{"id":362,"date":"2016-06-01T20:49:49","date_gmt":"2016-06-01T20:49:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=362"},"modified":"2016-10-03T20:51:51","modified_gmt":"2016-10-03T20:51:51","slug":"outcome-solve-one-step-linear-equations-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tallahassee-intermediatealgebra\/chapter\/outcome-solve-one-step-linear-equations-2\/","title":{"raw":"Real Numbers","rendered":"Real Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Classify a real number as a natural, whole, integer, rational, or irrational number.<\/li>\r\n \t<li>Define and use the commutative property of addition and multiplication<\/li>\r\n \t<li>Define and use the associative property of addition and multiplication<\/li>\r\n \t<li>Define and use the distributive property<\/li>\r\n \t<li>Define and use the identity property of addition and multiplication<\/li>\r\n \t<li>Define and use the inverse property of addition and multiplication<\/li>\r\n \t<li>Define and identify constants in an algebraic expression<\/li>\r\n \t<li>Evaluate algebraic expressions for different values<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe classes of numbers we will explore include:\r\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Natural numbers<\/span><\/h3>\r\nThe most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. The mathematical symbol for the set of all natural numbers is written as [latex]\\mathbb{N}[\/latex], and sometimes \u00a0[latex]\\mathbb{N_0}[\/latex]\u00a0 or \u00a0[latex]\\mathbb{N_1}[\/latex] when it is necessary to indicate whether the set should start with 0 or 1, respectively.\r\n<h3><span id=\"Integers\" class=\"mw-headline\">Integers<\/span><\/h3>\r\nWhen the set of negative numbers is combined with the set of natural numbers (including\u00a00), the result is defined as the set of integers,\u00a0[latex]\\mathbb{Z}[\/latex]\r\n<h3><span id=\"Rational_numbers\" class=\"mw-headline\">Rational numbers<\/span><\/h3>\r\n<div class=\"hatnote relarticle mainarticle\">A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator.<\/div>\r\n<h3>Real numbers<\/h3>\r\n<div class=\"hatnote relarticle mainarticle\">The real numbers include all the measuring numbers. The symbol for the real numbers is\u00a0 [latex]\\mathbb{R}[\/latex]. Real numbers are usually represented by using decimal numerals.<\/div>\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 [latex]\\{1, 2, 3, ...\\}[\/latex] 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: [latex]\\{0, 1, 2, 3,...\\}[\/latex].\r\n\r\nThe set of <strong>integers<\/strong> adds the opposites of the natural numbers to the set of whole numbers: [latex]\\{...-3, -2, -1, 0, 1, 2, 3,...\\}[\/latex]. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}{\\text{negative integers}}\\hfill &amp; {\\text{zero}}\\hfill &amp; {\\text{positive integers}}\\\\{\\dots ,-3,-2,-1,}\\hfill &amp; {0,}\\hfill &amp; {1,2,3,\\dots }\\end{array}[\/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<\/h3>\r\nWrite each of the following as a rational number.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>7<\/li>\r\n \t<li>0<\/li>\r\n \t<li>[latex]\u20138[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"725771\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"725771\"]\r\n\r\nWrite a fraction with the integer in the numerator and 1 in the denominator.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]7=\\frac{7}{1}[\/latex]<\/li>\r\n \t<li>[latex]0=\\frac{0}{1}[\/latex]<\/li>\r\n \t<li>[latex]-8=-\\frac{8}{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<\/h3>\r\nWrite each of the following rational numbers as either a terminating or repeating decimal.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-\\frac{5}{7}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{15}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{13}{25}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"88918\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"88918\"]\r\n\r\nWrite each fraction as a decimal by dividing the numerator by the denominator.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-\\frac{5}{7}=-0.\\overline{714285}[\/latex], a repeating decimal<\/li>\r\n \t<li>[latex]\\frac{15}{5}=3[\/latex] (or 3.0), a terminating decimal<\/li>\r\n \t<li>[latex]\\frac{13}{25}=0.52[\/latex],\u00a0a 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<\/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 style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\sqrt{25}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{33}{9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{11}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{17}{34}[\/latex]<\/li>\r\n \t<li>[latex]0.3033033303333\\dots[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"644924\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"644924\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\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]\\frac{33}{9}:[\/latex] Because it is a fraction, [latex]\\frac{33}{9}[\/latex] is a rational number. Next, simplify and divide.\r\n<div style=\"text-align: center;\">[latex]\\frac{33}{9}=\\frac{{{11}\\cdot{3}}}{{{3}\\cot{3}}}=\\frac{11}{3}=3.\\overline{6}[\/latex]<\/div>\r\nSo, [latex]\\frac{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]\\frac{17}{34}:[\/latex] Because it is a fraction, [latex]\\frac{17}{34}[\/latex] is a rational number. Simplify and divide.\r\n<div style=\"text-align: center;\">[latex]\\frac{17}{34}=\\frac{{1}{\\overline{)17}}}{\\underset{2}{\\overline{)34}}}=\\frac{1}{2}=0.5[\/latex]<\/div>\r\nSo, [latex]\\frac{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<h2>Real Numbers<\/h2>\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> as shown in Figure 1.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200208\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" data-media-type=\"image\/jpg\" \/> <b>Figure 1.<\/b> The real number line[\/caption]\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/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 style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-\\frac{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]0.615384615384\\dots[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"303752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"303752\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-\\frac{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]0.615384615384\\dots [\/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Sets of Numbers as Subsets<\/h2>\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&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200210\/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\" data-media-type=\"image\/jpg\" \/> <b>Figure 2.<\/b> 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<\/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]\\sqrt{36}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{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<\/ol>\r\n[reveal-answer q=\"400826\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"400826\"]\r\n<table style=\"width: 20%;\" summary=\"A table with six rows and six columns. The first entry in the first row is blank, but the rest of the entries read: N, W, I, Q, and Q'. (These are the sets of numbers from before.) The first entry in the second row reads: square root of thirty-six equals six. Then the second, third, fourth, and fifth columns are marked. The first entry in the third row reads: eight over three equals 2.6 with the 6 repeating forever. Then only the fifth column is marked. The first entry in the fourth row reads: square root of seventy-three. Then only the sixth column is marked. The first entry in the fifth row reads: negative six. Then the fourth and fifth columns are marked. The first entry in the sixth row reads: 3.2121121112\u2026. Then only the sixth column is marked.\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th><em>N<\/em><\/th>\r\n<th><em>W<\/em><\/th>\r\n<th><em>I<\/em><\/th>\r\n<th><em>Q<\/em><\/th>\r\n<th><em>Q'<\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1. [latex]\\sqrt{36}=6[\/latex]<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. [latex]\\frac{8}{3}=2.\\overline{6}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>X<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3. [latex]\\sqrt{73}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. [latex]\u20136[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>X<\/td>\r\n<td>X<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5. [latex]3.2121121112\\dots[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>X<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nhttps:\/\/youtu.be\/htP2goe31MM\r\n<h2>Use Properties of Real Numbers<\/h2>\r\nFor some activities we perform, the order of certain processes\u00a0does not matter, but the order of others do. 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 addition and multiplication.\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 class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nShow that numbers may be added in any order without affecting the sum. [latex]\\left(-2\\right)+7=5[\/latex]\r\n[reveal-answer q=\"279824\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"279824\"]\r\n\r\n[latex]7+\\left(-2\\right)=5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/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 class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nShow that numbers may be multiplied\u00a0in any order without affecting the product.[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44[latex]\r\n[reveal-answer q=\"112050\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"112050\"][latex]\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\" wp-image-980 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183526\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"61\" height=\"55\" \/>\r\n\r\nCaution! 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].\r\n\r\n<\/div>\r\n<h3>Associative Properties - Grouping<\/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 class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nShow that you can regroup numbers that are multiplied together and not affect the product.[latex]\\left(3\\cdot4\\right)\\cdot5=60[\/latex]\r\n[reveal-answer q=\"786302\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"786302\"][latex]3\\cdot\\left(4\\cdot5\\right)=60[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div style=\"text-align: left;\">The <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/div>\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 class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nShow that regrouping addition does not affect the sum. [latex][15+\\left(-9\\right)]+23=29[\/latex]\r\n[reveal-answer q=\"898684\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"898684\"][latex]15+[\\left(-9\\right)+23]=29[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nAre subtraction and division associative? Review these examples.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the associative property to explore whether subtraction and division are associative.\r\n\r\n1)[latex]8-\\left(3-15\\right)\\stackrel{?}{=}\\left(8-3\\right)-15[\/latex]\r\n\r\n2)[latex]64\\div\\left(8\\div4\\right)\\stackrel{?}{=}\\left(64\\div8\\right)\\div4[\/latex]\r\n[reveal-answer q=\"515666\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"515666\"]\r\n\r\n1)[latex]\\begin{array}{r}8-\\left(3-15\\right)\\stackrel{?}{=}\\left(8-3\\right)-15\\\\ 8-\\left(-12\\right)=5-15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\\\ 20\\neq-10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]\r\n\r\n2)[latex]\\begin{array}{r}64\\div\\left(8\\div4\\right)\\stackrel{?}{=}\\left(64\\div8\\right)\\div4\\\\64\\div2\\stackrel{?}{=}8\\div4 \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\ 32\\neq 2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]\r\n\r\nAs we can see, neither subtraction nor division is associative.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the distributive property to show that [latex]4\\cdot[12+(-7)]=20[\/latex]\r\n[reveal-answer q=\"907389\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"907389\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200212\/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. \" data-media-type=\"image\/jpg\" \/>Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by [latex]\u20137[\/latex], and adding the products.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo be more precise when describing this property, we say that multiplication distributes over addition.\r\n\r\nThe reverse is not true, as we can see in this example.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 6+\\left(3\\cdot 5\\right)&amp; \\stackrel{?}{=}&amp; \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ \\hfill 6+\\left(15\\right)&amp; \\stackrel{?}{=}&amp; \\left(9\\right)\\cdot \\left(11\\right)\\hfill \\\\ \\hfill 21&amp; \\ne &amp; \\text{ }99\\hfill \\end{array}[\/latex]<\/p>\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\n<div style=\"text-align: center;\">\r\n\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>[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>[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-1\\right)\\cdot\\left(5+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3]\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRewrite the last example by changing the sign of each term and adding the results.\r\n[reveal-answer q=\"719333\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"719333\"][latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-5-3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right) \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex][\/hidden-answer]\r\n\r\n<\/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.\r\n<div style=\"text-align: center;\"><\/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;\">[latex]a+0=a[\/latex]<\/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\n<div style=\"text-align: center;\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: left;\">Show that the identity property of addition and multiplication are true for [latex]-6, 23[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"587790\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"587790\"]<\/p>\r\n<p style=\"text-align: left;\">[latex]\\left(-6\\right)+0=-6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]23+0=23[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]-6\\cdot1=-6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]23\\cdot 1=23[\/latex]<\/p>\r\n<p style=\"text-align: left;\">There are no exceptions for these properties; they work for every real number, including 0 and 1.<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\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 \\frac{1}{a}=1[\/latex]<\/div>\r\n<div style=\"text-align: center;\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: left;\">1) Define the additive inverse of\u00a0[latex]a=-8[\/latex], and use it to illustrate the inverse property of addition.<\/p>\r\n<p style=\"text-align: left;\">2) Write the reciprocal of\u00a0[latex]a=-\\frac{2}{3}[\/latex], and use it to illustrate the inverse property of multiplication.<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"468875\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"468875\"]<\/p>\r\n<p style=\"text-align: left;\">1) The additive inverse is 8, and\u00a0[latex]\\left(-8\\right)+8=0[\/latex]<\/p>\r\n<p style=\"text-align: left;\">2) The reciprocal is [latex]-\\frac{3}{2}[\/latex]\u00a0and\u00a0[latex]\\left(-\\frac{2}{3}\\right)\\cdot \\left(-\\frac{3}{2}\\right)=1[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/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 style=\"width: 70%;\" 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<tbody>\r\n<tr>\r\n<th><\/th>\r\n<th>Addition<\/th>\r\n<th>Multiplication<\/th>\r\n<\/tr>\r\n<\/tbody>\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<td><\/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 style=\"text-align: center;\">[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 style=\"text-align: center;\">[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 [latex]\u2013a[\/latex], such that\r\n<div style=\"text-align: center;\">[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 style=\"text-align: center;\">[latex]a\\cdot \\left(\\frac{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<\/h3>\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]3\\left(6+4\\right)[\/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]\\frac{4}{7}\\cdot \\left(\\frac{2}{3}\\cdot \\frac{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=\"823624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"823624\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\3\\cdot\\left(6+4\\right)=3\\cdot6+3\\cdot4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Distributive property} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=18+12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=30\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\\\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{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\6-\\left(15+9\\right) \\hfill&amp; =6+[\\left(-15\\right)+\\left(-9\\right)] \\hfill&amp; \\text{Distributive property} \\\\ \\hfill&amp; =6+\\left(-24\\right) \\hfill&amp; \\text{Simplify} \\\\ \\hfill&amp; =-18 \\hfill&amp; \\text{Simplify}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\\\\\\\\\\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) \\hfill&amp; =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) \\hfill&amp; \\text{Commutative property of multiplication} \\\\ \\hfill&amp; =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3}\\hfill&amp; \\text{Associative property of multiplication} \\\\ \\hfill&amp; =1\\cdot\\frac{2}{3} \\hfill&amp; \\text{Inverse property of multiplication} \\\\ \\hfill&amp; =\\frac{2}{3} \\hfill&amp; \\text{Identity property of multiplication}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\100\\cdot[0.75+\\left(-2.38\\right)] \\hfill&amp; =100\\cdot0.75+100\\cdot\\left(-2.38\\right)\\hfill&amp; \\text{Distributive property} \\\\ \\hfill&amp; =75+\\left(-238\\right) \\hfill&amp; \\text{Simplify} \\\\ \\hfill&amp; =-163 \\hfill&amp; \\text{Simplify}\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/8SFm8Os_4C8\r\n<h2>Evaluate Algebraic Expressions<\/h2>\r\nIn 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[\/latex], 5 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{array}{r}\\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\\,\\,\\,\\,\\,\\,\\,\\\\\\text{ }\\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)\\end{array}[\/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\nIn the following example, we will practice identifying constants and variables in mathematical expressions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nList the constants and variables for each algebraic expression.\r\n<ol>\r\n \t<li>[latex]x+5[\/latex]<\/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=\"308507\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"308507\"]\r\n<table summary=\"A table with four rows and three columns. The first entry of the first row reads: expression, 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>Expression<\/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\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. In the next example we show how to substitute various types of numbers into a mathematical expression.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate the expression [latex]2x - 7[\/latex] for each value for <em>x.<\/em>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]x=0[\/latex]<\/li>\r\n \t<li>[latex]x=1[\/latex]<\/li>\r\n \t<li>[latex]x=\\frac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]x=-4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"664833\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"664833\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Substitute 0 for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }2x-7 \\hfill&amp; = 2\\left(0\\right)-7 \\\\ \\hfill&amp; =0-7 \\\\ \\hfill&amp; =-7\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 1 for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }2x-7 \\hfill&amp; = 2\\left(1\\right)-7 \\\\ \\hfill&amp; =2-7 \\\\ \\hfill&amp; =-5\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]\\frac{1}{2}[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }2x-7 \\hfill&amp; = 2\\left(\\frac{1}{2}\\right)-7 \\\\ \\hfill&amp; =1-7 \\\\ \\hfill&amp; =-6\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }2x-7 \\hfill&amp; = 2\\left(-4\\right)-7 \\\\ \\hfill&amp; =-8-7 \\\\ \\hfill&amp; =-15\\end{array}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow we will show more examples of evaluating a variety of mathematical expressions for various values.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate each expression for the given values.\r\n<ol style=\"list-style-type: lower-alpha;\">\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]\\frac{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=\"208163\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"208163\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }x+5\\hfill&amp;=\\left(-5\\right)+5 \\\\ \\hfill&amp;=0\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 10 for [latex]t[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }\\frac{t}{2t-1}\\hfill&amp; =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ \\hfill&amp; =\\frac{10}{20-1} \\\\ \\hfill&amp; =\\frac{10}{19}\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 5 for [latex]r[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }\\frac{4}{3}\\pi r^{3} \\hfill&amp; =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ \\hfill&amp; =\\frac{4}{3}\\pi\\left(125\\right) \\\\ \\hfill&amp; =\\frac{500}{3}\\pi\\end{array}[\/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{array}\\text{ }a+ab+b \\hfill&amp; =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ \\hfill&amp; =11-8-8 \\\\ \\hfill&amp; =-85\\end{array}[\/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{array}\\text{ }\\sqrt{2m^{3}n^{2}} \\hfill&amp; =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ \\hfill&amp; =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ \\hfill&amp; =\\sqrt{144} \\\\ \\hfill&amp; =12\\end{array}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we present more examples of evaluating a variety of expressions for given values.\r\n\r\nhttps:\/\/youtu.be\/MkRdwV4n91g\r\n<h2><\/h2>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Classify a real number as a natural, whole, integer, rational, or irrational number.<\/li>\n<li>Define and use the commutative property of addition and multiplication<\/li>\n<li>Define and use the associative property of addition and multiplication<\/li>\n<li>Define and use the distributive property<\/li>\n<li>Define and use the identity property of addition and multiplication<\/li>\n<li>Define and use the inverse property of addition and multiplication<\/li>\n<li>Define and identify constants in an algebraic expression<\/li>\n<li>Evaluate algebraic expressions for different values<\/li>\n<\/ul>\n<\/div>\n<p>The classes of numbers we will explore include:<\/p>\n<h3><span id=\"Natural_numbers\" class=\"mw-headline\">Natural numbers<\/span><\/h3>\n<p>The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. The mathematical symbol for the set of all natural numbers is written as [latex]\\mathbb{N}[\/latex], and sometimes \u00a0[latex]\\mathbb{N_0}[\/latex]\u00a0 or \u00a0[latex]\\mathbb{N_1}[\/latex] when it is necessary to indicate whether the set should start with 0 or 1, respectively.<\/p>\n<h3><span id=\"Integers\" class=\"mw-headline\">Integers<\/span><\/h3>\n<p>When the set of negative numbers is combined with the set of natural numbers (including\u00a00), the result is defined as the set of integers,\u00a0[latex]\\mathbb{Z}[\/latex]<\/p>\n<h3><span id=\"Rational_numbers\" class=\"mw-headline\">Rational numbers<\/span><\/h3>\n<div class=\"hatnote relarticle mainarticle\">A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator.<\/div>\n<h3>Real numbers<\/h3>\n<div class=\"hatnote relarticle mainarticle\">The real numbers include all the measuring numbers. The symbol for the real numbers is\u00a0 [latex]\\mathbb{R}[\/latex]. Real numbers are usually represented by using decimal numerals.<\/div>\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 [latex]\\{1, 2, 3, ...\\}[\/latex] 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: [latex]\\{0, 1, 2, 3,...\\}[\/latex].<\/p>\n<p>The set of <strong>integers<\/strong> adds the opposites of the natural numbers to the set of whole numbers: [latex]\\{...-3, -2, -1, 0, 1, 2, 3,...\\}[\/latex]. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}{\\text{negative integers}}\\hfill & {\\text{zero}}\\hfill & {\\text{positive integers}}\\\\{\\dots ,-3,-2,-1,}\\hfill & {0,}\\hfill & {1,2,3,\\dots }\\end{array}[\/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<\/h3>\n<p>Write each of the following as a rational number.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>7<\/li>\n<li>0<\/li>\n<li>[latex]\u20138[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725771\">Show Solution<\/span><\/p>\n<div id=\"q725771\" 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 style=\"list-style-type: lower-alpha;\">\n<li>[latex]7=\\frac{7}{1}[\/latex]<\/li>\n<li>[latex]0=\\frac{0}{1}[\/latex]<\/li>\n<li>[latex]-8=-\\frac{8}{1}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write each of the following rational numbers as either a terminating or repeating decimal.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]-\\frac{5}{7}[\/latex]<\/li>\n<li>[latex]\\frac{15}{5}[\/latex]<\/li>\n<li>[latex]\\frac{13}{25}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q88918\">Show Solution<\/span><\/p>\n<div id=\"q88918\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write each fraction as a decimal by dividing the numerator by the denominator.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]-\\frac{5}{7}=-0.\\overline{714285}[\/latex], a repeating decimal<\/li>\n<li>[latex]\\frac{15}{5}=3[\/latex] (or 3.0), a terminating decimal<\/li>\n<li>[latex]\\frac{13}{25}=0.52[\/latex],\u00a0a 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<\/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 style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\sqrt{25}[\/latex]<\/li>\n<li>[latex]\\frac{33}{9}[\/latex]<\/li>\n<li>[latex]\\sqrt{11}[\/latex]<\/li>\n<li>[latex]\\frac{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=\"q644924\">Show Solution<\/span><\/p>\n<div id=\"q644924\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\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]\\frac{33}{9}:[\/latex] Because it is a fraction, [latex]\\frac{33}{9}[\/latex] is a rational number. Next, simplify and divide.\n<div style=\"text-align: center;\">[latex]\\frac{33}{9}=\\frac{{{11}\\cdot{3}}}{{{3}\\cot{3}}}=\\frac{11}{3}=3.\\overline{6}[\/latex]<\/div>\n<p>So, [latex]\\frac{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]\\frac{17}{34}:[\/latex] Because it is a fraction, [latex]\\frac{17}{34}[\/latex] is a rational number. Simplify and divide.\n<div style=\"text-align: center;\">[latex]\\frac{17}{34}=\\frac{{1}{\\overline{)17}}}{\\underset{2}{\\overline{)34}}}=\\frac{1}{2}=0.5[\/latex]<\/div>\n<p>So, [latex]\\frac{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<h2>Real Numbers<\/h2>\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> as shown in Figure 1.<\/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-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200208\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> The real number line<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/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 style=\"list-style-type: lower-alpha;\">\n<li>[latex]-\\frac{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]0.615384615384\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q303752\">Show Solution<\/span><\/p>\n<div id=\"q303752\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]-\\frac{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]0.615384615384\\dots[\/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Sets of Numbers as Subsets<\/h2>\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<p>&nbsp;<\/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-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200210\/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\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2.<\/b> 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<\/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]\\sqrt{36}[\/latex]<\/li>\n<li>[latex]\\frac{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<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q400826\">Show Solution<\/span><\/p>\n<div id=\"q400826\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"width: 20%;\" summary=\"A table with six rows and six columns. The first entry in the first row is blank, but the rest of the entries read: N, W, I, Q, and Q'. (These are the sets of numbers from before.) The first entry in the second row reads: square root of thirty-six equals six. Then the second, third, fourth, and fifth columns are marked. The first entry in the third row reads: eight over three equals 2.6 with the 6 repeating forever. Then only the fifth column is marked. The first entry in the fourth row reads: square root of seventy-three. Then only the sixth column is marked. The first entry in the fifth row reads: negative six. Then the fourth and fifth columns are marked. The first entry in the sixth row reads: 3.2121121112\u2026. Then only the sixth column is marked.\">\n<thead>\n<tr>\n<th><\/th>\n<th><em>N<\/em><\/th>\n<th><em>W<\/em><\/th>\n<th><em>I<\/em><\/th>\n<th><em>Q<\/em><\/th>\n<th><em>Q&#8217;<\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1. [latex]\\sqrt{36}=6[\/latex]<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>2. [latex]\\frac{8}{3}=2.\\overline{6}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>X<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>3. [latex]\\sqrt{73}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>X<\/td>\n<\/tr>\n<tr>\n<td>4. [latex]\u20136[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td>X<\/td>\n<td>X<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>5. [latex]3.2121121112\\dots[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>X<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><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>Use Properties of Real Numbers<\/h2>\n<p>For some activities we perform, the order of certain processes\u00a0does not matter, but the order of others do. 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 addition and multiplication.<\/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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Show that numbers may be added in any order without affecting the sum. [latex]\\left(-2\\right)+7=5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q279824\">Show Answer<\/span><\/p>\n<div id=\"q279824\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]7+\\left(-2\\right)=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Show that numbers may be multiplied\u00a0in any order without affecting the product.[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44[latex]  [reveal-answer q=\"112050\"]Show Answer[\/reveal-answer]  [hidden-answer a=\"112050\"][latex]\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex][\/hidden-answer]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-980 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183526\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"61\" height=\"55\" \/><\/p>\n<p>Caution! 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<\/div>\n<h3>Associative Properties - Grouping<\/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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Show that you can regroup numbers that are multiplied together and not affect the product.[latex]\\left(3\\cdot4\\right)\\cdot5=60[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q786302\">Show Answer<\/span><\/p>\n<div id=\"q786302\" class=\"hidden-answer\" style=\"display: none\">[latex]3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"text-align: left;\">The <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/div>\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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Show that regrouping addition does not affect the sum. [latex][15+\\left(-9\\right)]+23=29[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q898684\">Show Answer<\/span><\/p>\n<div id=\"q898684\" class=\"hidden-answer\" style=\"display: none\">[latex]15+[\\left(-9\\right)+23]=29[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>Are subtraction and division associative? Review these examples.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the associative property to explore whether subtraction and division are associative.<\/p>\n<p>1)[latex]8-\\left(3-15\\right)\\stackrel{?}{=}\\left(8-3\\right)-15[\/latex]<\/p>\n<p>2)[latex]64\\div\\left(8\\div4\\right)\\stackrel{?}{=}\\left(64\\div8\\right)\\div4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q515666\">Show Answer<\/span><\/p>\n<div id=\"q515666\" class=\"hidden-answer\" style=\"display: none\">\n<p>1)[latex]\\begin{array}{r}8-\\left(3-15\\right)\\stackrel{?}{=}\\left(8-3\\right)-15\\\\ 8-\\left(-12\\right)=5-15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\\\ 20\\neq-10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>2)[latex]\\begin{array}{r}64\\div\\left(8\\div4\\right)\\stackrel{?}{=}\\left(64\\div8\\right)\\div4\\\\64\\div2\\stackrel{?}{=}8\\div4 \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\ 32\\neq 2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>As we can see, neither subtraction nor division is associative.<\/p>\n<\/div>\n<\/div>\n<\/div>\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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the distributive property to show that [latex]4\\cdot[12+(-7)]=20[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q907389\">Show Answer<\/span><\/p>\n<div id=\"q907389\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200212\/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.\" data-media-type=\"image\/jpg\" \/>Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by [latex]\u20137[\/latex], and adding the products.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To be more precise when describing this property, we say that multiplication distributes over addition.<\/p>\n<p>The reverse is not true, as we can see in this example.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 6+\\left(3\\cdot 5\\right)& \\stackrel{?}{=}& \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ \\hfill 6+\\left(15\\right)& \\stackrel{?}{=}& \\left(9\\right)\\cdot \\left(11\\right)\\hfill \\\\ \\hfill 21& \\ne & \\text{ }99\\hfill \\end{array}[\/latex]<\/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<div style=\"text-align: center;\">\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>[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>[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-1\\right)\\cdot\\left(5+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3]\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Rewrite the last example by changing the sign of each term and adding the results.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q719333\">Show Answer<\/span><\/p>\n<div id=\"q719333\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-5-3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right) \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/div>\n<\/div>\n<\/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.<\/p>\n<div style=\"text-align: center;\"><\/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;\">[latex]a+0=a[\/latex]<\/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<div style=\"text-align: center;\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p style=\"text-align: left;\">Show that the identity property of addition and multiplication are true for [latex]-6, 23[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q587790\">Show Answer<\/span><\/p>\n<div id=\"q587790\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]\\left(-6\\right)+0=-6[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]23+0=23[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]-6\\cdot1=-6[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]23\\cdot 1=23[\/latex]<\/p>\n<p style=\"text-align: left;\">There are no exceptions for these properties; they work for every real number, including 0 and 1.<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\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 \\frac{1}{a}=1[\/latex]<\/div>\n<div style=\"text-align: center;\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p style=\"text-align: left;\">1) Define the additive inverse of\u00a0[latex]a=-8[\/latex], and use it to illustrate the inverse property of addition.<\/p>\n<p style=\"text-align: left;\">2) Write the reciprocal of\u00a0[latex]a=-\\frac{2}{3}[\/latex], and use it to illustrate the inverse property of multiplication.<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q468875\">Show Answer<\/span><\/p>\n<div id=\"q468875\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">1) The additive inverse is 8, and\u00a0[latex]\\left(-8\\right)+8=0[\/latex]<\/p>\n<p style=\"text-align: left;\">2) The reciprocal is [latex]-\\frac{3}{2}[\/latex]\u00a0and\u00a0[latex]\\left(-\\frac{2}{3}\\right)\\cdot \\left(-\\frac{3}{2}\\right)=1[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/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 style=\"width: 70%;\" 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<tbody>\n<tr>\n<th><\/th>\n<th>Addition<\/th>\n<th>Multiplication<\/th>\n<\/tr>\n<\/tbody>\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<td><\/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 style=\"text-align: center;\">[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 style=\"text-align: center;\">[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 [latex]\u2013a[\/latex], such that<\/p>\n<div style=\"text-align: center;\">[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 style=\"text-align: center;\">[latex]a\\cdot \\left(\\frac{1}{a}\\right)=1[\/latex]<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]3\\left(6+4\\right)[\/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]\\frac{4}{7}\\cdot \\left(\\frac{2}{3}\\cdot \\frac{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=\"q823624\">Show Solution<\/span><\/p>\n<div id=\"q823624\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\begin{array}{l}\\\\\\\\3\\cdot\\left(6+4\\right)=3\\cdot6+3\\cdot4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Distributive property} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=18+12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=30\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\\\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{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\6-\\left(15+9\\right) \\hfill& =6+[\\left(-15\\right)+\\left(-9\\right)] \\hfill& \\text{Distributive property} \\\\ \\hfill& =6+\\left(-24\\right) \\hfill& \\text{Simplify} \\\\ \\hfill& =-18 \\hfill& \\text{Simplify}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\\\\\\\\\\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) \\hfill& =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) \\hfill& \\text{Commutative property of multiplication} \\\\ \\hfill& =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3}\\hfill& \\text{Associative property of multiplication} \\\\ \\hfill& =1\\cdot\\frac{2}{3} \\hfill& \\text{Inverse property of multiplication} \\\\ \\hfill& =\\frac{2}{3} \\hfill& \\text{Identity property of multiplication}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\100\\cdot[0.75+\\left(-2.38\\right)] \\hfill& =100\\cdot0.75+100\\cdot\\left(-2.38\\right)\\hfill& \\text{Distributive property} \\\\ \\hfill& =75+\\left(-238\\right) \\hfill& \\text{Simplify} \\\\ \\hfill& =-163 \\hfill& \\text{Simplify}\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Properties of Real Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8SFm8Os_4C8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Evaluate Algebraic Expressions<\/h2>\n<p>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[\/latex], 5 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{array}{r}\\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\\,\\,\\,\\,\\,\\,\\,\\\\\\text{ }\\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)\\end{array}[\/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.<br \/>\nIn the following example, we will practice identifying constants and variables in mathematical expressions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>List the constants and variables for each algebraic expression.<\/p>\n<ol>\n<li>[latex]x+5[\/latex]<\/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=\"q308507\">Show Solution<\/span><\/p>\n<div id=\"q308507\" class=\"hidden-answer\" style=\"display: none\">\n<table summary=\"A table with four rows and three columns. The first entry of the first row reads: expression, 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>Expression<\/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<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. In the next example we show how to substitute various types of numbers into a mathematical expression.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate the expression [latex]2x - 7[\/latex] for each value for <em>x.<\/em><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]x=0[\/latex]<\/li>\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=\\frac{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=\"q664833\">Show Solution<\/span><\/p>\n<div id=\"q664833\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Substitute 0 for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }2x-7 \\hfill& = 2\\left(0\\right)-7 \\\\ \\hfill& =0-7 \\\\ \\hfill& =-7\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute 1 for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }2x-7 \\hfill& = 2\\left(1\\right)-7 \\\\ \\hfill& =2-7 \\\\ \\hfill& =-5\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]\\frac{1}{2}[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }2x-7 \\hfill& = 2\\left(\\frac{1}{2}\\right)-7 \\\\ \\hfill& =1-7 \\\\ \\hfill& =-6\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }2x-7 \\hfill& = 2\\left(-4\\right)-7 \\\\ \\hfill& =-8-7 \\\\ \\hfill& =-15\\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Now we will show more examples of evaluating a variety of mathematical expressions for various values.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate each expression for the given values.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\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]\\frac{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=\"q208163\">Show Solution<\/span><\/p>\n<div id=\"q208163\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }x+5\\hfill&=\\left(-5\\right)+5 \\\\ \\hfill&=0\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute 10 for [latex]t[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }\\frac{t}{2t-1}\\hfill& =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ \\hfill& =\\frac{10}{20-1} \\\\ \\hfill& =\\frac{10}{19}\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute 5 for [latex]r[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}\\text{ }\\frac{4}{3}\\pi r^{3} \\hfill& =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ \\hfill& =\\frac{4}{3}\\pi\\left(125\\right) \\\\ \\hfill& =\\frac{500}{3}\\pi\\end{array}[\/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{array}\\text{ }a+ab+b \\hfill& =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ \\hfill& =11-8-8 \\\\ \\hfill& =-85\\end{array}[\/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{array}\\text{ }\\sqrt{2m^{3}n^{2}} \\hfill& =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ \\hfill& =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ \\hfill& =\\sqrt{144} \\\\ \\hfill& =12\\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we present more examples of evaluating a variety of expressions for given values.<\/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><\/h2>\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-362\">\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><li>Properties of Real Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8SFm8Os_4C8\">https:\/\/youtu.be\/8SFm8Os_4C8<\/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>Evaluate Various Algebraic Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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>College Algebra: Using Properties of Real Numbers. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/chapter\/using-properties-of-real-numbers\">https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/chapter\/using-properties-of-real-numbers<\/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>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>College Algebra: Evaluating Algebraic Expressions. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/chapter\/using-properties-of-real-numbers\">https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/chapter\/using-properties-of-real-numbers<\/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 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