4-16 \\\\ 28-16 \\\\ 12\\end{gathered}[\/latex]<\/div>\n<p>Therefore, [latex]24+6\\cdot \\dfrac{2}{3}-{4}^{2}=12[\/latex].<\/p>\n<p>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\">\n<h3>A General Note: Order of Operations<\/h3>\n<p>Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym <strong>PEMDAS<\/strong>:<\/p>\n<p><strong>P<\/strong>(arentheses)<\/p>\n<p><strong>E<\/strong>(xponents)<\/p>\n<p><strong>M<\/strong>(ultiplication) and <strong>D<\/strong>(ivision)<\/p>\n<p><strong>A<\/strong>(ddition) and <strong>S<\/strong>(ubtraction)<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a mathematical expression, simplify it using the order of operations.<\/h3>\n<ol>\n<li>Simplify any expressions within grouping symbols.<\/li>\n<li>Simplify any expressions containing exponents or radicals.<\/li>\n<li>Perform any multiplication and division in order, from left to right.<\/li>\n<li>Perform any addition and subtraction in order, from left to right.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Order of Operations<\/h3>\n<p>Use the order of operations to evaluate each of the following expressions.<\/p>\n<ol>\n<li>[latex]{\\left(3\\cdot 2\\right)}^{2}-4\\left(6+2\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{{5}^{2}-4}{7}-\\sqrt{11 - 2}[\/latex]<\/li>\n<li>[latex]6-|5 - 8|+3\\left(4 - 1\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{14 - 3\\cdot 2}{2\\cdot 5-{3}^{2}}[\/latex]<\/li>\n<li>[latex]7\\left(5\\cdot 3\\right)-2\\left[\\left(6 - 3\\right)-{4}^{2}\\right]+1[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q371324\">Show Solution<\/span><\/p>\n<div id=\"q371324\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(3\\cdot 2\\right)^{2} & =\\left(6\\right)^{2}-4\\left(8\\right) && \\text{Simplify parentheses} \\\\ & =36-4\\left(8\\right) && \\text{Simplify exponent} \\\\ & =36-32 && \\text{Simplify multiplication} \\\\ & =4 && \\text{Simplify subtraction}\\end{align}[\/latex]<\/p>\n<p>2.<\/p>\n<p class=\"p1\" style=\"text-align: center;\"><span class=\"s1\">[latex]\\begin{align}\\frac{5^{2}-4}{7}-\\sqrt{11-2} & =\\frac{5^{2}-4}{7}-\\sqrt{9} && \\text{Simplify grouping systems (radical)} \\\\ & =\\frac{5^{2}-4}{7}-3 && \\text{Simplify radical} \\\\ & =\\frac{25-4}{7}-3 && \\text{Simplify exponent} \\\\ & =\\frac{21}{7}-3 && \\text{Simplify subtraction in numerator} \\\\ & =3-3 && \\text{Simplify division} \\\\ & =0 && \\text{Simplify subtraction}\\end{align}[\/latex]<\/span><\/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>3.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}6-|5-8|+3\\left(4-1\\right) & =6-|-3|+3\\left(3\\right) && \\text{Simplify inside grouping system} \\\\ & =6-3+3\\left(3\\right) && \\text{Simplify absolute value} \\\\ & =6-3+9 && \\text{Simplify multiplication} \\\\ & =3+9 && \\text{Simplify subtraction} \\\\ & =12 && \\text{Simplify addition}\\end{align}[\/latex]<\/p>\n<p>4.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{14-3\\cdot2}{2\\cdot5-3^{2}} & =\\frac{14-3\\cdot2}{2\\cdot5-9} && \\text{Simplify exponent} \\\\ & =\\frac{14-6}{10-9} && \\text{Simplify products} \\\\ & =\\frac{8}{1} && \\text{Simplify quotient} \\\\ & =8 && \\text{Simplify quotient}\\end{align}[\/latex]<br \/>\nIn this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.<\/p>\n<p>5.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}7\\left(5\\cdot3\\right)-2[\\left(6-3\\right)-4^{2}]+1 & =7\\left(15\\right)-2[\\left(3\\right)-4^{2}]+1 && \\text{Simplify inside parentheses} \\\\ & 7\\left(15\\right)-2\\left(3-16\\right)+1 && \\text{Simplify exponent} \\\\ & =7\\left(15\\right)-2\\left(-13\\right)+1 && \\text{Subtract} \\\\ & =105+26+1 && \\text{Multiply} \\\\ & =132 && \\text{Add}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=259&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=99379&amp;theme=oea&amp;iframe_resize_id=mom11\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Watch the following video for more examples of using the order of operations to simplify an expression.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Order of Operations with a Fraction Containing a Square Root\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9suc63qB96o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using Properties of Real Numbers<\/h2>\n<p>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<h3>Commutative Properties<\/h3>\n<p>The <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a+b=b+a[\/latex]<\/div>\n<p>We can better see this relationship when using real numbers.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/div>\n<p>Similarly, the <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\n<p>Again, consider an example with real numbers.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/div>\n<p>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\\ne 5\\div 20[\/latex].<\/p>\n<h3>Associative Properties<\/h3>\n<p>The <strong>associative property of multiplication<\/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<div style=\"text-align: center;\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\n<p>Consider this example.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/div>\n<p>The <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\n<p>This property can be especially helpful when dealing with negative integers. Consider this example.<\/p>\n<div style=\"text-align: center;\">[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/div>\n<p>Are subtraction and division associative? Review these examples.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}8-\\left(3-15\\right) & \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) & \\stackrel{?}=5-15 \\\\ 20 & \\neq 20-10 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]\\begin{align}64\\div\\left(8\\div4\\right)&\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 & \\stackrel{?}{=}8\\div4 \\\\ 32 & \\neq 2 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<p>As we can see, neither subtraction nor division is associative.<\/p>\n<h3>Distributive Property<\/h3>\n<p>The <strong>distributive property<\/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<div style=\"text-align: center;\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\n<p>This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223815\/CNX_CAT_Figure_01_01_003.jpg\" 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.\" \/><\/p>\n<p>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>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<div style=\"text-align: center;\">[latex]\\begin{align} 6+\\left(3\\cdot 5\\right)& \\stackrel{?}{=} \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ 6+\\left(15\\right)& \\stackrel{?}{=} \\left(9\\right)\\cdot \\left(11\\right) \\\\ 21& \\ne 99 \\end{align}[\/latex]<\/div>\n<p>Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.<\/p>\n<p>A special case of the distributive property occurs when a sum of terms is subtracted.<\/p>\n<div style=\"text-align: center;\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\n<p>For example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms 12 and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.<\/p>\n<div style=\"text-align: center;\">[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\n<p>Now, distribute [latex]-1[\/latex] and simplify the result.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}12+\\left(-1\\right)\\cdot\\left(5+3\\right)&=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\&=12+(-5-3) \\\\&=12+\\left(-8\\right) \\\\&=4 \\end{align}[\/latex]<\/div>\n<p>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<div style=\"text-align: center;\">[latex]\\begin{align}12-\\left(5+3\\right) &=12+\\left(-5-3\\right) \\\\ &=12+\\left(-8\\right) \\\\ &=4\\end{align}[\/latex]<\/div>\n<h3>Identity Properties<\/h3>\n<p>The <strong>identity property of addition<\/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<div style=\"text-align: center;\">\u00a0[latex]a+0=a[\/latex]<\/div>\n<div><\/div>\n<p>The <strong>identity property of multiplication<\/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<div style=\"text-align: center;\">[latex]a\\cdot 1=a[\/latex]<\/div>\n<p>For example, we have [latex]\\left(-6\\right)+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<h3>Inverse Properties<\/h3>\n<p>The <strong>inverse property of addition<\/strong> states that, for every real number <em>a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted\u2212<em>a<\/em>, that, when added to the original number, results in the additive identity, 0.<\/p>\n<div style=\"text-align: center;\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<p>For example, if [latex]a=-8[\/latex], the additive inverse is 8, since [latex]\\left(-8\\right)+8=0[\/latex].<\/p>\n<p>The <strong>inverse property of multiplication<\/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>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<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=1[\/latex]<\/div>\n<p>For example, if [latex]a=-\\frac{2}{3}[\/latex], the reciprocal, denoted [latex]\\frac{1}{a}[\/latex], is [latex]-\\frac{3}{2}[\/latex]\u00a0because<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=\\left(-\\dfrac{2}{3}\\right)\\cdot \\left(-\\dfrac{3}{2}\\right)=1[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Properties of Real Numbers<\/h3>\n<p>The following properties hold for real numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>.<\/p>\n<table summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\n<thead>\n<tr>\n<th><\/th>\n<th>Addition<\/th>\n<th>Multiplication<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Commutative Property<\/strong><\/td>\n<td>[latex]a+b=b+a[\/latex]<\/td>\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Associative Property<\/strong><\/td>\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Distributive Property<\/strong><\/td>\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Identity Property<\/strong><\/td>\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em><\/p>\n<div>[latex]a+0=a[\/latex]<\/div>\n<\/td>\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em><\/p>\n<div>[latex]a\\cdot 1=a[\/latex]<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td><strong>Inverse Property<\/strong><\/td>\n<td>Every real number a has an additive inverse, or opposite, denoted <em>\u2013a<\/em>, such that<\/p>\n<div>[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<\/td>\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\frac{1}{a}[\/latex], such that<\/p>\n<div>[latex]a\\cdot \\left(\\dfrac{1}{a}\\right)=1[\/latex]<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Properties of Real Numbers<\/h3>\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<ol>\n<li>[latex]3\\cdot 6+3\\cdot 4[\/latex]<\/li>\n<li>[latex]\\left(5+8\\right)+\\left(-8\\right)[\/latex]<\/li>\n<li>[latex]6-\\left(15+9\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{7}\\cdot \\left(\\frac{2}{3}\\cdot \\dfrac{7}{4}\\right)[\/latex]<\/li>\n<li>[latex]100\\cdot \\left[0.75+\\left(-2.38\\right)\\right][\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q892710\">Show Solution<\/span><\/p>\n<div id=\"q892710\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p>[latex]\\begin{align}3\\cdot6+3\\cdot4 &=3\\cdot\\left(6+4\\right) && \\text{Distributive property} \\\\ &=3\\cdot10 && \\text{Simplify} \\\\ & =30 && \\text{Simplify}\\end{align}[\/latex]<\/p>\n<p>2.<\/p>\n<p>[latex]\\begin{align}\\left(5+8\\right)+\\left(-8\\right) &=5+\\left[8+\\left(-8\\right)\\right] &&\\text{Associative property of addition} \\\\ &=5+0 && \\text{Inverse property of addition} \\\\ &=5 &&\\text{Identity property of addition}\\end{align}[\/latex]<\/p>\n<p>3.<\/p>\n<p>[latex]\\begin{align}6-\\left(15+9\\right) & =6+(-15-9) && \\text{Distributive property} \\\\ & =6+\\left(-24\\right) && \\text{Simplify} \\\\ & =-18 && \\text{Simplify}\\end{align}[\/latex]<\/p>\n<p>4.<\/p>\n<p>[latex]\\begin{align}\\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) && \\text{Commutative property of multiplication} \\\\ & =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3} && \\text{Associative property of multiplication} \\\\ & =1\\cdot\\frac{2}{3} && \\text{Inverse property of multiplication} \\\\ & =\\frac{2}{3} && \\text{Identity property of multiplication}\\end{align}[\/latex]<\/p>\n<p>5.<\/p>\n<p>[latex]\\begin{align}100\\cdot[0.75+\\left(-2.38\\right)] & =100\\cdot0.75+100\\cdot\\left(-2.38\\right) && \\text{Distributive property} \\\\ & =75+\\left(-238\\right) && \\text{Simplify} \\\\ & =-163 && \\text{Simplify}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<ol>\n<li>[latex]\\left(-\\dfrac{23}{5}\\right)\\cdot \\left[11\\cdot \\left(-\\dfrac{5}{23}\\right)\\right][\/latex]<\/li>\n<li>[latex]5\\cdot \\left(6.2+0.4\\right)[\/latex]<\/li>\n<li>[latex]18-\\left(7 - 15\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{17}{18}+\\cdot \\left[\\dfrac{4}{9}+\\left(-\\dfrac{17}{18}\\right)\\right][\/latex]<\/li>\n<li>[latex]6\\cdot \\left(-3\\right)+6\\cdot 3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q881536\">Show Solution<\/span><\/p>\n<div id=\"q881536\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;<\/li>\n<li>33, distributive property;<\/li>\n<li>26, distributive property;<\/li>\n<li>[latex]\\dfrac{4}{9}[\/latex], commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;<\/li>\n<li>0, distributive property, inverse property of addition, identity property of addition<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92360&amp;theme=oea&amp;iframe_resize_id=mom115\" width=\"100%\" height=\"400\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92361&amp;theme=oea&amp;iframe_resize_id=mom120\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<h2>Evaluate and Simplify Algebraic Expressions<\/h2>\n<p>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}^{3}[\/latex], or [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]. In the expression [latex]x+5, 5[\/latex] is called a <strong>constant<\/strong> because it does not vary and <em>x<\/em> is called a <strong>variable<\/strong> because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.<\/p>\n<p>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<div style=\"text-align: center;\">[latex]\\begin{align}&\\left(-3\\right)^{5}=\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right) && x^{5}=x\\cdot x\\cdot x\\cdot x\\cdot x \\\\ &\\left(2\\cdot7\\right)^{3}=\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right) && \\left(yz\\right)^{3}=\\left(yz\\right)\\cdot\\left(yz\\right)\\cdot\\left(yz\\right)\\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<p>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>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 exercises\">\n<h3>Example: Describing Algebraic Expressions<\/h3>\n<p>List the constants and variables for each algebraic expression.<\/p>\n<ol>\n<li><em>x<\/em> + 5<\/li>\n<li>[latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/li>\n<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q790423\">Show Solution<\/span><\/p>\n<div id=\"q790423\" class=\"hidden-answer\" style=\"display: none\">\n<table summary=\"A table with four rows and three columns. The first entry of the first row is empty, but the second entry reads: Constants, and the third reads: Variables. The first entry of the second row reads: x plus five. The second column entry reads: five. The third column entry reads: x. The first entry of the third row reads: four-thirds pi times r cubed. The second column entry reads: four-thirds, pi. The third column entry reads: r. The first entry of the fourth row reads: the square root of two times m cubed times n squared. The second column entry reads: two. The third column entry reads: m, n.\">\n<thead>\n<tr>\n<th><\/th>\n<th>Constants<\/th>\n<th>Variables<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1. <em>x<\/em> + 5<\/td>\n<td>5<\/td>\n<td><em>x<\/em><\/td>\n<\/tr>\n<tr>\n<td>2. [latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/td>\n<td>[latex]\\frac{4}{3},\\pi[\/latex]<\/td>\n<td>[latex]r[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>3. [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/td>\n<td>2<\/td>\n<td>[latex]m,n[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109667&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating an Algebraic Expression at Different Values<\/h3>\n<p>Evaluate the expression [latex]2x - 7[\/latex] for each value for <em>x.<\/em><\/p>\n<ol>\n<li>[latex]x=0[\/latex]<\/li>\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]x=-4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q421675\">Show Solution<\/span><\/p>\n<div id=\"q421675\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute 0 for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(0\\right)-7 \\\\ & =0-7 \\\\ & =-7\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 1 for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(1\\right)-7 \\\\ & =2-7 \\\\ & =-5\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]\\dfrac{1}{2}[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(\\frac{1}{2}\\right)-7 \\\\ & =1-7 \\\\ & =-6\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(-4\\right)-7 \\\\ & =-8-7 \\\\ & =-15\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1976&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"200\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Algebraic Expressions<\/h3>\n<p>Evaluate each expression for the given values.<\/p>\n<ol>\n<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\n<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\n<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\n<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q182854\">Show Solution<\/span><\/p>\n<div id=\"q182854\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &=\\left(-5\\right)+5 \\\\ &=0\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 10 for [latex]t[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} & =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ & =\\frac{10}{20-1} \\\\ & =\\frac{10}{19}\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 5 for [latex]r[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} & =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ & =\\frac{4}{3}\\pi\\left(125\\right) \\\\ & =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 11 for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b & =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ & =11-8-8 \\\\ & =-85\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 2 for [latex]m[\/latex] and 3 for [latex]n[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} & =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ & =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ & =\\sqrt{144} \\\\ & =12\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=483&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom12\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92388&amp;theme=oea&amp;iframe_resize_id=mom12\" width=\"100%\" height=\"200\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom13\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109700&amp;theme=oea&amp;iframe_resize_id=mom13\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we present more examples of how to evaluate an expression for a given value.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Evaluate Various Algebraic Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/MkRdwV4n91g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Formulas<\/h2>\n<p>An <strong>equation<\/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 unique solution [latex]x=3[\/latex] because when we substitute 3 for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].<\/p>\n<p>A <strong>formula<\/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 exercises\">\n<h3>Example: Using a Formula<\/h3>\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\\left(r+h\\right)[\/latex]. 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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223817\/CNX_CAT_Figure_01_01_004.jpg\" alt=\"A right circular cylinder with an arrow extending from the center of the top circle outward to the edge, labeled: r. Another arrow beside the image going from top to bottom, labeled: h.\" width=\"487\" height=\"279\" \/><\/p>\n<p class=\"wp-caption-text\">Right circular cylinder<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q257174\">Show Solution<\/span><\/p>\n<div id=\"q257174\" class=\"hidden-answer\" style=\"display: none\">\n<p>Evaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}S&=2\\pi r\\left(r+h\\right) \\\\ & =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ & =2\\pi\\left(6\\right)\\left(15\\right) \\\\ & =180\\pi\\end{align}[\/latex]<\/div>\n<p>The surface area is [latex]180\\pi[\/latex] square inches.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223819\/CNX_CAT_Figure_01_01_005.jpg\" alt=\"\/ An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.\" width=\"487\" height=\"407\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<p>A photograph with length <em>L<\/em> and width <em>W<\/em> is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm<sup>2<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a matte for a photograph with length 32 cm and width 24 cm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q846181\">Show Solution<\/span><\/p>\n<div id=\"q846181\" class=\"hidden-answer\" style=\"display: none\">\n<p>1,152 cm<sup>2<\/sup><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110263&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<h2>Simplify Algebraic Expressions<\/h2>\n<p>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 exercises\">\n<h3>Example: Simplifying Algebraic Expressions<\/h3>\n<p>Simplify each algebraic expression.<\/p>\n<ol>\n<li>[latex]3x - 2y+x - 3y - 7[\/latex]<\/li>\n<li>[latex]2r - 5\\left(3-r\\right)+4[\/latex]<\/li>\n<li>[latex]\\left(4t-\\dfrac{5}{4}s\\right)-\\left(\\dfrac{2}{3}t+2s\\right)[\/latex]<\/li>\n<li>[latex]2mn - 5m+3mn+n[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q286046\">Show Solution<\/span><\/p>\n<div id=\"q286046\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\begin{align}3x-2y+x-3y-7 & =3x+x-2y-3y-7 && \\text{Commutative property of addition} \\\\ & =4x-5y-7 && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align}2r-5\\left(3-r\\right)+4 & =2r-15+5r+4 && \\text{Distributive property}\\\\&=2r+5r-15+4 && \\text{Commutative property of addition} \\\\ & =7r-11 && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} 4t-\\frac{5}{4}s -\\left(\\frac{2}{3}t+2s\\right) &=4t-\\frac{5}{4}s-\\frac{2}{3}t-2s &&\\text{Distributive property}\\\\&=4t-\\frac{2}{3}t-\\frac{5}{4}s-2s && \\text{Commutative property of addition}\\\\&=\\frac{12}{3}t-\\frac{2}{3}t-\\frac{5}{4}s-\\frac{8}{4}s && \\text{Common Denominators}\\\\ & =\\frac{10}{3}t-\\frac{13}{4}s && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align}mn-5m+3mn+n & =2mn+3mn-5m+n && \\text{Commutative property of addition} \\\\ & =5mn-5m+n && \\text{Simplify}\\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom30\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=50617&amp;theme=oea&amp;iframe_resize_id=mom30\" width=\"100%\" height=\"300\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom40\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1980&amp;theme=oea&amp;iframe_resize_id=mom40\" width=\"100%\" height=\"300\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom50\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3616&amp;theme=oea&amp;iframe_resize_id=mom50\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying a Formula<\/h3>\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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q921194\">Show Solution<\/span><\/p>\n<div id=\"q921194\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\begin{align}&P=L+W+L+W \\\\ &P=L+L+W+W && \\text{Commutative property of addition} \\\\ &P=2L+2W && \\text{Simplify} \\\\ &P=2\\left(L+W\\right) && \\text{Distributive property}\\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q600440\">Show Solution<\/span><\/p>\n<div id=\"q600440\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]A=P\\left(1+rt\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Writing Inequalities<\/h2>\n<p>Indicating an <strong>interval<\/strong>\u00a0such as [latex]x\\ge 4[\/latex] can be achieved in several ways.<\/p>\n<p>We can use a number line as shown below.\u00a0The blue ray begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225859\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" \/><\/p>\n<p>We can use <strong>set-builder notation<\/strong>: [latex]\\{x|x\\ge 4\\}[\/latex], which translates to &#8220;all real numbers <em>x <\/em>such that <em>x <\/em>is greater than or equal to 4.&#8221; Notice that braces are used to indicate a set.<\/p>\n<p>The third method is <strong>interval notation<\/strong>, where solution sets are indicated with parentheses or brackets. The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This is perhaps the most useful method as it applies to concepts studied later in this course and to other higher-level math courses.<\/p>\n<p>The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be &#8220;equaled.&#8221; A few examples of an <strong>interval<\/strong>, or a set of numbers in which a solution falls, are [latex]\\left[-2,6\\right)[\/latex], or all numbers between [latex]-2[\/latex] and [latex]6[\/latex], including [latex]-2[\/latex], but not including [latex]6[\/latex]; [latex]\\left(-1,0\\right)[\/latex], all real numbers between, but not including [latex]-1[\/latex] and [latex]0[\/latex]; and [latex]\\left(-\\infty ,1\\right][\/latex], all real numbers less than and including [latex]1[\/latex]. The table below\u00a0outlines the possibilities.<\/p>\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\n<thead>\n<tr>\n<th>Set Indicated<\/th>\n<th>Set-Builder Notation<\/th>\n<th>Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, but not including <em>a <\/em>or <em>b<\/em><\/td>\n<td>[latex]{x|a<x<b}[\/latex]<\/td>\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\n<td>[latex]\\{x|x>a\\}[\/latex]<\/td>\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\n<td>[latex]\\{x|x<b\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\{x|x\\ge a\\}[\/latex]<\/td>\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\{x|x\\le b\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\{x|a\\le x<b\\}[\/latex]<\/td>\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\{x|a<x\\le b\\}[\/latex]<\/td>\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\n<td>[latex]\\{x|a\\le x\\le b\\}[\/latex]<\/td>\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\n<td>[latex]\\{x|x<a\\text{ and }x>b\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers<\/td>\n<td>[latex]\\{x|x\\text{ is all real numbers}\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example: Using Interval Notation to Express All Real Numbers Greater Than or Equal to a Number<\/h3>\n<p>Use interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q143041\">Show Solution<\/span><\/p>\n<div id=\"q143041\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use interval notation to indicate all real numbers between and including [latex]-3[\/latex] and [latex]5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814810\">Show Solution<\/span><\/p>\n<div id=\"q814810\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left[-3,5\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=58&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92604&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Interval Notation to Express All Real Numbers Less Than or Equal to <em>a <\/em>or Greater Than or Equal to <em>b<\/em><\/h3>\n<p>Write the interval expressing all real numbers less than or equal to [latex]-1[\/latex] or greater than or equal to [latex]1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q797079\">Show Solution<\/span><\/p>\n<div id=\"q797079\" class=\"hidden-answer\" style=\"display: none\">\n<p>We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at [latex]-\\infty[\/latex] and ends at [latex]-1[\/latex], which is written as [latex]\\left(-\\infty ,-1\\right][\/latex].<\/p>\n<p>The second interval must show all real numbers greater than or equal to [latex]1[\/latex], which is written as [latex]\\left[1,\\infty \\right)[\/latex]. However, we want to combine these two sets. We accomplish this by inserting the union symbol, [latex]\\cup[\/latex], between the two intervals.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(-\\infty ,-1\\right]\\cup \\left[1,\\infty \\right)[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Express all real numbers less than [latex]-2[\/latex] or greater than or equal to 3 in interval notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q729196\">Show Solution<\/span><\/p>\n<div id=\"q729196\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\infty ,-2\\right)\\cup \\left[3,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2748&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<h2><\/h2>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>Rational numbers may be written as fractions or terminating or repeating decimals.<\/li>\n<li>Determine whether a number is rational or irrational by writing it as a decimal.<\/li>\n<li>The rational numbers and irrational numbers make up the set of real numbers. A number can be classified as natural, whole, integer, rational, or irrational.<\/li>\n<li>The order of operations is used to evaluate expressions.<\/li>\n<li>The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties.<\/li>\n<li>Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. They take on a numerical value when evaluated by replacing variables with constants.<\/li>\n<li>Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>algebraic expression\u00a0<\/strong>constants and variables combined using addition, subtraction, multiplication, and division<\/p>\n<p><strong>associative property of addition\u00a0<\/strong>the sum of three numbers may be grouped differently without affecting the result; in symbols, [latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/p>\n<p><strong>associative property of multiplication\u00a0<\/strong>the product of three numbers may be grouped differently without affecting the result; in symbols, [latex]a\\cdot \\left(b\\cdot c\\right)=\\left(a\\cdot b\\right)\\cdot c[\/latex]<\/p>\n<p><strong>base\u00a0<\/strong>in exponential notation, the expression that is being multiplied<\/p>\n<p><strong>commutative property of addition\u00a0<\/strong>two numbers may be added in either order without affecting the result; in symbols, [latex]a+b=b+a[\/latex]<\/p>\n<p><strong>commutative property of multiplication\u00a0<\/strong>two numbers may be multiplied in any order without affecting the result; in symbols, [latex]a\\cdot b=b\\cdot a[\/latex]<\/p>\n<p><strong>constant\u00a0<\/strong>a quantity that does not change value<\/p>\n<p><strong>distributive property\u00a0<\/strong>the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, [latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/p>\n<p><strong>equation\u00a0<\/strong>a mathematical statement indicating that two expressions are equal<\/p>\n<p><strong>exponent\u00a0<\/strong>in exponential notation, the raised number or variable that indicates how many times the base is being multiplied<\/p>\n<p><strong>exponential notation\u00a0<\/strong>a shorthand method of writing products of the same factor<\/p>\n<p><strong>formula\u00a0<\/strong>an equation expressing a relationship between constant and variable quantities<\/p>\n<p><strong>identity property of addition\u00a0<\/strong>there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, [latex]a+0=a[\/latex]<\/p>\n<p><strong>identity property of multiplication\u00a0<\/strong>there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, [latex]a\\cdot 1=a[\/latex]<\/p>\n<p><strong>integers\u00a0<\/strong>the set consisting of the natural numbers, their opposites, and 0: [latex]\\{\\dots ,-3,-2,-1,0,1,2,3,\\dots \\}[\/latex]<\/p>\n<p><strong>inverse property of addition\u00a0<\/strong>for every real number [latex]a[\/latex], there is a unique number, called the additive inverse (or opposite), denoted [latex]-a[\/latex], which, when added to the original number, results in the additive identity, 0; in symbols, [latex]a+\\left(-a\\right)=0[\/latex]<\/p>\n<p><strong>inverse property of multiplication\u00a0<\/strong>for every non-zero real number [latex]a[\/latex], there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\dfrac{1}{a}[\/latex], which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, [latex]a\\cdot \\dfrac{1}{a}=1[\/latex]<\/p>\n<p><strong>irrational numbers\u00a0<\/strong>the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers<\/p>\n<p><strong>natural numbers\u00a0<\/strong>the set of counting numbers: [latex]\\{1,2,3,\\dots \\}[\/latex]<\/p>\n<p><strong>order of operations\u00a0<\/strong>a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations<\/p>\n<p><strong>rational numbers\u00a0<\/strong>the set of all numbers of the form [latex]\\dfrac{m}{n}[\/latex], where [latex]m[\/latex] and [latex]n[\/latex] are integers and [latex]n\\ne 0[\/latex]. Any rational number may be written as a fraction or a terminating or repeating decimal.<\/p>\n<p><strong>real number line\u00a0<\/strong>a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.<\/p>\n<p><strong>real numbers\u00a0<\/strong>the sets of rational numbers and irrational numbers taken together<\/p>\n<p><strong>variable\u00a0<\/strong>a quantity that may change value<\/p>\n<p><strong>whole numbers\u00a0<\/strong>the set consisting of 0 plus the natural numbers: [latex]\\{0,1,2,3,\\dots \\}[\/latex]<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1739\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 92383, 109700, 110263, 109667. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 13740. <strong>Authored by<\/strong>: David Lippman. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 13741, 259. <strong>Authored by<\/strong>: James Sousa. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Identifying Sets of Real Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/htP2goe31MM\">https:\/\/youtu.be\/htP2goe31MM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplifying Expressions With Square Roots. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9suc63qB96o\">https:\/\/youtu.be\/9suc63qB96o<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 993379. <strong>Authored by<\/strong>: Desiree Davis. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 92360, 92361, 92388. <strong>Authored by<\/strong>: Michael Jenck. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Evaluating Algebraic Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/MkRdwV4n91g\">https:\/\/youtu.be\/MkRdwV4n91g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 50617. <strong>Authored by<\/strong>: Brenda Gardner. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 483. <strong>Authored by<\/strong>: Jeff Eldridge. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1976, 1980. <strong>Authored by<\/strong>: Lawrence Morales. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 3616. <strong>Authored by<\/strong>: Shawn Triplett. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":708740,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 92383, 109700, 110263, 109667\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 13740\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License, CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 13741, 259\",\"author\":\"James Sousa\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC- BY + GPL\"},{\"type\":\"cc\",\"description\":\"Identifying Sets of Real Numbers\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/htP2goe31MM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Simplifying Expressions With Square Roots\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/9suc63qB96o\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 993379\",\"author\":\"Desiree Davis\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 92360, 92361, 92388\",\"author\":\"Michael Jenck\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Evaluating Algebraic Expressions\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/MkRdwV4n91g\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 50617\",\"author\":\"Brenda Gardner\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 483\",\"author\":\"Jeff Eldridge\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 1976, 1980\",\"author\":\"Lawrence Morales\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 3616\",\"author\":\"Shawn Triplett\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1739","chapter","type-chapter","status-publish","hentry"],"part":1736,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1739","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/users\/708740"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1739\/revisions"}],"predecessor-version":[{"id":2559,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1739\/revisions\/2559"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/parts\/1736"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1739\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/media?parent=1739"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1739"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/contributor?post=1739"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/wp-json\/wp\/v2\/license?post=1739"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}