{"id":4736,"date":"2017-12-26T21:59:53","date_gmt":"2017-12-26T21:59:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/?post_type=chapter&#038;p=4736"},"modified":"2017-12-27T16:47:24","modified_gmt":"2017-12-27T16:47:24","slug":"real-numbers-review","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/real-numbers-review\/","title":{"raw":"Real Numbers Review","rendered":"Real Numbers Review"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Real numbers<\/li>\r\n \t<li>Add and subtract real numbers\r\n<ul>\r\n \t<li>Add\u00a0real numbers with the same and different signs<\/li>\r\n \t<li>Subtract real numbers with the same and different signs<\/li>\r\n \t<li>Simplify combinations that require both addition and subtraction of real numbers.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Multiply and divide real numbers\r\n<ul>\r\n \t<li>Multiply two or more real numbers.<\/li>\r\n \t<li>Divide real numbers<\/li>\r\n \t<li>Simplify expressions with both multiplication and division<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Properties of real numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\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>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223810\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" \/> The real number line[\/caption]\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Classifying Real Numbers<\/h3>\r\nClassify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?\r\n<ol>\r\n \t<li>[latex]-\\frac{10}{3}[\/latex]<\/li>\r\n \t<li>[latex]-6\\pi[\/latex]<\/li>\r\n \t<li>[latex]0.616161\\dots[\/latex]<\/li>\r\n \t<li>[latex] 0.13 [\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"705558\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"705558\"]\r\n<ol>\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]-6\\pi [\/latex] is negative and irrational. It lies to the left of 0.<\/li>\r\n \t<li>[latex]0.616161\\dots [\/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.<\/li>\r\n \t<li>[latex] 0.13 [\/latex] is a finite decimal and may be written as 13\/100. \u00a0So it is rational and positive.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nClassify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?\r\n<ol>\r\n \t<li>[latex]2\\pi[\/latex]<\/li>\r\n \t<li>[latex]-11.411411411\\dots [\/latex]<\/li>\r\n \t<li>[latex]\\frac{47}{19}[\/latex]<\/li>\r\n \t<li>[latex]6.210735[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"155954\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"155954\"]\r\n<ol>\r\n \t<li>positive, irrational; right<\/li>\r\n \t<li>negative, rational; left<\/li>\r\n \t<li>positive, rational; right<\/li>\r\n \t<li>positive, rational; right<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Sets of Numbers as Subsets<\/h3>\r\nBeginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223813\/CNX_CAT_Figure_01_01_001.jpg\" alt=\"A large box labeled: Real Numbers encloses five circles. Four of these circles enclose each other and the other is separate from the rest. The innermost circle contains: 1, 2, 3\u2026 N. The circle enclosing that circle contains: 0 W. The circle enclosing that circle contains: \u2026, -3, -2, -1 I. The outermost circle contains: m\/n, n not equal to zero Q. The separate circle contains: pi, square root of two, etc Q\u00b4.\" width=\"731\" height=\"352\" \/> Sets of numbers. \u00a0 <em>N<\/em>: the set of natural numbers \u00a0 <em>W<\/em>: the set of whole numbers \u00a0 <em>I<\/em>: the set of integers \u00a0 <em>Q<\/em>: the set of rational numbers \u00a0 <em>Q\u00b4<\/em>: the set of irrational numbers[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Sets of Numbers<\/h3>\r\nThe set of <strong>natural numbers<\/strong> includes the numbers used for counting: [latex]\\{1,2,3,\\dots\\}[\/latex].\r\n\r\nThe set of <strong>whole numbers<\/strong> is the set of natural numbers plus zero: [latex]\\{0,1,2,3,\\dots\\}[\/latex].\r\n\r\nThe set of <strong>integers<\/strong> adds the negative natural numbers to the set of whole numbers: [latex]\\{\\dots,-3,-2,-1,0,1,2,3,\\dots\\}[\/latex].\r\n\r\nThe set of <strong>rational numbers<\/strong> includes fractions written as [latex]\\{\\frac{m}{n}|m\\text{ and }n\\text{ are integers and }n\\ne 0\\}[\/latex].\r\n\r\nThe set of <strong>irrational numbers<\/strong> is the set of numbers that are not rational, are nonrepeating, and are nonterminating: [latex]\\{h|h\\text{ is not a rational number}\\}[\/latex].\r\n\r\n<\/div>\r\nThe ability to work comfortably with negative numbers is essential to success in algebra. For this reason we will do a quick review of adding, subtracting, multiplying and dividing integers. Integers are all the positive whole numbers, zero, and their opposites (negatives). As this is intended to be a review of integers, the descriptions and examples will not be as detailed as a normal lesson.\r\n<h2>Adding and Subtracting Real Numbers<\/h2>\r\nWhen adding integers we have two cases to consider. The first case is whether\u00a0the signs match (both positive or both negative). If the signs match, we will add the numbers together and keep the sign.\r\n\r\nIf the signs don\u2019t match (one positive and one negative number) we will subtract the numbers (as if they were all positive) and then use the sign from the larger number. This means if the larger number is positive, the answer is positive. If the larger number is negative, the answer is negative.\r\n<div class=\"textbox shaded\">\r\n<h3>To add two numbers with the same sign (both positive or both negative)<\/h3>\r\n<ul>\r\n \t<li><i>Add<\/i> their absolute values (without the [latex]+[\/latex] or [latex]-[\/latex] sign)<\/li>\r\n \t<li>Give the sum the same sign.<\/li>\r\n<\/ul>\r\n<h3>To add two numbers with different signs (one positive and one negative)<\/h3>\r\n<ul>\r\n \t<li>Find the<i> difference <\/i>of<i> <\/i>their absolute values. (Note that when you find the difference of the absolute values, you always subtract the lesser absolute value from the greater one.)<\/li>\r\n \t<li>Give the sum the same sign as the number with the greater absolute value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind\u00a0[latex]23\u201373[\/latex].\r\n\r\n[reveal-answer q=\"951238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951238\"]You can't use your usual method of subtraction because 73 is greater than 23.\u00a0Rewrite the subtraction as adding the opposite.\r\n<p style=\"text-align: center\">[latex]23+\\left(\u221273\\right)[\/latex]<\/p>\r\nThe addends have different signs, so find the difference of their absolute values.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left|23\\right|=23\\,\\,\\,\\text{and}\\,\\,\\,\\left|\u221273\\right|=73\\\\73-23=50\\end{array}[\/latex]<\/p>\r\nSince [latex]\\left|\u221273\\right|&gt;\\left|23\\right|[\/latex], the final answer is negative.\r\n<h4>Answer<\/h4>\r\n[latex]23\u201373=\u221250[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nAnother way to think about subtracting is to think about the distance between the two numbers on the number line. In the example below, [latex]382[\/latex] is to the <i>right<\/i> of 0 by [latex]382[\/latex] units, and [latex]\u221293[\/latex] is to the <i>left<\/i> of 0 by 93 units. The distance between them is the sum of their distances to 0: [latex]382+93[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170754\/image050.gif\" alt=\"A number line from negative 93 to 382. Negative 93 is 93 units from 0 and 382 is 382 units from 0. The total distance from negative 93 to 382 can be found by adding them together. 382+93=475 units.\" width=\"575\" height=\"147\" \/>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind\u00a0[latex]382\u2013\\left(\u221293\\right)[\/latex].\r\n\r\n[reveal-answer q=\"342295\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"342295\"]You are subtracting a negative, so think of this as taking the negative sign away. This becomes an addition problem. [latex]-93[\/latex] becomes [latex]+93[\/latex]\r\n<p style=\"text-align: center\">[latex]382+93=475[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]382\u2013(\u221293)=475[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video explains how to subtract two signed integers.\r\n\r\nhttps:\/\/youtu.be\/ciuIKFCtWWU\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex]-\\frac{3}{7}-\\frac{6}{7}+\\frac{2}{7}[\/latex]\r\n\r\n[reveal-answer q=\"11416\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"11416\"]Add the first two and give the result a negative sign:\r\n\r\nSince the signs of the first two are the same, find the sum of the absolute values of the fractions\r\n\r\nSince both numbers are negative, the sum is negative. If you owe money, then borrow more, the amount you owe becomes larger.\r\n<p style=\"text-align: center\">[latex]\\left| -\\frac{3}{7} \\right|=\\frac{3}{7}[\/latex] and [latex]\\left| -\\frac{6}{7} \\right|=\\frac{6}{7}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{3}{7}+\\frac{6}{7}=\\frac{9}{7}\\\\\\\\-\\frac{3}{7}-\\frac{6}{7} =-\\frac{9}{7}\\end{array}[\/latex]<\/p>\r\nNow add the third number. The signs are different, so find the <em>difference<\/em> of their absolute values.\r\n<p style=\"text-align: center\">[latex] \\left| -\\frac{9}{7} \\right|=\\frac{9}{7}[\/latex] and [latex] \\left| \\frac{2}{7} \\right|=\\frac{2}{7}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex] \\frac{9}{7}-\\frac{2}{7}=\\frac{7}{7}[\/latex]<\/p>\r\nSince [latex]\\left|\\frac{-9}{7}\\right|&gt;\\left|\\frac{2}{7}\\right|[\/latex], the sign of the final sum is the same as the sign of [latex]-\\frac{9}{7}[\/latex].<i>\r\n<\/i>\r\n<p style=\"text-align: center\">[latex] -\\frac{9}{7}+\\frac{2}{7}=-\\frac{7}{7}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]-\\frac{3}{7}+\\left(-\\frac{6}{7}\\right)+\\frac{2}{7}=-\\frac{7}{7}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see an example of how to add three fractions with a common denominator that have different signs.\r\n\r\nhttps:\/\/youtu.be\/P972VVbR98k\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]27.832+(\u22123.06)[\/latex]. When you add decimals, remember to line up the decimal points so you are adding tenths to tenths, hundredths to hundredths, and so on.\r\n\r\n[reveal-answer q=\"545871\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"545871\"]Since the addends have different signs, subtract their absolute values.\r\n<p style=\"text-align: center\">[latex] \\begin{array}{r}\\underline{\\begin{array}{r}27.832\\\\-\\text{ }3.06\\,\\,\\,\\end{array}}\\\\24.772\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\left|-3.06\\right|=3.06[\/latex]<\/p>\r\nThe sum has the same sign as 27.832 whose absolute value is greater.\r\n<h4>Answer<\/h4>\r\n[latex]27.832+\\left(-3.06\\right)=24.772[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video are\u00a0examples of adding and subtracting decimals with different signs.\r\n\r\nhttps:\/\/youtu.be\/3FHZQ5iKcpI\r\n<h2>Multiplying and Dividing Real Numbers<\/h2>\r\nMultiplication and division are <strong>inverse operations<\/strong>, just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal. Inverse operations \"undo\" each other.\r\n<h2>Multiply Real Numbers<\/h2>\r\nMultiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven't learned what effect a negative sign has on the product.\r\n\r\nWith whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product [latex]3\\cdot4[\/latex] as 3 jumps of 4 units each.\r\n\r\n<img id=\"Picture 262\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170800\/image054.gif\" alt=\"A number line showing 3 times 4 is 12. From the 0, the right-facing person jumps 4 units at a time, and jumps 3 times. The person lands on 12.\" width=\"521\" height=\"130\" \/>\r\n\r\nSo to multiply [latex]3(\u22124)[\/latex], you can face left (toward the negative side) and make three \u201cjumps\u201d forward (in a negative direction).\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170801\/image055.jpg\" alt=\"A number line representing 3 times negative 4 equals negative 12. A left-facing person jumps left 4 spaces 3 times so that the person lands on negative 12.\" width=\"516\" height=\"136\" \/>\r\n\r\nThe product of a positive number and a negative number (or a negative and a positive) is negative.\r\n<div class=\"textbox shaded\">\r\n<h3>The Product of a Positive Number and a Negative Number<\/h3>\r\nTo multiply a <strong>positive number<\/strong> and a <strong>negative number<\/strong>, multiply their absolute values. The product is <strong>negative<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex]\u22123.8(0.6)[\/latex].\r\n[reveal-answer q=\"456211\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"456211\"]Multiply the absolute values as you normally would.\u00a0Place the decimal point by counting place values.\u00a03.8 has 1 place after the decimal point, and 0.6 has 1 place after the decimal point, so the product has [latex]1+1[\/latex] or 2 places after the decimal point.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3.8\\\\\\underline{\\times\\,\\,\\,0.6}\\\\2.28\\end{array}[\/latex]<\/p>\r\nThe product of a negative and a positive is negative.\r\n<h4>Answer<\/h4>\r\n[latex]\u22123.8(0.6)=\u22122.28[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video contains examples of how to multiply decimal numbers with different signs.\r\n\r\nhttps:\/\/youtu.be\/7gY0S3LUUyQ\r\n<div class=\"textbox shaded\">\r\n<h3>The Product of Two Numbers with the Same Sign (both positive or both negative)<\/h3>\r\nTo multiply two <strong>positive numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.\r\n\r\nTo multiply two <strong>negative numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex] ~\\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)[\/latex]\r\n\r\n[reveal-answer q=\"322816\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"322816\"]Multiply the absolute values of the numbers.\u00a0First, multiply the numerators together to get the product's numerator. Then, multiply the denominators together to get the product's denominator. Rewrite in lowest terms, if needed.\r\n<p style=\"text-align: center\">[latex] \\left( \\frac{3}{4} \\right)\\left( \\frac{2}{5} \\right)=\\frac{6}{20}=\\frac{3}{10}[\/latex]<\/p>\r\nThe product of two negative numbers is positive.\r\n<h4>Answer<\/h4>\r\n[latex] \\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)=\\frac{3}{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows examples of multiplying two signed fractions, including simplification of the answer.\r\n\r\nhttps:\/\/youtu.be\/yUdJ46pTblo\r\n\r\nTo summarize:\r\n<ul>\r\n \t<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\r\n \t<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\r\n \t<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\r\n \t<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\r\n<\/ul>\r\nYou can see that the product of two negative numbers is a positive number. So, if you are multiplying more than two numbers, you can count the number of negative factors.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying More Than Two Negative Numbers<\/h3>\r\nIf there are an <strong>even<\/strong> number (0, 2, 4, ...) of negative factors to multiply, the product is <strong>positive<\/strong>.\r\nIf there are an <strong>odd<\/strong> number (1, 3, 5, ...) of negative factors, the product is <strong>negative<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex].\r\n\r\n[reveal-answer q=\"149062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149062\"]Multiply the absolute values of the numbers.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3(6)(2)(3)(1)\\\\18(2)(3)(1)\\\\36(3)(1)\\\\108(1)\\\\108\\end{array}[\/latex]<\/p>\r\nCount the number of negative factors. There are three [latex]\\left(\u22126,\u22123,\u22121\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex]<\/p>\r\nSince there are an odd number of negative factors, the product is negative.\r\n<h4>Answer<\/h4>\r\n[latex]3(\u22126)(2)(\u22123)(\u22121)=\u2212108[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video contains examples of multiplying more than two signed integers.\r\n\r\nhttps:\/\/youtu.be\/rx8F9SPd0HE\r\n<h2>Divide Real Numbers<\/h2>\r\nYou may remember that when you divided fractions, you multiplied by the <strong>reciprocal<\/strong>. <i>Reciprocal <\/i>is another name for the multiplicative inverse (just as <i>opposite <\/i>is another name for additive inverse).\r\n\r\nAn easy way to find the multiplicative inverse is to just \u201cflip\u201d the numerator and denominator as you did to find the reciprocal. Here are some examples:\r\n<ul>\r\n \t<li>The reciprocal of [latex]\\frac{4}{9}[\/latex]\u00a0is [latex] \\frac{9}{4}[\/latex]because [latex]\\frac{4}{9}\\left(\\frac{9}{4}\\right)=\\frac{36}{36}=1[\/latex].<\/li>\r\n \t<li>The reciprocal of 3 is [latex]\\frac{1}{3}[\/latex]\u00a0because [latex]\\frac{3}{1}\\left(\\frac{1}{3}\\right)=\\frac{3}{3}=1[\/latex].<\/li>\r\n \t<li>The reciprocal of [latex]-\\frac{5}{6}[\/latex]\u00a0is [latex]\\frac{-6}{5}[\/latex]\u00a0because [latex]-\\frac{5}{6}\\left( -\\frac{6}{5} \\right)=\\frac{30}{30}=1[\/latex].<\/li>\r\n \t<li>The reciprocal of 1 is 1 as [latex]1(1)=1[\/latex].<\/li>\r\n<\/ul>\r\nWhen you divided by positive fractions, you learned to multiply by the reciprocal. You also do this to divide <strong>real numbers<\/strong>.\r\n\r\nThink about dividing a bag of 26 marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has <i>one half<\/i> of the marbles.\r\n<p style=\"text-align: center\">[latex] 26\\div 2=26\\left( \\frac{1}{2} \\right)=13[\/latex]<\/p>\r\nNotice that 2 and [latex] \\frac{1}{2}[\/latex] are reciprocals.\r\n\r\nTry again, dividing a bag of 36 marbles into smaller bags.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Number of bags<\/th>\r\n<th>Dividing by number of bags<\/th>\r\n<th>Multiplying by reciprocal<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<thead><\/thead>\r\n<tbody>\r\n<tr>\r\n<td>3<\/td>\r\n<td>[latex]\\frac{36}{3}=12[\/latex]<\/td>\r\n<td>[latex] 36\\left( \\frac{1}{3} \\right)=\\frac{36}{3}=\\frac{12(3)}{3}=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>[latex]\\frac{36}{4}=9[\/latex]<\/td>\r\n<td>[latex]36\\left(\\frac{1}{4}\\right)=\\frac{36}{4}=\\frac{9\\left(4\\right)}{4}=9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td>[latex]\\frac{36}{6}=6[\/latex]<\/td>\r\n<td>[latex]36\\left(\\frac{1}{6}\\right)=\\frac{36}{6}=\\frac{6\\left(6\\right)}{6}=6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDividing by a number is the same as multiplying by its reciprocal. (That is, you use the reciprocal of the <strong>divisor<\/strong>, the second number in the division problem.)\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex] 28\\div \\frac{4}{3}[\/latex]\r\n\r\n[reveal-answer q=\"210216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"210216\"]Rewrite the division as multiplication by the reciprocal. The reciprocal of [latex] \\frac{4}{3}[\/latex] is [latex]\\frac{3}{4} [\/latex].\r\n<p style=\"text-align: center\">[latex] 28\\div \\frac{4}{3}=28\\left( \\frac{3}{4} \\right)[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center\">[latex]\\frac{28}{1}\\left(\\frac{3}{4}\\right)=\\frac{28\\left(3\\right)}{4}=\\frac{4\\left(7\\right)\\left(3\\right)}{4}=7\\left(3\\right)=21[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]28\\div\\frac{4}{3}=21[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow let's see what this means when one or more of the numbers is negative. A number and its reciprocal have the same sign. Since division is rewritten as multiplication using the reciprocal of the divisor, and taking the reciprocal doesn\u2019t change any of the signs, division follows the same rules as multiplication.\r\n<div class=\"textbox shaded\">\r\n<h3>Rules of Division<\/h3>\r\nWhen dividing, rewrite the problem as multiplication using the reciprocal of the divisor as the second factor.\r\n\r\nWhen one number is <strong>positive<\/strong> and the other is <strong>negative<\/strong>, the <strong>quotient<\/strong> is <strong>negative<\/strong>.\r\n\r\nWhen <em>both<\/em> numbers are <strong>negative<\/strong>, the quotient is <strong>positive<\/strong>.\r\n\r\nWhen <em>both<\/em> numbers are <strong>positive<\/strong>, the quotient is <strong>positive<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex]24\\div\\left(-\\frac{5}{6}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"716581\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"716581\"]Rewrite the division as multiplication by the reciprocal.\r\n<p style=\"text-align: center\">[latex] 24\\div \\left( -\\frac{5}{6} \\right)=24\\left( -\\frac{6}{5} \\right)[\/latex]<\/p>\r\nMultiply. Since one number is positive and one is negative, the product is negative.\r\n<p style=\"text-align: center\">[latex] \\frac{24}{1}\\left( -\\frac{6}{5} \\right)=-\\frac{144}{5}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] 24\\div \\left( -\\frac{5}{6} \\right)=-\\frac{144}{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex] 4\\,\\left( -\\frac{2}{3} \\right)\\,\\div \\left( -6 \\right)[\/latex]\r\n\r\n[reveal-answer q=\"557653\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"557653\"]Rewrite the division as multiplication by the reciprocal.\r\n<p style=\"text-align: center\">[latex] \\frac{4}{1}\\left( -\\frac{2}{3} \\right)\\left( -\\frac{1}{6} \\right)[\/latex]<\/p>\r\nMultiply. There is an even number of negative numbers, so the product is positive.\r\n<p style=\"text-align: center\">[latex]\\frac{4\\left(2\\right)\\left(1\\right)}{3\\left(6\\right)}=\\frac{8}{18}[\/latex]<\/p>\r\nWrite the fraction in lowest terms.\r\n<h4>Answer<\/h4>\r\n[latex] 4\\left( -\\frac{2}{3} \\right)\\div \\left( -6 \\right)=\\frac{4}{9}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video explains how to divide signed fractions.\r\n\r\nhttps:\/\/youtu.be\/OPHdadhDJoI\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"103\" height=\"91\" \/>\r\n\r\nRemember that a fraction bar also indicates division, so a negative sign in front of a fraction goes with the numerator, the denominator, or the whole fraction: [latex]-\\frac{3}{4}=\\frac{-3}{4}=\\frac{3}{-4}[\/latex].\r\n\r\nIn each case, the overall fraction is negative because there's only one negative in the division.\r\n\r\n<\/div>\r\n<h2>Properties of Real Numbers<\/h2>\r\n<\/div>\r\nFor some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.\r\n<h3>Commutative Properties<\/h3>\r\nThe <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.\r\n<div style=\"text-align: center\">[latex]a+b=b+a[\/latex]<\/div>\r\nWe can better see this relationship when using real numbers.\r\n<div style=\"text-align: center\">[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/div>\r\nSimilarly, the <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.\r\n<div style=\"text-align: center\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\r\nAgain, consider an example with real numbers.\r\n<div style=\"text-align: center\">[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/div>\r\nIt is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex]. Similarly, [latex]20\\div 5\\ne 5\\div 20[\/latex].\r\n<h3>Associative Properties<\/h3>\r\nThe <strong>associative property of multiplication<\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.\r\n<div style=\"text-align: center\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\r\nConsider this example.\r\n<div style=\"text-align: center\">[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/div>\r\nThe <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.\r\n<div style=\"text-align: center\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\r\nThis property can be especially helpful when dealing with negative integers. Consider this example.\r\n<div style=\"text-align: center\">[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/div>\r\nAre subtraction and division associative? Review these examples.\r\n<div style=\"text-align: center\">[latex]\\begin{array}\\text{ }8-\\left(3-15\\right) \\hfill&amp; \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) \\hfill&amp; =5-15 \\\\ 20 \\hfill&amp; \\neq 20-10 \\\\ \\text{ }\\end{array}[\/latex]<\/div>\r\n<div style=\"text-align: center\">[latex]\\begin{array}\\text{ }64\\div\\left(8\\div4\\right)\\hfill&amp;\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 \\hfill&amp; \\stackrel{?}{=}8\\div4 \\\\ 32 \\hfill&amp; \\neq 2\\end{array}[\/latex]<\/div>\r\nAs we can see, neither subtraction nor division is associative.\r\n<h3>Distributive Property<\/h3>\r\nThe <strong>distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.\r\n<div style=\"text-align: center\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\r\nThis property combines both addition and multiplication (and is the only property to do so). Let us consider an example.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223815\/CNX_CAT_Figure_01_01_003.jpg\" alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.\" \/>\r\n\r\nNote that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by \u20137, and adding the products.\r\n\r\nTo be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.\r\n<div style=\"text-align: center\">[latex]\\begin{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]<\/div>\r\nMultiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.\r\n\r\nA special case of the distributive property occurs when a sum of terms is subtracted.\r\n<div style=\"text-align: center\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\r\nFor example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms 12 and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.\r\n<div style=\"text-align: center\">[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\r\nNow, distribute [latex]-1[\/latex] and simplify the result.\r\n<div style=\"text-align: center\">[latex]\\begin{array}12-\\left(5+3\\right) \\hfill&amp; =12+\\left(-1\\right)\\cdot\\left(5+3\\right) \\\\ \\hfill&amp; =12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\ \\hfill&amp; =12+\\left(-8\\right) \\\\ \\hfill&amp; =4 \\end{array}[\/latex]<\/div>\r\nThis seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.\r\n<div style=\"text-align: center\">[latex]\\begin{array}12-\\left(5+3\\right) \\hfill&amp; =12+\\left(-5-3\\right) \\\\ \\hfill&amp; =12+\\left(-8\\right) \\\\ \\hfill&amp; =4\\end{array}[\/latex]<\/div>\r\n<h3>Identity Properties<\/h3>\r\nThe <strong>identity property of addition<\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.\r\n<div style=\"text-align: center\">[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\nFor example, we have [latex]\\left(-6\\right)+0=-6[\/latex] and [latex]23\\cdot 1=23[\/latex]. There are no exceptions for these properties; they work for every real number, including 0 and 1.\r\n<h3>Inverse Properties<\/h3>\r\nThe <strong>inverse property of addition<\/strong> states that, for every real number <em>a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted\u2212<em>a<\/em>, that, when added to the original number, results in the additive identity, 0.\r\n<div style=\"text-align: center\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\r\nFor example, if [latex]a=-8[\/latex], the additive inverse is 8, since [latex]\\left(-8\\right)+8=0[\/latex].\r\n\r\nThe <strong>inverse property of multiplication<\/strong> holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number <em>a<\/em>, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\frac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, 1.\r\n<div style=\"text-align: center\">[latex]a\\cdot \\frac{1}{a}=1[\/latex]<\/div>\r\nFor example, if [latex]a=-\\frac{2}{3}[\/latex], the reciprocal, denoted [latex]\\frac{1}{a}[\/latex], is [latex]-\\frac{3}{2}[\/latex]\u00a0because\r\n<div style=\"text-align: center\">[latex]a\\cdot \\frac{1}{a}=\\left(-\\frac{2}{3}\\right)\\cdot \\left(-\\frac{3}{2}\\right)=1[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Properties of Real Numbers<\/h3>\r\nThe following properties hold for real numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>.\r\n<table summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Addition<\/th>\r\n<th>Multiplication<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong>Commutative Property<\/strong><\/td>\r\n<td>[latex]a+b=b+a[\/latex]<\/td>\r\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Associative Property<\/strong><\/td>\r\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\r\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Distributive Property<\/strong><\/td>\r\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Identity Property<\/strong><\/td>\r\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em>\r\n<div>[latex]a+0=a[\/latex]<\/div><\/td>\r\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em>\r\n<div>[latex]a\\cdot 1=a[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Inverse Property<\/strong><\/td>\r\n<td>Every real number a has an additive inverse, or opposite, denoted <em>\u2013a<\/em>, such that\r\n<div>[latex]a+\\left(-a\\right)=0[\/latex]<\/div><\/td>\r\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\frac{1}{a}[\/latex], such that\r\n<div>[latex]a\\cdot \\left(\\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: Using Properties of Real Numbers<\/h3>\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n<ol>\r\n \t<li>[latex]3\\cdot 6+3\\cdot 4[\/latex]<\/li>\r\n \t<li>[latex]\\left(5+8\\right)+\\left(-8\\right)[\/latex]<\/li>\r\n \t<li>[latex]6-\\left(15+9\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\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=\"892710\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"892710\"]\r\n<ol>\r\n \t<li>[latex]\\begin{array}\\text{ }3\\cdot6+3\\cdot4 \\hfill&amp; =3\\cdot\\left(6+4\\right) \\hfill&amp; \\text{Distributive property} \\\\ \\hfill&amp; =3\\cdot10 \\hfill&amp; \\text{Simplify} \\\\ \\hfill&amp; =30 \\hfill&amp; \\text{Simplify}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}\\text{ }\\left(5+8\\right)+\\left(-8\\right) \\hfill&amp; =5+\\left[8+\\left(-8\\right)\\right] \\hfill&amp; \\text{Associative property of addition} \\\\ &amp;\\hfill =5+0 \\hfill&amp; \\text{Inverse property of addition} \\\\ \\hfill&amp; =5 \\hfill&amp; \\text{Identity property of addition}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}6-\\left(15+9\\right) \\hfill&amp; =6+[15\\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}\\text{ }\\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}\\text{ }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\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n<ol>\r\n \t<li>[latex]\\left(-\\frac{23}{5}\\right)\\cdot \\left[11\\cdot \\left(-\\frac{5}{23}\\right)\\right][\/latex]<\/li>\r\n \t<li>[latex]5\\cdot \\left(6.2+0.4\\right)[\/latex]<\/li>\r\n \t<li>[latex]18-\\left(7 - 15\\right)[\/latex]<\/li>\r\n \t<li>[latex]6\\cdot \\left(-3\\right)+6\\cdot 3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"881536\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"881536\"]\r\n<ol>\r\n \t<li>commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;<\/li>\r\n \t<li>33, distributive property;<\/li>\r\n \t<li>26, distributive property;<\/li>\r\n \t<li>0, distributive property, inverse property of addition, identity property of addition<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92360&amp;theme=oea&amp;iframe_resize_id=mom115\" width=\"100%\" height=\"400\"><\/iframe>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92361&amp;theme=oea&amp;iframe_resize_id=mom120\" width=\"100%\" height=\"400\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Real numbers<\/li>\n<li>Add and subtract real numbers\n<ul>\n<li>Add\u00a0real numbers with the same and different signs<\/li>\n<li>Subtract real numbers with the same and different signs<\/li>\n<li>Simplify combinations that require both addition and subtraction of real numbers.<\/li>\n<\/ul>\n<\/li>\n<li>Multiply and divide real numbers\n<ul>\n<li>Multiply two or more real numbers.<\/li>\n<li>Divide real numbers<\/li>\n<li>Simplify expressions with both multiplication and division<\/li>\n<\/ul>\n<\/li>\n<li>Properties of real numbers<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\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>.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223810\/CNX_CAT_Figure_01_01_002.jpg\" alt=\"A number line that is marked from negative five to five\" width=\"487\" height=\"49\" \/><\/p>\n<p class=\"wp-caption-text\">The real number line<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Classifying Real Numbers<\/h3>\n<p>Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?<\/p>\n<ol>\n<li>[latex]-\\frac{10}{3}[\/latex]<\/li>\n<li>[latex]-6\\pi[\/latex]<\/li>\n<li>[latex]0.616161\\dots[\/latex]<\/li>\n<li>[latex]0.13[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q705558\">Solution<\/span><\/p>\n<div id=\"q705558\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\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]-6\\pi[\/latex] is negative and irrational. It lies to the left of 0.<\/li>\n<li>[latex]0.616161\\dots[\/latex] is a repeating decimal so it is rational and positive. It lies to the right of 0.<\/li>\n<li>[latex]0.13[\/latex] is a finite decimal and may be written as 13\/100. \u00a0So it is rational and positive.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?<\/p>\n<ol>\n<li>[latex]2\\pi[\/latex]<\/li>\n<li>[latex]-11.411411411\\dots[\/latex]<\/li>\n<li>[latex]\\frac{47}{19}[\/latex]<\/li>\n<li>[latex]6.210735[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q155954\">Solution<\/span><\/p>\n<div id=\"q155954\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>positive, irrational; right<\/li>\n<li>negative, rational; left<\/li>\n<li>positive, rational; right<\/li>\n<li>positive, rational; right<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h3>Sets of Numbers as Subsets<\/h3>\n<p>Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223813\/CNX_CAT_Figure_01_01_001.jpg\" alt=\"A large box labeled: Real Numbers encloses five circles. Four of these circles enclose each other and the other is separate from the rest. The innermost circle contains: 1, 2, 3\u2026 N. The circle enclosing that circle contains: 0 W. The circle enclosing that circle contains: \u2026, -3, -2, -1 I. The outermost circle contains: m\/n, n not equal to zero Q. The separate circle contains: pi, square root of two, etc Q\u00b4.\" width=\"731\" height=\"352\" \/><\/p>\n<p class=\"wp-caption-text\">Sets of numbers. \u00a0 <em>N<\/em>: the set of natural numbers \u00a0 <em>W<\/em>: the set of whole numbers \u00a0 <em>I<\/em>: the set of integers \u00a0 <em>Q<\/em>: the set of rational numbers \u00a0 <em>Q\u00b4<\/em>: the set of irrational numbers<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Sets of Numbers<\/h3>\n<p>The set of <strong>natural numbers<\/strong> includes the numbers used for counting: [latex]\\{1,2,3,\\dots\\}[\/latex].<\/p>\n<p>The set of <strong>whole numbers<\/strong> is the set of natural numbers plus zero: [latex]\\{0,1,2,3,\\dots\\}[\/latex].<\/p>\n<p>The set of <strong>integers<\/strong> adds the negative natural numbers to the set of whole numbers: [latex]\\{\\dots,-3,-2,-1,0,1,2,3,\\dots\\}[\/latex].<\/p>\n<p>The set of <strong>rational numbers<\/strong> includes fractions written as [latex]\\{\\frac{m}{n}|m\\text{ and }n\\text{ are integers and }n\\ne 0\\}[\/latex].<\/p>\n<p>The set of <strong>irrational numbers<\/strong> is the set of numbers that are not rational, are nonrepeating, and are nonterminating: [latex]\\{h|h\\text{ is not a rational number}\\}[\/latex].<\/p>\n<\/div>\n<p>The ability to work comfortably with negative numbers is essential to success in algebra. For this reason we will do a quick review of adding, subtracting, multiplying and dividing integers. Integers are all the positive whole numbers, zero, and their opposites (negatives). As this is intended to be a review of integers, the descriptions and examples will not be as detailed as a normal lesson.<\/p>\n<h2>Adding and Subtracting Real Numbers<\/h2>\n<p>When adding integers we have two cases to consider. The first case is whether\u00a0the signs match (both positive or both negative). If the signs match, we will add the numbers together and keep the sign.<\/p>\n<p>If the signs don\u2019t match (one positive and one negative number) we will subtract the numbers (as if they were all positive) and then use the sign from the larger number. This means if the larger number is positive, the answer is positive. If the larger number is negative, the answer is negative.<\/p>\n<div class=\"textbox shaded\">\n<h3>To add two numbers with the same sign (both positive or both negative)<\/h3>\n<ul>\n<li><i>Add<\/i> their absolute values (without the [latex]+[\/latex] or [latex]-[\/latex] sign)<\/li>\n<li>Give the sum the same sign.<\/li>\n<\/ul>\n<h3>To add two numbers with different signs (one positive and one negative)<\/h3>\n<ul>\n<li>Find the<i> difference <\/i>of<i> <\/i>their absolute values. (Note that when you find the difference of the absolute values, you always subtract the lesser absolute value from the greater one.)<\/li>\n<li>Give the sum the same sign as the number with the greater absolute value.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find\u00a0[latex]23\u201373[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q951238\">Show Solution<\/span><\/p>\n<div id=\"q951238\" class=\"hidden-answer\" style=\"display: none\">You can&#8217;t use your usual method of subtraction because 73 is greater than 23.\u00a0Rewrite the subtraction as adding the opposite.<\/p>\n<p style=\"text-align: center\">[latex]23+\\left(\u221273\\right)[\/latex]<\/p>\n<p>The addends have different signs, so find the difference of their absolute values.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left|23\\right|=23\\,\\,\\,\\text{and}\\,\\,\\,\\left|\u221273\\right|=73\\\\73-23=50\\end{array}[\/latex]<\/p>\n<p>Since [latex]\\left|\u221273\\right|>\\left|23\\right|[\/latex], the final answer is negative.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]23\u201373=\u221250[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Another way to think about subtracting is to think about the distance between the two numbers on the number line. In the example below, [latex]382[\/latex] is to the <i>right<\/i> of 0 by [latex]382[\/latex] units, and [latex]\u221293[\/latex] is to the <i>left<\/i> of 0 by 93 units. The distance between them is the sum of their distances to 0: [latex]382+93[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170754\/image050.gif\" alt=\"A number line from negative 93 to 382. Negative 93 is 93 units from 0 and 382 is 382 units from 0. The total distance from negative 93 to 382 can be found by adding them together. 382+93=475 units.\" width=\"575\" height=\"147\" \/><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find\u00a0[latex]382\u2013\\left(\u221293\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q342295\">Show Solution<\/span><\/p>\n<div id=\"q342295\" class=\"hidden-answer\" style=\"display: none\">You are subtracting a negative, so think of this as taking the negative sign away. This becomes an addition problem. [latex]-93[\/latex] becomes [latex]+93[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]382+93=475[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]382\u2013(\u221293)=475[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video explains how to subtract two signed integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 2:  Subtracting Integers (Two Digit Integers)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ciuIKFCtWWU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]-\\frac{3}{7}-\\frac{6}{7}+\\frac{2}{7}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q11416\">Show Solution<\/span><\/p>\n<div id=\"q11416\" class=\"hidden-answer\" style=\"display: none\">Add the first two and give the result a negative sign:<\/p>\n<p>Since the signs of the first two are the same, find the sum of the absolute values of the fractions<\/p>\n<p>Since both numbers are negative, the sum is negative. If you owe money, then borrow more, the amount you owe becomes larger.<\/p>\n<p style=\"text-align: center\">[latex]\\left| -\\frac{3}{7} \\right|=\\frac{3}{7}[\/latex] and [latex]\\left| -\\frac{6}{7} \\right|=\\frac{6}{7}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{3}{7}+\\frac{6}{7}=\\frac{9}{7}\\\\\\\\-\\frac{3}{7}-\\frac{6}{7} =-\\frac{9}{7}\\end{array}[\/latex]<\/p>\n<p>Now add the third number. The signs are different, so find the <em>difference<\/em> of their absolute values.<\/p>\n<p style=\"text-align: center\">[latex]\\left| -\\frac{9}{7} \\right|=\\frac{9}{7}[\/latex] and [latex]\\left| \\frac{2}{7} \\right|=\\frac{2}{7}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\frac{9}{7}-\\frac{2}{7}=\\frac{7}{7}[\/latex]<\/p>\n<p>Since [latex]\\left|\\frac{-9}{7}\\right|>\\left|\\frac{2}{7}\\right|[\/latex], the sign of the final sum is the same as the sign of [latex]-\\frac{9}{7}[\/latex].<i><br \/>\n<\/i><\/p>\n<p style=\"text-align: center\">[latex]-\\frac{9}{7}+\\frac{2}{7}=-\\frac{7}{7}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]-\\frac{3}{7}+\\left(-\\frac{6}{7}\\right)+\\frac{2}{7}=-\\frac{7}{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see an example of how to add three fractions with a common denominator that have different signs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Find the Sum and Difference of Three Signed Fractions (Common Denom)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/P972VVbR98k?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]27.832+(\u22123.06)[\/latex]. When you add decimals, remember to line up the decimal points so you are adding tenths to tenths, hundredths to hundredths, and so on.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q545871\">Show Solution<\/span><\/p>\n<div id=\"q545871\" class=\"hidden-answer\" style=\"display: none\">Since the addends have different signs, subtract their absolute values.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{\\begin{array}{r}27.832\\\\-\\text{ }3.06\\,\\,\\,\\end{array}}\\\\24.772\\end{array}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\left|-3.06\\right|=3.06[\/latex]<\/p>\n<p>The sum has the same sign as 27.832 whose absolute value is greater.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]27.832+\\left(-3.06\\right)=24.772[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video are\u00a0examples of adding and subtracting decimals with different signs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Adding Signed Decimals\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3FHZQ5iKcpI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiplying and Dividing Real Numbers<\/h2>\n<p>Multiplication and division are <strong>inverse operations<\/strong>, just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal. Inverse operations &#8220;undo&#8221; each other.<\/p>\n<h2>Multiply Real Numbers<\/h2>\n<p>Multiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven&#8217;t learned what effect a negative sign has on the product.<\/p>\n<p>With whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product [latex]3\\cdot4[\/latex] as 3 jumps of 4 units each.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" id=\"Picture 262\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170800\/image054.gif\" alt=\"A number line showing 3 times 4 is 12. From the 0, the right-facing person jumps 4 units at a time, and jumps 3 times. The person lands on 12.\" width=\"521\" height=\"130\" \/><\/p>\n<p>So to multiply [latex]3(\u22124)[\/latex], you can face left (toward the negative side) and make three \u201cjumps\u201d forward (in a negative direction).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170801\/image055.jpg\" alt=\"A number line representing 3 times negative 4 equals negative 12. A left-facing person jumps left 4 spaces 3 times so that the person lands on negative 12.\" width=\"516\" height=\"136\" \/><\/p>\n<p>The product of a positive number and a negative number (or a negative and a positive) is negative.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Product of a Positive Number and a Negative Number<\/h3>\n<p>To multiply a <strong>positive number<\/strong> and a <strong>negative number<\/strong>, multiply their absolute values. The product is <strong>negative<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]\u22123.8(0.6)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q456211\">Show Solution<\/span><\/p>\n<div id=\"q456211\" class=\"hidden-answer\" style=\"display: none\">Multiply the absolute values as you normally would.\u00a0Place the decimal point by counting place values.\u00a03.8 has 1 place after the decimal point, and 0.6 has 1 place after the decimal point, so the product has [latex]1+1[\/latex] or 2 places after the decimal point.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3.8\\\\\\underline{\\times\\,\\,\\,0.6}\\\\2.28\\end{array}[\/latex]<\/p>\n<p>The product of a negative and a positive is negative.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\u22123.8(0.6)=\u22122.28[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video contains examples of how to multiply decimal numbers with different signs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex:  Multiplying Signed Decimals\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/7gY0S3LUUyQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<h3>The Product of Two Numbers with the Same Sign (both positive or both negative)<\/h3>\n<p>To multiply two <strong>positive numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.<\/p>\n<p>To multiply two <strong>negative numbers<\/strong>, multiply their absolute values. The product is <strong>positive<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]~\\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q322816\">Show Solution<\/span><\/p>\n<div id=\"q322816\" class=\"hidden-answer\" style=\"display: none\">Multiply the absolute values of the numbers.\u00a0First, multiply the numerators together to get the product&#8217;s numerator. Then, multiply the denominators together to get the product&#8217;s denominator. Rewrite in lowest terms, if needed.<\/p>\n<p style=\"text-align: center\">[latex]\\left( \\frac{3}{4} \\right)\\left( \\frac{2}{5} \\right)=\\frac{6}{20}=\\frac{3}{10}[\/latex]<\/p>\n<p>The product of two negative numbers is positive.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left( -\\frac{3}{4} \\right)\\left( -\\frac{2}{5} \\right)=\\frac{3}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows examples of multiplying two signed fractions, including simplification of the answer.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex:  Multiplying Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yUdJ46pTblo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>To summarize:<\/p>\n<ul>\n<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\n<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>positive<\/strong>.<\/li>\n<li><strong>negative <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0positive<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\n<li><strong>positive <\/strong>[latex]\\cdot[\/latex]<strong><i>\u00a0negative<\/i>:<\/strong> The product is <strong>negative<\/strong>.<\/li>\n<\/ul>\n<p>You can see that the product of two negative numbers is a positive number. So, if you are multiplying more than two numbers, you can count the number of negative factors.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying More Than Two Negative Numbers<\/h3>\n<p>If there are an <strong>even<\/strong> number (0, 2, 4, &#8230;) of negative factors to multiply, the product is <strong>positive<\/strong>.<br \/>\nIf there are an <strong>odd<\/strong> number (1, 3, 5, &#8230;) of negative factors, the product is <strong>negative<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149062\">Show Solution<\/span><\/p>\n<div id=\"q149062\" class=\"hidden-answer\" style=\"display: none\">Multiply the absolute values of the numbers.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3(6)(2)(3)(1)\\\\18(2)(3)(1)\\\\36(3)(1)\\\\108(1)\\\\108\\end{array}[\/latex]<\/p>\n<p>Count the number of negative factors. There are three [latex]\\left(\u22126,\u22123,\u22121\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]3(\u22126)(2)(\u22123)(\u22121)[\/latex]<\/p>\n<p>Since there are an odd number of negative factors, the product is negative.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3(\u22126)(2)(\u22123)(\u22121)=\u2212108[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video contains examples of multiplying more than two signed integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex:  Multiplying Three or More Integers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rx8F9SPd0HE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Divide Real Numbers<\/h2>\n<p>You may remember that when you divided fractions, you multiplied by the <strong>reciprocal<\/strong>. <i>Reciprocal <\/i>is another name for the multiplicative inverse (just as <i>opposite <\/i>is another name for additive inverse).<\/p>\n<p>An easy way to find the multiplicative inverse is to just \u201cflip\u201d the numerator and denominator as you did to find the reciprocal. Here are some examples:<\/p>\n<ul>\n<li>The reciprocal of [latex]\\frac{4}{9}[\/latex]\u00a0is [latex]\\frac{9}{4}[\/latex]because [latex]\\frac{4}{9}\\left(\\frac{9}{4}\\right)=\\frac{36}{36}=1[\/latex].<\/li>\n<li>The reciprocal of 3 is [latex]\\frac{1}{3}[\/latex]\u00a0because [latex]\\frac{3}{1}\\left(\\frac{1}{3}\\right)=\\frac{3}{3}=1[\/latex].<\/li>\n<li>The reciprocal of [latex]-\\frac{5}{6}[\/latex]\u00a0is [latex]\\frac{-6}{5}[\/latex]\u00a0because [latex]-\\frac{5}{6}\\left( -\\frac{6}{5} \\right)=\\frac{30}{30}=1[\/latex].<\/li>\n<li>The reciprocal of 1 is 1 as [latex]1(1)=1[\/latex].<\/li>\n<\/ul>\n<p>When you divided by positive fractions, you learned to multiply by the reciprocal. You also do this to divide <strong>real numbers<\/strong>.<\/p>\n<p>Think about dividing a bag of 26 marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has <i>one half<\/i> of the marbles.<\/p>\n<p style=\"text-align: center\">[latex]26\\div 2=26\\left( \\frac{1}{2} \\right)=13[\/latex]<\/p>\n<p>Notice that 2 and [latex]\\frac{1}{2}[\/latex] are reciprocals.<\/p>\n<p>Try again, dividing a bag of 36 marbles into smaller bags.<\/p>\n<table>\n<tbody>\n<tr>\n<th>Number of bags<\/th>\n<th>Dividing by number of bags<\/th>\n<th>Multiplying by reciprocal<\/th>\n<\/tr>\n<\/tbody>\n<thead><\/thead>\n<tbody>\n<tr>\n<td>3<\/td>\n<td>[latex]\\frac{36}{3}=12[\/latex]<\/td>\n<td>[latex]36\\left( \\frac{1}{3} \\right)=\\frac{36}{3}=\\frac{12(3)}{3}=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>[latex]\\frac{36}{4}=9[\/latex]<\/td>\n<td>[latex]36\\left(\\frac{1}{4}\\right)=\\frac{36}{4}=\\frac{9\\left(4\\right)}{4}=9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>[latex]\\frac{36}{6}=6[\/latex]<\/td>\n<td>[latex]36\\left(\\frac{1}{6}\\right)=\\frac{36}{6}=\\frac{6\\left(6\\right)}{6}=6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Dividing by a number is the same as multiplying by its reciprocal. (That is, you use the reciprocal of the <strong>divisor<\/strong>, the second number in the division problem.)<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]28\\div \\frac{4}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q210216\">Show Solution<\/span><\/p>\n<div id=\"q210216\" class=\"hidden-answer\" style=\"display: none\">Rewrite the division as multiplication by the reciprocal. The reciprocal of [latex]\\frac{4}{3}[\/latex] is [latex]\\frac{3}{4}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]28\\div \\frac{4}{3}=28\\left( \\frac{3}{4} \\right)[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{28}{1}\\left(\\frac{3}{4}\\right)=\\frac{28\\left(3\\right)}{4}=\\frac{4\\left(7\\right)\\left(3\\right)}{4}=7\\left(3\\right)=21[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]28\\div\\frac{4}{3}=21[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<p>Now let&#8217;s see what this means when one or more of the numbers is negative. A number and its reciprocal have the same sign. Since division is rewritten as multiplication using the reciprocal of the divisor, and taking the reciprocal doesn\u2019t change any of the signs, division follows the same rules as multiplication.<\/p>\n<div class=\"textbox shaded\">\n<h3>Rules of Division<\/h3>\n<p>When dividing, rewrite the problem as multiplication using the reciprocal of the divisor as the second factor.<\/p>\n<p>When one number is <strong>positive<\/strong> and the other is <strong>negative<\/strong>, the <strong>quotient<\/strong> is <strong>negative<\/strong>.<\/p>\n<p>When <em>both<\/em> numbers are <strong>negative<\/strong>, the quotient is <strong>positive<\/strong>.<\/p>\n<p>When <em>both<\/em> numbers are <strong>positive<\/strong>, the quotient is <strong>positive<\/strong>.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]24\\div\\left(-\\frac{5}{6}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q716581\">Show Solution<\/span><\/p>\n<div id=\"q716581\" class=\"hidden-answer\" style=\"display: none\">Rewrite the division as multiplication by the reciprocal.<\/p>\n<p style=\"text-align: center\">[latex]24\\div \\left( -\\frac{5}{6} \\right)=24\\left( -\\frac{6}{5} \\right)[\/latex]<\/p>\n<p>Multiply. Since one number is positive and one is negative, the product is negative.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{24}{1}\\left( -\\frac{6}{5} \\right)=-\\frac{144}{5}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]24\\div \\left( -\\frac{5}{6} \\right)=-\\frac{144}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]4\\,\\left( -\\frac{2}{3} \\right)\\,\\div \\left( -6 \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q557653\">Show Solution<\/span><\/p>\n<div id=\"q557653\" class=\"hidden-answer\" style=\"display: none\">Rewrite the division as multiplication by the reciprocal.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{4}{1}\\left( -\\frac{2}{3} \\right)\\left( -\\frac{1}{6} \\right)[\/latex]<\/p>\n<p>Multiply. There is an even number of negative numbers, so the product is positive.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{4\\left(2\\right)\\left(1\\right)}{3\\left(6\\right)}=\\frac{8}{18}[\/latex]<\/p>\n<p>Write the fraction in lowest terms.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]4\\left( -\\frac{2}{3} \\right)\\div \\left( -6 \\right)=\\frac{4}{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video explains how to divide signed fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Ex 1:  Dividing Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OPHdadhDJoI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"103\" height=\"91\" \/><\/p>\n<p>Remember that a fraction bar also indicates division, so a negative sign in front of a fraction goes with the numerator, the denominator, or the whole fraction: [latex]-\\frac{3}{4}=\\frac{-3}{4}=\\frac{3}{-4}[\/latex].<\/p>\n<p>In each case, the overall fraction is negative because there&#8217;s only one negative in the division.<\/p>\n<\/div>\n<h2>Properties of Real Numbers<\/h2>\n<\/div>\n<p>For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.<\/p>\n<h3>Commutative Properties<\/h3>\n<p>The <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.<\/p>\n<div style=\"text-align: center\">[latex]a+b=b+a[\/latex]<\/div>\n<p>We can better see this relationship when using real numbers.<\/p>\n<div style=\"text-align: center\">[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/div>\n<p>Similarly, the <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.<\/p>\n<div style=\"text-align: center\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\n<p>Again, consider an example with real numbers.<\/p>\n<div style=\"text-align: center\">[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/div>\n<p>It is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex]. Similarly, [latex]20\\div 5\\ne 5\\div 20[\/latex].<\/p>\n<h3>Associative Properties<\/h3>\n<p>The <strong>associative property of multiplication<\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.<\/p>\n<div style=\"text-align: center\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\n<p>Consider this example.<\/p>\n<div style=\"text-align: center\">[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/div>\n<p>The <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/p>\n<div style=\"text-align: center\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\n<p>This property can be especially helpful when dealing with negative integers. Consider this example.<\/p>\n<div style=\"text-align: center\">[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/div>\n<p>Are subtraction and division associative? Review these examples.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{array}\\text{ }8-\\left(3-15\\right) \\hfill& \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) \\hfill& =5-15 \\\\ 20 \\hfill& \\neq 20-10 \\\\ \\text{ }\\end{array}[\/latex]<\/div>\n<div style=\"text-align: center\">[latex]\\begin{array}\\text{ }64\\div\\left(8\\div4\\right)\\hfill&\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 \\hfill& \\stackrel{?}{=}8\\div4 \\\\ 32 \\hfill& \\neq 2\\end{array}[\/latex]<\/div>\n<p>As we can see, neither subtraction nor division is associative.<\/p>\n<h3>Distributive Property<\/h3>\n<p>The <strong>distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.<\/p>\n<div style=\"text-align: center\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\n<p>This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223815\/CNX_CAT_Figure_01_01_003.jpg\" alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.\" \/><\/p>\n<p>Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by \u20137, and adding the products.<\/p>\n<p>To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{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]<\/div>\n<p>Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.<\/p>\n<p>A special case of the distributive property occurs when a sum of terms is subtracted.<\/p>\n<div style=\"text-align: center\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\n<p>For example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms 12 and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.<\/p>\n<div style=\"text-align: center\">[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\n<p>Now, distribute [latex]-1[\/latex] and simplify the result.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{array}12-\\left(5+3\\right) \\hfill& =12+\\left(-1\\right)\\cdot\\left(5+3\\right) \\\\ \\hfill& =12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\ \\hfill& =12+\\left(-8\\right) \\\\ \\hfill& =4 \\end{array}[\/latex]<\/div>\n<p>This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{array}12-\\left(5+3\\right) \\hfill& =12+\\left(-5-3\\right) \\\\ \\hfill& =12+\\left(-8\\right) \\\\ \\hfill& =4\\end{array}[\/latex]<\/div>\n<h3>Identity Properties<\/h3>\n<p>The <strong>identity property of addition<\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.<\/p>\n<div style=\"text-align: center\">[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<p>For example, we have [latex]\\left(-6\\right)+0=-6[\/latex] and [latex]23\\cdot 1=23[\/latex]. There are no exceptions for these properties; they work for every real number, including 0 and 1.<\/p>\n<h3>Inverse Properties<\/h3>\n<p>The <strong>inverse property of addition<\/strong> states that, for every real number <em>a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted\u2212<em>a<\/em>, that, when added to the original number, results in the additive identity, 0.<\/p>\n<div style=\"text-align: center\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<p>For example, if [latex]a=-8[\/latex], the additive inverse is 8, since [latex]\\left(-8\\right)+8=0[\/latex].<\/p>\n<p>The <strong>inverse property of multiplication<\/strong> holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number <em>a<\/em>, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\frac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, 1.<\/p>\n<div style=\"text-align: center\">[latex]a\\cdot \\frac{1}{a}=1[\/latex]<\/div>\n<p>For example, if [latex]a=-\\frac{2}{3}[\/latex], the reciprocal, denoted [latex]\\frac{1}{a}[\/latex], is [latex]-\\frac{3}{2}[\/latex]\u00a0because<\/p>\n<div style=\"text-align: center\">[latex]a\\cdot \\frac{1}{a}=\\left(-\\frac{2}{3}\\right)\\cdot \\left(-\\frac{3}{2}\\right)=1[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Properties of Real Numbers<\/h3>\n<p>The following properties hold for real numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>.<\/p>\n<table summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\n<thead>\n<tr>\n<th><\/th>\n<th>Addition<\/th>\n<th>Multiplication<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Commutative Property<\/strong><\/td>\n<td>[latex]a+b=b+a[\/latex]<\/td>\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Associative Property<\/strong><\/td>\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Distributive Property<\/strong><\/td>\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Identity Property<\/strong><\/td>\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em><\/p>\n<div>[latex]a+0=a[\/latex]<\/div>\n<\/td>\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em><\/p>\n<div>[latex]a\\cdot 1=a[\/latex]<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td><strong>Inverse Property<\/strong><\/td>\n<td>Every real number a has an additive inverse, or opposite, denoted <em>\u2013a<\/em>, such that<\/p>\n<div>[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<\/td>\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\frac{1}{a}[\/latex], such that<\/p>\n<div>[latex]a\\cdot \\left(\\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: Using Properties of Real Numbers<\/h3>\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<ol>\n<li>[latex]3\\cdot 6+3\\cdot 4[\/latex]<\/li>\n<li>[latex]\\left(5+8\\right)+\\left(-8\\right)[\/latex]<\/li>\n<li>[latex]6-\\left(15+9\\right)[\/latex]<\/li>\n<li>[latex]\\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=\"q892710\">Solution<\/span><\/p>\n<div id=\"q892710\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\begin{array}\\text{ }3\\cdot6+3\\cdot4 \\hfill& =3\\cdot\\left(6+4\\right) \\hfill& \\text{Distributive property} \\\\ \\hfill& =3\\cdot10 \\hfill& \\text{Simplify} \\\\ \\hfill& =30 \\hfill& \\text{Simplify}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}\\text{ }\\left(5+8\\right)+\\left(-8\\right) \\hfill& =5+\\left[8+\\left(-8\\right)\\right] \\hfill& \\text{Associative property of addition} \\\\ &\\hfill =5+0 \\hfill& \\text{Inverse property of addition} \\\\ \\hfill& =5 \\hfill& \\text{Identity property of addition}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}6-\\left(15+9\\right) \\hfill& =6+[15\\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}\\text{ }\\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}\\text{ }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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<ol>\n<li>[latex]\\left(-\\frac{23}{5}\\right)\\cdot \\left[11\\cdot \\left(-\\frac{5}{23}\\right)\\right][\/latex]<\/li>\n<li>[latex]5\\cdot \\left(6.2+0.4\\right)[\/latex]<\/li>\n<li>[latex]18-\\left(7 - 15\\right)[\/latex]<\/li>\n<li>[latex]6\\cdot \\left(-3\\right)+6\\cdot 3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q881536\">Solution<\/span><\/p>\n<div id=\"q881536\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;<\/li>\n<li>33, distributive property;<\/li>\n<li>26, distributive property;<\/li>\n<li>0, distributive property, inverse property of addition, identity property of addition<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92360&amp;theme=oea&amp;iframe_resize_id=mom115\" width=\"100%\" height=\"400\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92361&amp;theme=oea&amp;iframe_resize_id=mom120\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n\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-4736\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 2: Subtracting Integers (Two Digit Integers). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ciuIKFCtWWU\">https:\/\/youtu.be\/ciuIKFCtWWU<\/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>Find the Sum and Difference of Three Signed Fractions (Common Denom). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/P972VVbR98k\">https:\/\/youtu.be\/P972VVbR98k<\/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>Unit 9: Real Numbers, 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>Ex: Multiplying Three or More Integers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/rx8F9SPd0HE\">https:\/\/youtu.be\/rx8F9SPd0HE<\/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>Ex: Multiplying Signed Fractions . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/yUdJ46pTblo\">https:\/\/youtu.be\/yUdJ46pTblo<\/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>Ex 1: Dividing Signed Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/OPHdadhDJoI\">https:\/\/youtu.be\/OPHdadhDJoI<\/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. <strong>Authored by<\/strong>: College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Identifying Sets 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\/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><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":60342,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 2: Subtracting Integers (Two Digit Integers)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ciuIKFCtWWU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Find the Sum and Difference of Three Signed Fractions (Common Denom)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/P972VVbR98k\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 9: Real Numbers, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Multiplying Three or More Integers\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/rx8F9SPd0HE\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Multiplying Signed Fractions \",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/yUdJ46pTblo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Dividing Signed Fractions\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/OPHdadhDJoI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Identifying Sets of Real Numbers\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/htP2goe31MM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4736","chapter","type-chapter","status-publish","hentry"],"part":249,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4736","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/users\/60342"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4736\/revisions"}],"predecessor-version":[{"id":4804,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4736\/revisions\/4804"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/249"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4736\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/media?parent=4736"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4736"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/contributor?post=4736"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/wp-json\/wp\/v2\/license?post=4736"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}