{"id":591,"date":"2021-08-30T12:15:35","date_gmt":"2021-08-30T12:15:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=591"},"modified":"2026-03-27T16:11:13","modified_gmt":"2026-03-27T16:11:13","slug":"1-2-operations-on-integers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-2-operations-on-integers\/","title":{"raw":"1.2.1: Addition and Subtraction of Integers","rendered":"1.2.1: Addition and Subtraction of Integers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Simplify absolute values<\/li>\r\n \t<li>Simplify expressions using addition of integers<\/li>\r\n \t<li>Simplify expressions using subtraction of integers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>KEY words<\/h1>\r\n<ul>\r\n \t<li><strong>Opposites<\/strong>: A number the same distance from 0 but on the opposite side of the number line<\/li>\r\n \t<li><strong>Absolute value<\/strong>:\u00a0the distance from 0 of a number<\/li>\r\n \t<li><strong>Sum<\/strong>: the result of adding two or more numbers<\/li>\r\n \t<li><strong>Difference<\/strong>: the result of subtracting two numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Absolute Value<\/h2>\r\nNumbers such as [latex]5[\/latex] and [latex]-5[\/latex] are <em><strong>opposites<\/strong><\/em> because they are the same distance from [latex]0[\/latex] on the real number line. They are both five units from [latex]0[\/latex]. <strong>The distance between [latex]0[\/latex] and any number on the number line is called the absolute value of that number.<\/strong>\r\n\r\nBecause distance is never negative, the absolute value of any number is never negative.\r\n\r\nThe symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[\/latex] is written as [latex]\\big |5\\big |[\/latex], and the absolute value of [latex]-5[\/latex] is written as [latex]\\big |-5\\big |[\/latex] as shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220135\/CNX_BMath_Figure_03_01_019.png\" alt=\"This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Absolute Value<\/h3>\r\nThe absolute value of a number is its distance from [latex]0[\/latex] on the number line.\r\nThe absolute value of a number [latex]n[\/latex] is written as [latex]\\big |n\\big |[\/latex].\r\n\r\n[latex]\\big |n\\big |\\ge 0[\/latex] for all numbers\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>\u00a0[latex]\\big |3\\big |[\/latex]<\/li>\r\n \t<li>\u00a0[latex]\\big |-44\\big |[\/latex]<\/li>\r\n \t<li>\u00a0[latex]\\big |0\\big |[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1165118025864\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\big |3\\big |[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex] is [latex]3[\/latex] units from zero.<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1171842979184\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\big |-44\\big |[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\u221244[\/latex] is [latex]44[\/latex] units from zero.<\/td>\r\n<td>[latex]44[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1171842583307\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>3.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\big |0\\big |[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex] is already at zero.<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144930&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144931&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nIn the video below we show another example of how to find the absolute value of an integer.\r\n\r\nhttps:\/\/youtu.be\/I8bTqGmkqGI\r\n<h2>Addition of Integers<\/h2>\r\nOne way to think of positive and negative integers that may be helpful is to think about the numbers in terms of money.\r\n\r\nThink of positive numbers as money you have and negative numbers as money you spend.\u00a0 This will help you determine if your answer is positive or negative. \u00a0(-4) + 7 would be spending $4 and having $7, once you settle up, you still have $3.\u00a0 So the answer would be positive 3.\r\n\r\nAnother example is (-3) + (-5).\u00a0 This means you spend $3 and you spend an additional $5, so you have spent $8, which would be represented by -8.\r\n\r\nIf you have 6 + (-10) and we think in terms of money, you have $6 but you spend $10.\u00a0 Once you settle up, you are in debt $4.\u00a0 This gives you an answer of -4.\r\n\r\nAnother way to think about integers is yards gained or lost in a football game. Positive numbers represent yards gained, while negative numbers represent yards lost. (-4) + 7 means we lose 4 yards then gain 7 yards, for a net gain of 3 yards. So,\u00a0(-4) + 7 = 3.\r\n\r\nLikewise, (-3) + (-5) means we lose 3 yards then lose another 5 yards for a net loss of 8 yards. So,\u00a0(-3) + (-5) = -8. And,\u00a06 + (-10) means we gain 6 yards the lose 10 yards for a net loss of 4 yards. So,\u00a06 + (-10) = -4.\r\n<div>\r\n\r\nOf course when the numbers are much bigger we need a generalized method of adding two integers. \u00a0We can use absolute values to help us.\r\n<div class=\"textbox shaded\">\r\n<h3>ADDITION OF INTEGERS<\/h3>\r\nWhen the signs are the same, add the absolute value of each number and keep the common sign.\r\n\r\nWhen the signs are different, subtract the absolute values and keep the sign of the number with the larger absolute value.\r\n\r\nThe result from adding two or more numbers is called the sum.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]19+\\left(-47\\right)[\/latex]<\/li>\r\n \t<li>[latex](-32)+40[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution:<\/h4>\r\n<ol>\r\n \t<li>Since the signs are different we subtract the absolute values: [latex]47-19=28[\/latex]<span style=\"font-size: 1rem; text-align: initial;\">. Then since \u00a0[latex]\\big |-47\\big |[\/latex] is greater than [latex]\\big |19\\big |[\/latex], and -47 is negative the answer will be negative.\u00a0<\/span>[latex]19+(-47)= -28[\/latex]<\/li>\r\n \t<li>The signs are different so we subtract the absolute values: [latex]40-32=8[\/latex]. <span style=\"font-size: 1rem; text-align: initial;\">Then since \u00a0[latex]\\big |40\\big |[\/latex] is greater than [latex]\\big |-32\\big |[\/latex], and 40 is positive the answer will be positive.\u00a0<\/span>[latex](-32)+40=8[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145013&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\left(-14\\right)+\\left(-36\\right)[\/latex]\r\n\r\nSolution:\r\nSince the signs are the same, we add the absolute values. The answer will be negative because we are adding only negatives.\r\n<p style=\"padding-left: 30px;\">[latex]\\left(-14\\right)+\\left(-36\\right)= -50[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145014&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\nWe know that [latex]2+3=3+2[\/latex]. But is\u00a0[latex]-2+3=3+\\left(-2\\right)[\/latex]? Well\u00a0[latex]-2+3=1[\/latex] and \u00a0[latex]3+\\left(-2\\right)=1[\/latex]. So,\u00a0[latex]-2+3=3+\\left(-2\\right)[\/latex]. In fact, this is true for all integer values and is called the\u00a0<strong><em>commutative property of addition<\/em><\/strong><strong>.<\/strong>\u00a0T<span style=\"font-size: 1em;\">he order<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0that we add integers\u00a0doesn't matter. <\/span>\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">It is also true that adding zero to any integer has no effect on the integer. For example, [latex]-5+0=-5[\/latex]. Because [latex]0[\/latex] does not change the identity of any integer it is added to, [latex]0[\/latex] is called the\u00a0<\/span><em style=\"font-size: 1rem; text-align: initial;\"><strong>additive identity<\/strong><\/em><span style=\"font-size: 1rem; text-align: initial;\">.<\/span>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3>COMMUTATIVE PROPERTY OF ADDITION<\/h3>\r\nFor any integers [latex]a[\/latex] and [latex]b[\/latex], [latex]a+b=b+a[\/latex].\r\n\r\nThe order in which we add integers does not matter.\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3>additive identity<\/h3>\r\nFor any integers [latex]a[\/latex]], [latex]a+0=a[\/latex].\r\n\r\nAdding zero to an integer does not change the value of the integer.\r\n\r\n<\/div>\r\n<\/div>\r\n<h3>Adding more than two integers<\/h3>\r\nWhen we add more than two integers, we add them two at a time.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAdd [latex]-12+4+\\left(-5\\right)[\/latex]\r\n\r\n<strong>Solution<\/strong>\r\n\r\nAdding from left to right: [latex]-12+4+\\left(-5\\right)=-8+\\left(-5\\right)=-13[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nAdd [latex]6+\\left(-10\\right)+\\left(-7\\right)+5[\/latex]\r\n\r\n[reveal-answer q=\"382878\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"382878\"]\r\n\r\n[latex]6+\\left(-10\\right)+\\left(-7\\right)+5=-6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnother way to solve the problem\u00a0[latex]6+\\left(-10\\right)+\\left(-7\\right)+5[\/latex] is to add up all the positive and negatives separately then add the sums. This requires reorganizing the problem using the commutative property to:\u00a0[latex]6+5+\\left(-10\\right)+\\left(-7\\right)[\/latex]. Then\u00a0[latex]6+5=11[\/latex] and\u00a0[latex]\\left(-10\\right)+\\left(-7\\right)=-17[\/latex]. Adding these sums gives:\u00a0[latex]11+\\left(-17\\right)=-6[\/latex]. The same answer as before. Regrouping the numbers is an example of the\u00a0<strong><i>associative property of addition<\/i><\/strong>.\r\n<div class=\"textbox shaded\">\r\n<h3>THE ASSOCIATIVE PROPERTY OF ADDITION<\/h3>\r\nFor any integers [latex]a,\\,b, \\, c,\\: \\left(a+b\\right)+c=a+\\left(b+c\\right)[\/latex]\r\n\r\nWe can regroup the integers to get the same sum.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nAdd: [latex]-16+28+\\left(-12\\right)+19[\/latex]\r\n\r\n[reveal-answer q=\"525850\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"525850\"]\r\n\r\n[latex]\\:\\:\\left(-16\\left+(-12\\right)\\right)+\\left(28+19\\right)[\/latex]\r\n\r\n=\u00a0[latex]\\left(-28\\right)+\\left(47\\right)[\/latex]\r\n\r\n= [latex]19[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Subtraction of Integers<\/h2>\r\nIf we continue to think about integers as money, we can start to understand the subtraction process. If you have $5 and spend $3, you will have $2 left over. That can be written as [latex]5-3=2[\/latex].\r\n\r\nBut if you have $5 and you spend $12, you end up in debt by $7. The debt can be expressed as a negative number. This can be written as [latex]5-12=-7[\/latex].\r\n\r\nWhat if you are already in debt and you continue to spend money. Suppose you are in debt by $10 and you spend $7 more. Now you are further in debt, and your debt totals $17. This can be written as [latex]-10-7=-17[\/latex].\r\n\r\nWe can also think about the subtraction of money as the\u00a0<em>addition of debt.<\/em> This leads us to start to think of subtraction as the addition of opposites. You will often see this idea, the Subtraction Property, written as follows:\r\n<div class=\"textbox shaded\">\r\n<h3>Subtraction Property<\/h3>\r\nSubtracting a number is equivalent to adding the opposite of the number.\r\n<p style=\"text-align: center;\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/p>\r\nThe result of subtracting two numbers is called a difference.\r\n\r\n<\/div>\r\nConsider these examples.\r\n\r\nIf we have $6 and spend $4, we have $2 left.\r\n\r\nIf we have $6 and we add a debt of $4, we have $2 left.\r\n\r\nThis illustrates that [latex]6 - 4[\/latex] is equivalent to [latex]6+\\left(-4\\right)[\/latex].\r\n\r\nOf course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already know how to subtract [latex]6 - 4[\/latex]. But knowing that [latex]6 - 4[\/latex] gives the same answer as [latex]6+\\left(-4\\right)[\/latex] helps when we are subtracting negative numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]13 - 8\\text{ and }13+\\left(-8\\right)[\/latex]<\/li>\r\n \t<li>[latex]-17 - 9\\text{ and }-17+\\left(-9\\right)[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1168469839872\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]13 - 8[\/latex] and [latex]13+\\left(-8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract to simplify.<\/td>\r\n<td>[latex]13 - 8=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add to simplify.<\/td>\r\n<td>[latex]13+\\left(-8\\right)=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]13[\/latex] <em>minus<\/em>\u00a0[latex]8=[\/latex]\r\n\r\n<span style=\"font-family: inherit; font-size: inherit;\">[latex]13[\/latex]\u00a0<\/span><span style=\"font-family: inherit; font-size: inherit;\"><em>plus<\/em> [latex]\u22128[\/latex]\u00a0<\/span><span style=\"font-family: inherit; font-size: inherit;\">.<\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1164272171485\" class=\"unnumbered unstyled\" style=\"width: 85%; height: 176px;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"height: 12px;\">2.<\/th>\r\n<td style=\"height: 12px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px;\"><\/td>\r\n<td style=\"height: 24px;\">[latex]-17 - 9[\/latex] and [latex]-17+\\left(-9\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px;\">Subtract to simplify.<\/td>\r\n<td style=\"height: 12px;\">[latex]-17 - 9=-26[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px;\">Add to simplify.<\/td>\r\n<td style=\"height: 12px;\">[latex]-17+\\left(-9\\right)=-26[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 116px;\">\r\n<td style=\"height: 116px;\">[latex]\u221217[\/latex] <em>minus<\/em> [latex]9=[\/latex]\r\n\r\n<span style=\"font-family: inherit; font-size: inherit;\">[latex]\u221217[\/latex] <em>plus<\/em><\/span>\u00a0[latex]\u22129[\/latex]<\/td>\r\n<td style=\"height: 116px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145193[\/ohm_question]\r\n\r\n<\/div>\r\nNow consider what happens when we subtract a negative.\r\n\r\nSuppose we have $8 and we subtract a $5 debt (which means we gain money). Subtracting a debt ultimately means that we're adding to the amount of money that we have.\r\n\r\n[latex]8-\\left(-5\\right)[\/latex] gives the same result as [latex]8+5[\/latex]. Subtracting a negative number is like adding a positive. In other words, subtracting a negative is equivalent to adding the opposite, a positive.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\left(-74\\right)-\\left(-58\\right)[\/latex].\r\n[reveal-answer q=\"397191\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"397191\"]\r\n\r\nFirst write the subtraction as addition of the opposite: [latex]\\left(-74\\right)\\color{red}{-}\\left(\\color{red}{-}58\\right)\\,= \\left(-74\\right)\\color{red}{+}\\left(\\color{red}{+}58\\right)[\/latex]\r\n\r\nThen use absolute values to add the integers: \u00a0[latex]\\left(-74\\right)+\\left(58\\right)[\/latex]\r\n\r\nSubtract the absolute values: [latex]74-58=16[\/latex]\r\n\r\nThe sign of the answer is negative because [latex]\\big |-74\\big |[\/latex] is greater than\u00a0[latex]\\big |58\\big |[\/latex]\r\n\r\nSo,\u00a0[latex]\\left(-74\\right)-\\left(-58\\right)=-16[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145197[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show another example of subtracting two digit integers.\r\n\r\nhttps:\/\/youtu.be\/IfiN-mJZu2E\r\n<h3>Properties<\/h3>\r\nWhen we looked at the addition of integers, we discovered certain properties. The additive identity, [latex]0[\/latex]; the commutative property of addition; the associative property of addition.\r\n\r\n[latex]-3+5=2[\/latex] and [latex]5+(-3)=2[\/latex],\u00a0the commutative property of addition.\r\n\r\n[latex]-4+(3+(-5))=-4+(-2)=-6[\/latex] and [latex](-4+(3))+(-5)=-1+(-5)=-6[\/latex],\u00a0the associative property of addition.\r\n\r\nBut, beware, for if we leave the problems as subtraction, the properties do not apply!\r\n\r\n[latex]3-7=-4[\/latex] but [latex]7-3=4[\/latex]. The commutative property does not work for subtraction!\r\n\r\n[latex](8-3)-5=5-5=0[\/latex] but [latex]8-(3-5)=8-(-2)=10[\/latex]. The associative property does not work for subtraction!\r\n\r\n<strong>We must write subtraction as addition of the opposite for the properties of addition to apply.<\/strong>","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Simplify absolute values<\/li>\n<li>Simplify expressions using addition of integers<\/li>\n<li>Simplify expressions using subtraction of integers<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>KEY words<\/h1>\n<ul>\n<li><strong>Opposites<\/strong>: A number the same distance from 0 but on the opposite side of the number line<\/li>\n<li><strong>Absolute value<\/strong>:\u00a0the distance from 0 of a number<\/li>\n<li><strong>Sum<\/strong>: the result of adding two or more numbers<\/li>\n<li><strong>Difference<\/strong>: the result of subtracting two numbers<\/li>\n<\/ul>\n<\/div>\n<h2>Absolute Value<\/h2>\n<p>Numbers such as [latex]5[\/latex] and [latex]-5[\/latex] are <em><strong>opposites<\/strong><\/em> because they are the same distance from [latex]0[\/latex] on the real number line. They are both five units from [latex]0[\/latex]. <strong>The distance between [latex]0[\/latex] and any number on the number line is called the absolute value of that number.<\/strong><\/p>\n<p>Because distance is never negative, the absolute value of any number is never negative.<\/p>\n<p>The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[\/latex] is written as [latex]\\big |5\\big |[\/latex], and the absolute value of [latex]-5[\/latex] is written as [latex]\\big |-5\\big |[\/latex] as shown below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220135\/CNX_BMath_Figure_03_01_019.png\" alt=\"This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Absolute Value<\/h3>\n<p>The absolute value of a number is its distance from [latex]0[\/latex] on the number line.<br \/>\nThe absolute value of a number [latex]n[\/latex] is written as [latex]\\big |n\\big |[\/latex].<\/p>\n<p>[latex]\\big |n\\big |\\ge 0[\/latex] for all numbers<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>\u00a0[latex]\\big |3\\big |[\/latex]<\/li>\n<li>\u00a0[latex]\\big |-44\\big |[\/latex]<\/li>\n<li>\u00a0[latex]\\big |0\\big |[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1165118025864\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\big |3\\big |[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex] is [latex]3[\/latex] units from zero.<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1171842979184\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\big |-44\\big |[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\u221244[\/latex] is [latex]44[\/latex] units from zero.<\/td>\n<td>[latex]44[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1171842583307\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<th>3.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\big |0\\big |[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex] is already at zero.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144930&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144931&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>In the video below we show another example of how to find the absolute value of an integer.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1: Determine the Absolute Value of an Integer\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/I8bTqGmkqGI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Addition of Integers<\/h2>\n<p>One way to think of positive and negative integers that may be helpful is to think about the numbers in terms of money.<\/p>\n<p>Think of positive numbers as money you have and negative numbers as money you spend.\u00a0 This will help you determine if your answer is positive or negative. \u00a0(-4) + 7 would be spending $4 and having $7, once you settle up, you still have $3.\u00a0 So the answer would be positive 3.<\/p>\n<p>Another example is (-3) + (-5).\u00a0 This means you spend $3 and you spend an additional $5, so you have spent $8, which would be represented by -8.<\/p>\n<p>If you have 6 + (-10) and we think in terms of money, you have $6 but you spend $10.\u00a0 Once you settle up, you are in debt $4.\u00a0 This gives you an answer of -4.<\/p>\n<p>Another way to think about integers is yards gained or lost in a football game. Positive numbers represent yards gained, while negative numbers represent yards lost. (-4) + 7 means we lose 4 yards then gain 7 yards, for a net gain of 3 yards. So,\u00a0(-4) + 7 = 3.<\/p>\n<p>Likewise, (-3) + (-5) means we lose 3 yards then lose another 5 yards for a net loss of 8 yards. So,\u00a0(-3) + (-5) = -8. And,\u00a06 + (-10) means we gain 6 yards the lose 10 yards for a net loss of 4 yards. So,\u00a06 + (-10) = -4.<\/p>\n<div>\n<p>Of course when the numbers are much bigger we need a generalized method of adding two integers. \u00a0We can use absolute values to help us.<\/p>\n<div class=\"textbox shaded\">\n<h3>ADDITION OF INTEGERS<\/h3>\n<p>When the signs are the same, add the absolute value of each number and keep the common sign.<\/p>\n<p>When the signs are different, subtract the absolute values and keep the sign of the number with the larger absolute value.<\/p>\n<p>The result from adding two or more numbers is called the sum.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]19+\\left(-47\\right)[\/latex]<\/li>\n<li>[latex](-32)+40[\/latex]<\/li>\n<\/ol>\n<h4>Solution:<\/h4>\n<ol>\n<li>Since the signs are different we subtract the absolute values: [latex]47-19=28[\/latex]<span style=\"font-size: 1rem; text-align: initial;\">. Then since \u00a0[latex]\\big |-47\\big |[\/latex] is greater than [latex]\\big |19\\big |[\/latex], and -47 is negative the answer will be negative.\u00a0<\/span>[latex]19+(-47)= -28[\/latex]<\/li>\n<li>The signs are different so we subtract the absolute values: [latex]40-32=8[\/latex]. <span style=\"font-size: 1rem; text-align: initial;\">Then since \u00a0[latex]\\big |40\\big |[\/latex] is greater than [latex]\\big |-32\\big |[\/latex], and 40 is positive the answer will be positive.\u00a0<\/span>[latex](-32)+40=8[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145013&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\left(-14\\right)+\\left(-36\\right)[\/latex]<\/p>\n<p>Solution:<br \/>\nSince the signs are the same, we add the absolute values. The answer will be negative because we are adding only negatives.<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\left(-14\\right)+\\left(-36\\right)= -50[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145014&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<p>We know that [latex]2+3=3+2[\/latex]. But is\u00a0[latex]-2+3=3+\\left(-2\\right)[\/latex]? Well\u00a0[latex]-2+3=1[\/latex] and \u00a0[latex]3+\\left(-2\\right)=1[\/latex]. So,\u00a0[latex]-2+3=3+\\left(-2\\right)[\/latex]. In fact, this is true for all integer values and is called the\u00a0<strong><em>commutative property of addition<\/em><\/strong><strong>.<\/strong>\u00a0T<span style=\"font-size: 1em;\">he order<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0that we add integers\u00a0doesn&#8217;t matter. <\/span><\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">It is also true that adding zero to any integer has no effect on the integer. For example, [latex]-5+0=-5[\/latex]. Because [latex]0[\/latex] does not change the identity of any integer it is added to, [latex]0[\/latex] is called the\u00a0<\/span><em style=\"font-size: 1rem; text-align: initial;\"><strong>additive identity<\/strong><\/em><span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<div>\n<div class=\"textbox shaded\">\n<h3>COMMUTATIVE PROPERTY OF ADDITION<\/h3>\n<p>For any integers [latex]a[\/latex] and [latex]b[\/latex], [latex]a+b=b+a[\/latex].<\/p>\n<p>The order in which we add integers does not matter.<\/p>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<h3>additive identity<\/h3>\n<p>For any integers [latex]a[\/latex]], [latex]a+0=a[\/latex].<\/p>\n<p>Adding zero to an integer does not change the value of the integer.<\/p>\n<\/div>\n<\/div>\n<h3>Adding more than two integers<\/h3>\n<p>When we add more than two integers, we add them two at a time.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Add [latex]-12+4+\\left(-5\\right)[\/latex]<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Adding from left to right: [latex]-12+4+\\left(-5\\right)=-8+\\left(-5\\right)=-13[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Add [latex]6+\\left(-10\\right)+\\left(-7\\right)+5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q382878\">Show Answer<\/span><\/p>\n<div id=\"q382878\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]6+\\left(-10\\right)+\\left(-7\\right)+5=-6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Another way to solve the problem\u00a0[latex]6+\\left(-10\\right)+\\left(-7\\right)+5[\/latex] is to add up all the positive and negatives separately then add the sums. This requires reorganizing the problem using the commutative property to:\u00a0[latex]6+5+\\left(-10\\right)+\\left(-7\\right)[\/latex]. Then\u00a0[latex]6+5=11[\/latex] and\u00a0[latex]\\left(-10\\right)+\\left(-7\\right)=-17[\/latex]. Adding these sums gives:\u00a0[latex]11+\\left(-17\\right)=-6[\/latex]. The same answer as before. Regrouping the numbers is an example of the\u00a0<strong><i>associative property of addition<\/i><\/strong>.<\/p>\n<div class=\"textbox shaded\">\n<h3>THE ASSOCIATIVE PROPERTY OF ADDITION<\/h3>\n<p>For any integers [latex]a,\\,b, \\, c,\\: \\left(a+b\\right)+c=a+\\left(b+c\\right)[\/latex]<\/p>\n<p>We can regroup the integers to get the same sum.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Add: [latex]-16+28+\\left(-12\\right)+19[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q525850\">Show Answer<\/span><\/p>\n<div id=\"q525850\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\:\\:\\left(-16\\left+(-12\\right)\\right)+\\left(28+19\\right)[\/latex]<\/p>\n<p>=\u00a0[latex]\\left(-28\\right)+\\left(47\\right)[\/latex]<\/p>\n<p>= [latex]19[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Subtraction of Integers<\/h2>\n<p>If we continue to think about integers as money, we can start to understand the subtraction process. If you have $5 and spend $3, you will have $2 left over. That can be written as [latex]5-3=2[\/latex].<\/p>\n<p>But if you have $5 and you spend $12, you end up in debt by $7. The debt can be expressed as a negative number. This can be written as [latex]5-12=-7[\/latex].<\/p>\n<p>What if you are already in debt and you continue to spend money. Suppose you are in debt by $10 and you spend $7 more. Now you are further in debt, and your debt totals $17. This can be written as [latex]-10-7=-17[\/latex].<\/p>\n<p>We can also think about the subtraction of money as the\u00a0<em>addition of debt.<\/em> This leads us to start to think of subtraction as the addition of opposites. You will often see this idea, the Subtraction Property, written as follows:<\/p>\n<div class=\"textbox shaded\">\n<h3>Subtraction Property<\/h3>\n<p>Subtracting a number is equivalent to adding the opposite of the number.<\/p>\n<p style=\"text-align: center;\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/p>\n<p>The result of subtracting two numbers is called a difference.<\/p>\n<\/div>\n<p>Consider these examples.<\/p>\n<p>If we have $6 and spend $4, we have $2 left.<\/p>\n<p>If we have $6 and we add a debt of $4, we have $2 left.<\/p>\n<p>This illustrates that [latex]6 - 4[\/latex] is equivalent to [latex]6+\\left(-4\\right)[\/latex].<\/p>\n<p>Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already know how to subtract [latex]6 - 4[\/latex]. But knowing that [latex]6 - 4[\/latex] gives the same answer as [latex]6+\\left(-4\\right)[\/latex] helps when we are subtracting negative numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]13 - 8\\text{ and }13+\\left(-8\\right)[\/latex]<\/li>\n<li>[latex]-17 - 9\\text{ and }-17+\\left(-9\\right)[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469839872\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]13 - 8[\/latex] and [latex]13+\\left(-8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract to simplify.<\/td>\n<td>[latex]13 - 8=5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add to simplify.<\/td>\n<td>[latex]13+\\left(-8\\right)=5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]13[\/latex] <em>minus<\/em>\u00a0[latex]8=[\/latex]<\/p>\n<p><span style=\"font-family: inherit; font-size: inherit;\">[latex]13[\/latex]\u00a0<\/span><span style=\"font-family: inherit; font-size: inherit;\"><em>plus<\/em> [latex]\u22128[\/latex]\u00a0<\/span><span style=\"font-family: inherit; font-size: inherit;\">.<\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1164272171485\" class=\"unnumbered unstyled\" style=\"width: 85%; height: 176px;\" summary=\".\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"height: 12px;\">2.<\/th>\n<td style=\"height: 12px;\"><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\"><\/td>\n<td style=\"height: 24px;\">[latex]-17 - 9[\/latex] and [latex]-17+\\left(-9\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px;\">Subtract to simplify.<\/td>\n<td style=\"height: 12px;\">[latex]-17 - 9=-26[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px;\">Add to simplify.<\/td>\n<td style=\"height: 12px;\">[latex]-17+\\left(-9\\right)=-26[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 116px;\">\n<td style=\"height: 116px;\">[latex]\u221217[\/latex] <em>minus<\/em> [latex]9=[\/latex]<\/p>\n<p><span style=\"font-family: inherit; font-size: inherit;\">[latex]\u221217[\/latex] <em>plus<\/em><\/span>\u00a0[latex]\u22129[\/latex]<\/td>\n<td style=\"height: 116px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145193\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145193&theme=oea&iframe_resize_id=ohm145193&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now consider what happens when we subtract a negative.<\/p>\n<p>Suppose we have $8 and we subtract a $5 debt (which means we gain money). Subtracting a debt ultimately means that we're adding to the amount of money that we have.<\/p>\n<p>[latex]8-\\left(-5\\right)[\/latex] gives the same result as [latex]8+5[\/latex]. Subtracting a negative number is like adding a positive. In other words, subtracting a negative is equivalent to adding the opposite, a positive.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\left(-74\\right)-\\left(-58\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q397191\">Show Solution<\/span><\/p>\n<div id=\"q397191\" class=\"hidden-answer\" style=\"display: none\">\n<p>First write the subtraction as addition of the opposite: [latex]\\left(-74\\right)\\color{red}{-}\\left(\\color{red}{-}58\\right)\\,= \\left(-74\\right)\\color{red}{+}\\left(\\color{red}{+}58\\right)[\/latex]<\/p>\n<p>Then use absolute values to add the integers: \u00a0[latex]\\left(-74\\right)+\\left(58\\right)[\/latex]<\/p>\n<p>Subtract the absolute values: [latex]74-58=16[\/latex]<\/p>\n<p>The sign of the answer is negative because [latex]\\big |-74\\big |[\/latex] is greater than\u00a0[latex]\\big |58\\big |[\/latex]<\/p>\n<p>So,\u00a0[latex]\\left(-74\\right)-\\left(-58\\right)=-16[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145197\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145197&theme=oea&iframe_resize_id=ohm145197&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of subtracting two digit integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Subtract Two Digit Integers (Pos-Neg) Formal Rules and Number Line (Pos Sum)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/IfiN-mJZu2E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Properties<\/h3>\n<p>When we looked at the addition of integers, we discovered certain properties. The additive identity, [latex]0[\/latex]; the commutative property of addition; the associative property of addition.<\/p>\n<p>[latex]-3+5=2[\/latex] and [latex]5+(-3)=2[\/latex],\u00a0the commutative property of addition.<\/p>\n<p>[latex]-4+(3+(-5))=-4+(-2)=-6[\/latex] and [latex](-4+(3))+(-5)=-1+(-5)=-6[\/latex],\u00a0the associative property of addition.<\/p>\n<p>But, beware, for if we leave the problems as subtraction, the properties do not apply!<\/p>\n<p>[latex]3-7=-4[\/latex] but [latex]7-3=4[\/latex]. The commutative property does not work for subtraction!<\/p>\n<p>[latex](8-3)-5=5-5=0[\/latex] but [latex]8-(3-5)=8-(-2)=10[\/latex]. The associative property does not work for subtraction!<\/p>\n<p><strong>We must write subtraction as addition of the opposite for the properties of addition to apply.<\/strong><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-591\">\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>Addition; Subtraction . <strong>Authored by<\/strong>: Roxanne Brinkerhoff and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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>Simplify Absolute Value Expressions. <strong>Authored by<\/strong>: James Sousa. <strong>Provided by<\/strong>: Mathispower4u.com. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/I8bTqGmkqGI\">https:\/\/youtu.be\/I8bTqGmkqGI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 145193, 145197, . <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, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\">http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":422605,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Addition; Subtraction \",\"author\":\"Roxanne Brinkerhoff and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Simplify Absolute Value Expressions\",\"author\":\"James Sousa\",\"organization\":\"Mathispower4u.com\",\"url\":\" https:\/\/youtu.be\/I8bTqGmkqGI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID: 145193, 145197, \",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\",\"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-591","chapter","type-chapter","status-publish","hentry"],"part":587,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/591","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/422605"}],"version-history":[{"count":19,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/591\/revisions"}],"predecessor-version":[{"id":3208,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/591\/revisions\/3208"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/587"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/591\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=591"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=591"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=591"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=591"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}