{"id":881,"date":"2018-12-11T13:22:36","date_gmt":"2018-12-11T18:22:36","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/integers-2\/"},"modified":"2018-12-11T13:22:36","modified_gmt":"2018-12-11T18:22:36","slug":"integers-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/integers-2\/","title":{"raw":"Integers","rendered":"Integers"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Simplify expressions with absolute value<\/li><li>Add and subtract integers<\/li><li>Multiply and divide integers<\/li><li>Simplify expressions with integers<\/li><li>Evaluate variable expressions with integers<\/li><li>Translate phrases to expressions with integers<\/li><li>Use integers in applications<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167835364461\" class=\"be-prepared\"><p id=\"fs-id1167835343769\">A more thorough introduction to the topics covered in this section can be found in the <em data-effect=\"italics\">Elementary Algebra<\/em> chapter, Foundations.<\/p><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832051980\"><h3 data-type=\"title\">Simplify Expressions with Absolute Value<\/h3><p id=\"fs-id1167835321040\">A <span data-type=\"term\">negative numbers<\/span> is a number less than 0. The negative numbers are to the left of zero on the number line. See <a href=\"#CNX_IntAlg_Figure_01_02_001\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_02_001\"><div class=\"bc-figcaption figcaption\">The number line shows the location of positive and negative numbers.<\/div><span data-type=\"media\" id=\"fs-id1167835325238\" data-alt=\"Figure shows a horizontal line marked with numbers at equal distances. At the center of the line is 0. To the right of this, starting from the number closest to 0 are 1, 2, 3 and 4. These are labeled positive numbers. To the left of 0, starting from the number closest to 0 are minus 1, minus 2, minus 3 and minus 4. These are labeled negative numbers.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a horizontal line marked with numbers at equal distances. At the center of the line is 0. To the right of this, starting from the number closest to 0 are 1, 2, 3 and 4. These are labeled positive numbers. To the left of 0, starting from the number closest to 0 are minus 1, minus 2, minus 3 and minus 4. These are labeled negative numbers.\"><\/span><\/div><p id=\"fs-id1167834062128\">You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers \\(2\\) and \\(-2\\) are the same distance from zero, each one is called the <span data-type=\"term\">opposite<\/span> of the other. The opposite of \\(2\\) is \\(-2,\\) and the opposite of \\(-2\\) is \\(2.\\)<\/p><div data-type=\"note\" id=\"fs-id1167835326232\"><div data-type=\"title\">Opposite<\/div><p id=\"fs-id1167834186346\">The <strong data-effect=\"bold\">opposite<\/strong> of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.<\/p><\/div><p id=\"fs-id1167835368578\"><a href=\"#CNX_IntAlg_Figure_01_02_002\" class=\"autogenerated-content\">(Figure)<\/a> illustrates the definition.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_02_002\"><div class=\"bc-figcaption figcaption\">The opposite of 3 is \\(-3\\).<\/div><span data-type=\"media\" id=\"fs-id1167834195463\" data-alt=\"Figure shows a number line with the numbers 3 and minus 3 highlighted. These are equidistant from 0, both being 3 numbers away from 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with the numbers 3 and minus 3 highlighted. These are equidistant from 0, both being 3 numbers away from 0.\"><\/span><\/div><div data-type=\"note\" id=\"fs-id1167835596363\"><div data-type=\"title\">Opposite Notation<\/div><div data-type=\"equation\" id=\"fs-id1167835347808\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{}\\\\ \\\\ \\text{\u2212}a\\phantom{\\rule{0.2em}{0ex}}\\text{means the opposite of the number}\\phantom{\\rule{0.2em}{0ex}}a\\hfill \\\\ \\text{The notation}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2212}a\\phantom{\\rule{0.2em}{0ex}}\\text{is read as \u201cthe opposite of}\\phantom{\\rule{0.2em}{0ex}}a\\text{.\u201d}\\hfill \\end{array}\\)<\/div><\/div><p id=\"fs-id1167835307051\">We saw that numbers such as 3 and \\(-3\\) are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the <span data-type=\"term\">absolute value<\/span> of that number.<\/p><div data-type=\"note\" id=\"fs-id1167831871666\"><div data-type=\"title\">Absolute Value<\/div><p id=\"fs-id1167835304200\">The <strong data-effect=\"bold\">absolute value<\/strong> of a number is its distance from 0 on the number line.<\/p><p id=\"fs-id1167834138250\">The absolute value of a number \\(n\\) is written as \\(|n|\\) and \\(|n|\\ge 0\\) for all numbers.<\/p><p id=\"fs-id1167834345757\">Absolute values are always greater than or equal to zero.<\/p><\/div><p id=\"fs-id1167835325056\">For example,<\/p><div data-type=\"equation\" id=\"fs-id1167834063148\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}-5\\phantom{\\rule{0.2em}{0ex}}\\text{is 5 units away from 0, so}\\phantom{\\rule{0.2em}{0ex}}|-5|=5.\\hfill \\\\ \\text{5 is 5 units away from 0, so}\\phantom{\\rule{0.2em}{0ex}}|5|=5.\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167834464240\"><a href=\"#CNX_IntAlg_Figure_01_02_003\" class=\"autogenerated-content\">(Figure)<\/a> illustrates this idea.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_02_003\"><div class=\"bc-figcaption figcaption\">The numbers 5 and \\(-5\\) are 5 units away from 0.<\/div><span data-type=\"media\" id=\"fs-id1167834439195\" data-alt=\"Figure shows a number line showing the numbers 0, 5 and minus 5. 5 and minus 5 are equidistant from 0, both being 5 units away from 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_003_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line showing the numbers 0, 5 and minus 5. 5 and minus 5 are equidistant from 0, both being 5 units away from 0.\"><\/span><\/div><p id=\"fs-id1167832031212\">The absolute value of a number is never negative because distance cannot be negative. The only number with absolute value equal to zero is the number zero itself because the distance from 0 to 0 on the number line is zero units.<\/p><p id=\"fs-id1167834063255\">In the next example, we\u2019ll order expressions with absolute values.<\/p><div data-type=\"example\" id=\"fs-id1167835595046\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835362334\"><div data-type=\"problem\" id=\"fs-id1167834219221\"><p id=\"fs-id1167835217963\">Fill in \\(&lt;,&gt;,\\) or \\(=\\) for each of the following pairs of numbers:<\/p><p id=\"fs-id1167831883212\"><span class=\"token\">\u24d0<\/span>\\(|-5|__-|-5|\\)<span class=\"token\">\u24d1<\/span>\\(8__-|-8|\\)<span class=\"token\">\u24d2<\/span>\\(-9__-|-9|\\)<span class=\"token\">\u24d3<\/span>\\(\\text{\u2212}\\left(-16\\right)__|-16|.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835301570\"><p id=\"fs-id1167835380308\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{}\\\\ \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.5em}{0ex}}|-5|&amp; \\hfill __\\hfill &amp; -|-5|\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.5em}{0ex}}5&amp; \\hfill __\\hfill &amp; -5\\hfill \\\\ \\text{Order.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.5em}{0ex}}5&amp; \\hfill &gt;\\hfill &amp; -5\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.5em}{0ex}}|-5|&amp; \\hfill &gt;\\hfill &amp; -|-5|\\hfill \\end{array}\\)<p id=\"fs-id1167835342749\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccccc}&amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{3.4em}{0ex}}8&amp; \\hfill __\\hfill &amp; -|-8|\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{3.4em}{0ex}}8&amp; \\hfill __\\hfill &amp; -8\\hfill \\\\ \\text{Order.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{3.4em}{0ex}}8&amp; \\hfill &gt;\\hfill &amp; -8\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{3.4em}{0ex}}8&amp; \\hfill &gt;\\hfill &amp; -|-8|\\hfill \\end{array}\\)<p id=\"fs-id1167831880002\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccccc}&amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.8em}{0ex}}-9&amp; \\hfill __\\hfill &amp; -|-9|\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.8em}{0ex}}-9&amp; \\hfill __\\hfill &amp; -9\\hfill \\\\ \\text{Order.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.8em}{0ex}}-9&amp; \\hfill =\\hfill &amp; -9\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.8em}{0ex}}-9&amp; \\hfill =\\hfill &amp; \\text{\u2212}|-9|\\hfill \\end{array}\\)<p id=\"fs-id1167835333080\"><span class=\"token\">\u24d3<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccccc}&amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}\\text{\u2212}\\left(-16\\right)&amp; \\hfill __\\hfill &amp; |-16|\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}16&amp; \\hfill __\\hfill &amp; 16\\hfill \\\\ \\text{Order.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}16&amp; \\hfill =\\hfill &amp; 16\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}\\text{\u2212}\\left(-16\\right)&amp; \\hfill =\\hfill &amp; |-16|\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835365755\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835370336\"><div data-type=\"problem\" id=\"fs-id1167834195431\"><p id=\"fs-id1167831928877\">Fill in \\(&lt;,&gt;,\\) or \\(=\\) for each of the following pairs of numbers:<\/p><p id=\"fs-id1167832056480\"><span class=\"token\">\u24d0<\/span>\\(-9__-|-9|\\)<span class=\"token\">\u24d1<\/span>\\(2__-|-2|\\)<span class=\"token\">\u24d2<\/span>\\(-8__|-8|\\)<span class=\"token\">\u24d3<\/span>\\(\\text{\u2212}\\left(-9\\right)__|-9|.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835381034\"><p id=\"fs-id1167832052135\"><span class=\"token\">\u24d0<\/span>\\(&gt;\\)<span class=\"token\">\u24d1<\/span>\\(&gt;\\)<span class=\"token\">\u24d2<\/span>\\(&lt;\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(=\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835225722\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835355749\"><div data-type=\"problem\" id=\"fs-id1167835417963\"><p id=\"fs-id1167831076591\">Fill in \\(&lt;,&gt;,\\) or \\(=\\) for each of the following pairs of numbers:<\/p><p id=\"fs-id1167835321729\"><span class=\"token\">\u24d0<\/span>\\(7__-|-7|\\)<span class=\"token\">\u24d1<\/span>\\(\\text{\u2212}\\left(-10\\right)__|-10|\\)<span class=\"token\">\u24d2<\/span>\\(|-4|__-|-4|\\)<span class=\"token\">\u24d3<\/span>\\(-1__|-1|.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835198777\"><p id=\"fs-id1167835367362\"><span class=\"token\">\u24d0<\/span>\\(&gt;\\)<span class=\"token\">\u24d1<\/span>\\(=\\)<span class=\"token\">\u24d2<\/span>\\(&gt;\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(&lt;\\)<\/div><\/div><\/div><p id=\"fs-id1167832052677\">We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.<\/p><div data-type=\"note\" id=\"fs-id1167834433550\"><div data-type=\"title\">Grouping Symbols<\/div><div data-type=\"equation\" id=\"fs-id1167835321488\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccccccc}\\text{Parentheses}\\hfill &amp; &amp; \\left(\\phantom{\\rule{0.2em}{0ex}}\\right)\\hfill &amp; &amp; \\text{Braces}\\hfill &amp; &amp; \\left\\{\\phantom{\\rule{0.2em}{0ex}}\\right\\}\\hfill \\\\ \\text{Brackets}\\hfill &amp; &amp; \\left[\\phantom{\\rule{0.2em}{0ex}}\\right]\\hfill &amp; &amp; \\text{Absolute value}\\hfill &amp; &amp; |\\phantom{\\rule{0.2em}{0ex}}|\\hfill \\end{array}\\)<\/div><\/div><p id=\"fs-id1167835166543\">In the next example, we simplify the expressions inside absolute value bars first just like we do with parentheses.<\/p><div data-type=\"example\" id=\"fs-id1167835319324\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834180019\"><div data-type=\"problem\" id=\"fs-id1167834430895\"><p id=\"fs-id1167832058609\">Simplify: \\(24-|19-3\\left(6-2\\right)|.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835306143\"><p id=\"fs-id1167832054257\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}24-|19-3\\left(6-2\\right)|\\hfill \\\\ \\text{Work inside parentheses first:}\\hfill &amp; &amp; &amp; \\\\ \\text{subtract 2 from 6.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}24-|19-3\\left(4\\right)|\\hfill \\\\ \\text{Multiply 3(4).}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}24-|19-12|\\hfill \\\\ \\text{Subtract inside the absolute value bars.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}24-|7|\\hfill \\\\ \\text{Take the absolute value.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}24-7\\hfill \\\\ \\text{Subtract.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}17\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835206644\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835180537\"><div data-type=\"problem\" id=\"fs-id1167832055359\"><p id=\"fs-id1167834124820\">Simplify:\\(19-|11-4\\left(3-1\\right)|.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835259043\"><p id=\"fs-id1167835258413\">16<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835164568\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835609424\"><div data-type=\"problem\" id=\"fs-id1167826880456\"><p id=\"fs-id1167835346341\">Simplify: \\(9-|8-4\\left(7-5\\right)|.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835365656\"><p id=\"fs-id1167835262356\">9<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835511674\"><h3 data-type=\"title\">Add and Subtract Integers<\/h3><p id=\"fs-id1167831908724\">So far, we have only used the counting numbers and the whole numbers.<\/p><div data-type=\"equation\" id=\"fs-id1167835355753\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\text{Counting numbers}\\hfill &amp; &amp; &amp; 1,2,3\\dots \\hfill \\\\ \\text{Whole numbers}\\hfill &amp; &amp; &amp; 0,1,2,3\\dots .\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835335608\">Our work with opposites gives us a way to define the <span data-type=\"term\">integers<\/span>. The whole numbers and their opposites are called the integers. The integers are the numbers \\(\\dots -3,-2,-1,0,1,2,3\\text{\u2026}\\)<\/p><div data-type=\"note\" id=\"fs-id1167835307207\"><div data-type=\"title\">Integers<\/div><p id=\"fs-id1167835232918\">The whole numbers and their opposites are called the <strong data-effect=\"bold\">integers<\/strong>.<\/p><p id=\"fs-id1167835304330\">The integers are the numbers<\/p><div data-type=\"equation\" id=\"fs-id1167834192221\" class=\"unnumbered\" data-label=\"\">\\(\\dots -3,-2,-1,0,1,2,3\\text{\u2026,}\\)<\/div><\/div><p id=\"fs-id1167835302203\">Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.<\/p><p id=\"fs-id1167835213624\">We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.<\/p><p id=\"fs-id1167835311335\">We let one color (blue) represent positive. The other color (red) will represent the negatives.<\/p><span data-type=\"media\" id=\"fs-id1167835309360\" data-alt=\"Figure show two circles labeled positive blue and negative red.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_004_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure show two circles labeled positive blue and negative red.\"><\/span><p id=\"fs-id1167835309409\">If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.<\/p><span data-type=\"media\" id=\"fs-id1167835309392\" data-alt=\"Figure shows a blue circle and a red circle encircled in a larger shape. This is labeled 1 plus minus 1 equals 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_005_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a blue circle and a red circle encircled in a larger shape. This is labeled 1 plus minus 1 equals 0.\"><\/span><p id=\"fs-id1167835350247\">We will use the counters to show how to add:<\/p><div data-type=\"equation\" id=\"fs-id1167832076598\" class=\"unnumbered\" data-label=\"\">\\(5+3\\phantom{\\rule{3em}{0ex}}-5+\\left(-3\\right)\\phantom{\\rule{3em}{0ex}}-5+3\\phantom{\\rule{3em}{0ex}}5+\\left(-3\\right)\\)<\/div><p id=\"fs-id1167835326907\">The first example, \\(5+3,\\) adds 5 positives and 3 positives\u2014both positives.<\/p><p id=\"fs-id1167835303389\">The second example, \\(-5+\\left(-3\\right),\\) adds 5 negatives and 3 negatives\u2014both negatives.<\/p><p id=\"fs-id1167835191059\">When the signs are the same, the counters are all the same color, and so we add them. In each case we get 8\u2014either 8 positives or 8 negatives.<\/p><span data-type=\"media\" id=\"fs-id1167835190851\" data-alt=\"Figure on the left is labeled 5 plus 3. It shows 8 blue circles. 5 plus 3 equals 8. Figure on the right is labeled minus 5 plus open parentheses minus 3 close parentheses. It shows 8 blue circles labeled 8 negatives. Minus 5 plus open parentheses minus 3 close parentheses equals minus 8.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_006_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure on the left is labeled 5 plus 3. It shows 8 blue circles. 5 plus 3 equals 8. Figure on the right is labeled minus 5 plus open parentheses minus 3 close parentheses. It shows 8 blue circles labeled 8 negatives. Minus 5 plus open parentheses minus 3 close parentheses equals minus 8.\"><\/span><p id=\"fs-id1167831934032\">So what happens when the signs are different? Let\u2019s add \\(-5+3\\) and \\(5+\\left(-3\\right).\\)<\/p><p id=\"fs-id1167835346415\">When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.<\/p><span data-type=\"media\" id=\"fs-id1167835346024\" data-alt=\"Figure on the left is labeled minus 5 plus 3. It has 5 red circles and 3 blue circles. Three pairs of red and blue circles are formed. More negatives means the sum is negative. The figure on the right is labeled 5 plus minus 3. It has 5 blue and 3 red circles. Three pairs of red and blue circles are formed. More positives means the sum is positive.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_007_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure on the left is labeled minus 5 plus 3. It has 5 red circles and 3 blue circles. Three pairs of red and blue circles are formed. More negatives means the sum is negative. The figure on the right is labeled 5 plus minus 3. It has 5 blue and 3 red circles. Three pairs of red and blue circles are formed. More positives means the sum is positive.\"><\/span><div data-type=\"example\" id=\"fs-id1167835365552\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834190186\"><div data-type=\"problem\" id=\"fs-id1167835214274\"><p id=\"fs-id1167835345490\">Add: <span class=\"token\">\u24d0<\/span> \\(-1+\\left(-4\\right)\\) <span class=\"token\">\u24d1<\/span> \\(-1+5\\) <span class=\"token\">\u24d2<\/span> \\(1+\\left(-5\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835214138\"><p id=\"fs-id1167832055795\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167835166183\" summary=\"Figure shows five red circles. It is labeled minus 1 minus 4. 1 negative plus 4 negatives is 5 negatives. The answer is minus 5.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834195654\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td><\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835355963\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">1 negative plus 4 negatives is 5 negatives<\/td><td><\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834121084\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_008c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835232646\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167835254685\" summary=\"Figure is labeled minus 1 plus 5. It has 5 blue circles and 1 red circle. 1 pair of red and blue circles is formed. There are more positives, so the sum is positive. The answer is 4.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834183995\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835214048\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">There are more positives, so the sum is positive.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228881\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835167257\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167835329511\" summary=\"The figure is labeled 1 plus minus 5. There is 1 blue circle and 5 red circles. 1 pair of blue and red circles is formed. There are more negatives, so the sum is negative. The answer is minus 4.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835354908\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834274063\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_010b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">There are more negatives, so the sum is negative.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835421621\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_010c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835303770\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835420812\"><div data-type=\"problem\" id=\"fs-id1167835319023\"><p id=\"fs-id1167835418075\">Add: <span class=\"token\">\u24d0<\/span> \\(-2+\\left(-4\\right)\\) <span class=\"token\">\u24d1<\/span> \\(-2+4\\) <span class=\"token\">\u24d2<\/span> \\(2+\\left(-4\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835334779\"><p id=\"fs-id1167835360011\"><span class=\"token\">\u24d0<\/span>\\(-6\\)<span class=\"token\">\u24d1<\/span> 2 <span class=\"token\">\u24d2<\/span> \\(-2\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826814134\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835380916\"><div data-type=\"problem\" id=\"fs-id1167834186295\"><p id=\"fs-id1167826977942\">Add: <span class=\"token\">\u24d0<\/span> \\(-2+\\left(-5\\right)\\) <span class=\"token\">\u24d1<\/span> \\(-2+5\\) <span class=\"token\">\u24d2<\/span> \\(2+\\left(-5\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835269016\"><p id=\"fs-id1167835283690\"><span class=\"token\">\u24d0<\/span>\\(-7\\)<span class=\"token\">\u24d1<\/span> 3 <span class=\"token\">\u24d2<\/span> \\(-3\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835361703\">We will continue to use counters to model the subtraction. Perhaps when you were younger, you read \\(\u201c5-3\u201d\\) as \u201c5 take away 3.\u201d When you use counters, you can think of subtraction the same way!<\/p><p id=\"fs-id1167835189607\">We will use the counters to show to subtract:<\/p><div data-type=\"equation\" id=\"fs-id1167835478988\" class=\"unnumbered\" data-label=\"\">\\(5-3\\phantom{\\rule{3em}{0ex}}-5-\\left(-3\\right)\\phantom{\\rule{3em}{0ex}}-5-3\\phantom{\\rule{3em}{0ex}}5-\\left(-3\\right)\\)<\/div><p id=\"fs-id1167835304878\">The first example, \\(5-3,\\) we subtract 3 positives from 5 positives and end up with 2 positives.<\/p><p id=\"fs-id1167832053392\">In the second example, \\(-5-\\left(-3\\right),\\) we subtract 3 negatives from 5 negatives and end up with 2 negatives.<\/p><p id=\"fs-id1167834536601\">Each example used counters of only one color, and the \u201ctake away\u201d model of subtraction was easy to apply.<\/p><span data-type=\"media\" id=\"fs-id1167834282821\" data-alt=\"Figure on the left is labeled 5 minus 3 equals 2. There are 5 blue circles. Three of these are encircled and an arrow indicates that they are taken away. The figure on the right is labeled minus 5 minus open parentheses minus 3 close parentheses equals minus 2. There are 5 red circles. Three of these are encircled and an arrow indicates that they are taken away.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_011_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure on the left is labeled 5 minus 3 equals 2. There are 5 blue circles. Three of these are encircled and an arrow indicates that they are taken away. The figure on the right is labeled minus 5 minus open parentheses minus 3 close parentheses equals minus 2. There are 5 red circles. Three of these are encircled and an arrow indicates that they are taken away.\"><\/span><p id=\"fs-id1167835348640\">What happens when we have to subtract one positive and one negative number? We\u2019ll need to use both blue and red counters as well as some neutral pairs. If we don\u2019t have the number of counters needed to take away, we add neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels\u2014the value is the same, but it looks different.<\/p><p id=\"fs-id1167835370956\">Let\u2019s look at \\(-5-3\\) and \\(5-\\left(-3\\right).\\)<\/p><table id=\"fs-id1167834536310\" summary=\"Calculations of two expressions are shown. The expression on the left is minus 5 minus 3. The expression on the right is 5 minus open parentheses minus 3 close parentheses. We first model the first number. So, we have 5 red counters on the left and 5 white counters on the right. We now add three neutral pairs for both expressions. So we have 8 red and 3 white counters on the left and 8 white and 3 red counters on the right. We now remove the number of counters modeled by the second number. Three white counters are removed from the left and three red counters are removed from the right. We now have eight red counters on the left and 8 white counters on the right. Hence, minus 5 minus 3, on the left, is minus 8, whereas 5 minus minus 3, on the right, is 8.\" class=\"unnumbered unstyled can-break\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831911244\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835621605\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Model the first number.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831883380\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835306654\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">We now add the needed neutral pairs.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835370213\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835353953\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">We remove the number of counters modeled by the second number.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835418298\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832055009\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Count what is left.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835358812\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834314791\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834064550\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834183933\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831872542\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834505122\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><div data-type=\"example\" id=\"fs-id1167835514389\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835514391\"><div data-type=\"problem\" id=\"fs-id1167835187188\"><p id=\"fs-id1167835187190\">Subtract: <span class=\"token\">\u24d0<\/span> \\(3-1\\) <span class=\"token\">\u24d1<\/span> \\(-3-\\left(-1\\right)\\) <span class=\"token\">\u24d2<\/span> \\(-3-1\\) <span class=\"token\">\u24d3<\/span> \\(3-\\left(-1\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834227489\"><p id=\"fs-id1167831882978\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167835479134\" summary=\"The expression is 3 minus 1. Three white counters are shown. An arrow indicates that 1 is taken away from these, and 2 remain. The answer is 2.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\">\\(\\phantom{\\rule{0.3em}{0ex}}\\)<span data-type=\"media\" id=\"fs-id1167835510839\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_018a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835358825\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_018b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Take 1 positive from 3 positives and get 2 positives.<\/td><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835511311\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_018c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167834535740\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167832076029\" summary=\"The expression is minus 3 minus minus 1. Three red counters are shown. An arrow indicates that 1 is taken away from these, and 2 remain. The answer is minus 2.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826978713\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_019a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834539540\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_019b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Take 1 positive from 3 negatives and get 2 negatives.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832066812\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_019c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167834120450\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167834280273\" summary=\"The expression is minus 3 minus 1. 4 red and 1 blue counters are shown. An arrow indicates that 1 blue counter is taken away from these. 4 red counters remain. The answer is minus 4.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.8em}{0ex}}\\)<span data-type=\"media\" id=\"fs-id1167831138153\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_013a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834422963\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_013c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Take 1 positive from the one added neutral pair.<\/td><td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167835301849\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_013b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831239549\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_013d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835367421\"><span class=\"token\">\u24d3<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167834539525\" summary=\"The expression is 3 minus minus 1. 4 blue and 1 red counters are shown. An arrow indicates that 1 red counter is taken away from these. 4 blue counters remain. The answer is 4.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.5em}{0ex}}\\)<span data-type=\"media\" id=\"fs-id1167834539309\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_014a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834538618\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_014c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Take 1 negative from the one added neutral pair.<\/td><td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167832054944\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_014b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834190252\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_014d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834536304\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831882604\"><div data-type=\"problem\" id=\"fs-id1167831882606\"><p id=\"fs-id1167835340804\">Subtract: <span class=\"token\">\u24d0<\/span> \\(6-4\\) <span class=\"token\">\u24d1<\/span> \\(-6-\\left(-4\\right)\\) <span class=\"token\">\u24d2<\/span> \\(-6-4\\) <span class=\"token\">\u24d3<\/span> \\(6-\\left(-4\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834183611\"><p id=\"fs-id1167834539501\"><span class=\"token\">\u24d0<\/span> 2 <span class=\"token\">\u24d1<\/span> \\(-2\\) <span class=\"token\">\u24d2<\/span> \\(-10\\) <span class=\"token\">\u24d3<\/span> 10<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831882121\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835346029\"><div data-type=\"problem\" id=\"fs-id1167835346032\"><p id=\"fs-id1167835346034\">Subtract: <span class=\"token\">\u24d0<\/span> \\(7-4\\) <span class=\"token\">\u24d1<\/span> \\(-7-\\left(-4\\right)\\) <span class=\"token\">\u24d2<\/span> \\(-7-4\\) <span class=\"token\">\u24d3<\/span> \\(7-\\left(-4\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835367941\"><p id=\"fs-id1167835367944\"><span class=\"token\">\u24d0<\/span> 3 <span class=\"token\">\u24d1<\/span> \\(-3\\) <span class=\"token\">\u24d2<\/span> \\(-11\\) <span class=\"token\">\u24d3<\/span> 11<\/p><\/div><\/div><\/div><p id=\"fs-id1167834464452\">Have you noticed that <em data-effect=\"italics\">subtraction of signed numbers can be done by adding the opposite<\/em>? In the last example, \\(-3-1\\) is the same as \\(-3+\\left(-1\\right)\\) and \\(3-\\left(-1\\right)\\) is the same as \\(3+1.\\) You will often see this idea, the Subtraction Property, written as follows:<\/p><div data-type=\"note\" id=\"fs-id1167835322024\"><div data-type=\"title\">Subtraction Property<\/div><div data-type=\"equation\" id=\"fs-id1167835322029\" class=\"unnumbered\" data-label=\"\">\\(a-b=a+\\left(\\text{\u2212}b\\right)\\)<\/div><p id=\"fs-id1167835281540\">Subtracting a number is the same as adding its opposite.<\/p><\/div><div data-type=\"example\" id=\"fs-id1167834539392\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834539394\"><div data-type=\"problem\" id=\"fs-id1167834182501\"><p id=\"fs-id1167834182503\">Simplify: <span class=\"token\">\u24d0<\/span> \\(13-8\\) and \\(13+\\left(-8\\right)\\) <span class=\"token\">\u24d1<\/span> \\(-17-9\\) and \\(-17+\\left(-9\\right)\\) <span class=\"token\">\u24d2<\/span> \\(9-\\left(-15\\right)\\) and \\(9+15\\) <span class=\"token\">\u24d3<\/span> \\(-7-\\left(-4\\right)\\) and \\(-7+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835378105\"><p id=\"fs-id1167835307054\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4.2em}{0ex}}13-8\\hfill &amp; \\hfill \\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}\\hfill &amp; \\hfill 13+\\left(-8\\right)\\hfill \\\\ \\text{Subtract.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4.2em}{0ex}}5\\hfill &amp; &amp; \\hfill 5\\hfill \\end{array}\\)<p id=\"fs-id1167835254052\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{3.5em}{0ex}}-17-9\\hfill &amp; \\hfill \\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}\\hfill &amp; \\hfill -17+\\left(-9\\right)\\hfill \\\\ \\text{Subtract.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3.5em}{0ex}}-26\\hfill &amp; &amp; \\hfill -26\\hfill \\end{array}\\)<p id=\"fs-id1167834346258\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{2.8em}{0ex}}9-\\left(-15\\right)\\hfill &amp; \\hfill \\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}\\hfill &amp; \\hfill 9+15\\hfill \\\\ \\text{Subtract.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.8em}{0ex}}24\\hfill &amp; &amp; \\hfill 24\\hfill \\end{array}\\)<p id=\"fs-id1167834555060\"><span class=\"token\">\u24d3<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{2.7em}{0ex}}-7-\\left(-4\\right)\\hfill &amp; \\hfill \\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}\\hfill &amp; \\hfill -7+4\\hfill \\\\ \\text{Subtract.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.7em}{0ex}}-3\\hfill &amp; &amp; \\hfill -3\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832075833\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834252708\"><div data-type=\"problem\" id=\"fs-id1167834252710\"><p id=\"fs-id1167835369538\">Simplify: <span class=\"token\">\u24d0<\/span> \\(21-13\\) and \\(21+\\left(-13\\right)\\) <span class=\"token\">\u24d1<\/span> \\(-11-7\\) and \\(-11+\\left(-7\\right)\\) <span class=\"token\">\u24d2<\/span> \\(6-\\left(-13\\right)\\) and \\(6+13\\) <span class=\"token\">\u24d3<\/span> \\(-5-\\left(-1\\right)\\) and \\(-5+1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834506085\"><p id=\"fs-id1167834506087\"><span class=\"token\">\u24d0<\/span>\\(8,8\\)<span class=\"token\">\u24d1<\/span>\\(-18,\\)\\(-18\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(19,19\\)<span class=\"token\">\u24d3<\/span>\\(-4,\\)\\(-4\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835379147\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835380208\"><div data-type=\"problem\" id=\"fs-id1167835380210\"><p id=\"fs-id1167834538449\">Simplify: <span class=\"token\">\u24d0<\/span> \\(15-7\\) and \\(15+\\left(-7\\right)\\) <span class=\"token\">\u24d1<\/span> \\(-14-8\\) and \\(-14+\\left(-8\\right)\\) <span class=\"token\">\u24d2<\/span> \\(4-\\left(-19\\right)\\) and \\(4+19\\) <span class=\"token\">\u24d3<\/span> \\(-4-\\left(-7\\right)\\) and \\(-4+7.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834505550\"><p id=\"fs-id1167835379157\"><span class=\"token\">\u24d0<\/span>\\(8,8\\)<span class=\"token\">\u24d1<\/span>\\(-22,\\)\\(-22\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(23,23\\)<span class=\"token\">\u24d3<\/span>\\(3,3\\)<\/div><\/div><\/div><p id=\"fs-id1167834539673\">What happens when there are more than three integers? We just use the order of operations as usual.<\/p><div data-type=\"example\" id=\"fs-id1167835416831\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835416833\"><div data-type=\"problem\" id=\"fs-id1167835416835\"><p id=\"fs-id1167834555195\">Simplify: \\(7-\\left(-4-3\\right)-9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831880380\"><p id=\"fs-id1167831887723\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}7-\\left(-4-3\\right)-9\\hfill \\\\ \\text{Simplify inside the parentheses first.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}7-\\left(-7\\right)-9\\hfill \\\\ \\text{Subtract left to right.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}14-9\\hfill \\\\ \\text{Subtract.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}5\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831908823\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831908827\"><div data-type=\"problem\" id=\"fs-id1167835216901\"><p id=\"fs-id1167835216903\">Simplify: \\(8-\\left(-3-1\\right)-9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831881377\"><p id=\"fs-id1167831881379\">3<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832067665\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835349053\"><div data-type=\"problem\" id=\"fs-id1167835349055\"><p id=\"fs-id1167835349057\">Simplify: \\(12-\\left(-9-6\\right)-14.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835306347\"><p id=\"fs-id1167835306349\">13<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834094558\"><h3 data-type=\"title\">Multiply and Divide Integers<\/h3><p id=\"fs-id1167826804678\">Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we are using the model just to help us discover the pattern.<\/p><p id=\"fs-id1167834505365\">We remember that \\(a\u00b7b\\) means <strong data-effect=\"bold\">add<\/strong> <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em> <strong data-effect=\"bold\">times<\/strong>.<\/p><span data-type=\"media\" id=\"fs-id1167834222458\" data-alt=\"The figure on the left is labeled 5 dot 3. Here, we need to add 5, 3 times. Three rows of five blue counters each are shown. This makes 15 positives. Hence, 5 times 3 is 15. The figure on the right is labeled minus 5 open parentheses 3 close parentheses. Here we need to add minus 5, 3 times. Three rows of five red counters each are shown. This makes 15 negatives. Hence, minus 5 times 3 is minus 15.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_020_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure on the left is labeled 5 dot 3. Here, we need to add 5, 3 times. Three rows of five blue counters each are shown. This makes 15 positives. Hence, 5 times 3 is 15. The figure on the right is labeled minus 5 open parentheses 3 close parentheses. Here we need to add minus 5, 3 times. Three rows of five red counters each are shown. This makes 15 negatives. Hence, minus 5 times 3 is minus 15.\"><\/span><p id=\"fs-id1167834183356\">The next two examples are more interesting. What does it mean to multiply 5 by \\(-3?\\) It means subtract \\(5,3\\) times. Looking at subtraction as \u201ctaking away\u201d, it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace.<\/p><span data-type=\"media\" id=\"fs-id1167831881942\" data-alt=\"The figure on the left is labeled 5 open parentheses minus 3 close parentheses. We need to take away 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 negatives. Hence, 5 times minus 3 is minus 15. The figure on the right is labeled open parentheses minus 5 close parentheses open parentheses minus 3 close parentheses. We need to take away minus 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 positives. Hence, minus 5 times minus 3 is 15.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_021_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure on the left is labeled 5 open parentheses minus 3 close parentheses. We need to take away 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 negatives. Hence, 5 times minus 3 is minus 15. The figure on the right is labeled open parentheses minus 5 close parentheses open parentheses minus 3 close parentheses. We need to take away minus 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 positives. Hence, minus 5 times minus 3 is 15.\"><\/span><p id=\"fs-id1167834556774\">In summary:<\/p><div data-type=\"equation\" id=\"fs-id1167834556777\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}5\u00b73=15\\hfill &amp; &amp; &amp; -5\\left(3\\right)=-15\\hfill \\\\ 5\\left(-3\\right)=-15\\hfill &amp; &amp; &amp; \\left(-5\\right)\\left(-3\\right)=15\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835304702\">Notice that for multiplication of two signed numbers, when the<\/p><div data-type=\"equation\" id=\"fs-id1167835304705\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\text{signs are the}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{same}}\\text{, the product is}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{positive}}\\text{.}\\hfill \\\\ \\text{signs are}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{different}}\\text{, the product is}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{negative}}\\text{.}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167831910778\">What about division? Division is the inverse operation of multiplication. So, \\(15\u00f73=5\\) because \\(5\u00b73=15.\\) In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.<\/p><div data-type=\"equation\" id=\"fs-id1167834183503\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccccc}5\u00b73=15\\hfill &amp; \\text{so}\\hfill &amp; 15\u00f73=5\\hfill &amp; &amp; &amp; -5\\left(3\\right)=-15\\hfill &amp; \\text{so}\\hfill &amp; \\phantom{\\rule{0.2em}{0ex}}-15\u00f73=-5\\hfill \\\\ \\left(-5\\right)\\left(-3\\right)=15\\hfill &amp; \\text{so}\\hfill &amp; 15\u00f7\\left(-3\\right)=-5\\hfill &amp; &amp; &amp; 5\\left(-3\\right)=-15\\hfill &amp; \\text{so}\\hfill &amp; -15\u00f7\\left(-3\\right)=5\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167832076237\">Division follows the same rules as multiplication with regard to signs.<\/p><div data-type=\"note\" id=\"fs-id1167832076240\"><div data-type=\"title\">Multiplication and Division of Signed Numbers<\/div><p id=\"fs-id1167835468553\">For multiplication and division of two signed numbers:<\/p><table id=\"fs-id1167835468556\" summary=\".\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Same signs<\/th><th data-valign=\"top\" data-align=\"left\">Result<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Two positives<\/td><td data-valign=\"top\" data-align=\"left\">Positive<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Two negatives<\/td><td data-valign=\"top\" data-align=\"left\">Positive<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167834505247\">If the signs are the same, the result is positive.<\/p><table id=\"fs-id1167834505250\" summary=\".\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\">Different signs<\/th><th data-valign=\"top\" data-align=\"left\">Result<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Positive and negative<\/td><td data-valign=\"top\" data-align=\"left\">Negative<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Negative and positive<\/td><td data-valign=\"top\" data-align=\"left\">Negative<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835232461\">If the signs are different, the result is negative.<\/p><\/div><div data-type=\"example\" id=\"fs-id1167834536325\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834536327\"><div data-type=\"problem\" id=\"fs-id1167834536329\"><p id=\"fs-id1167834535837\">Multiply or divide: <span class=\"token\">\u24d0<\/span> \\(-100\u00f7\\left(-4\\right)\\) <span class=\"token\">\u24d1<\/span> \\(7\u00b76\\) <span class=\"token\">\u24d2<\/span> \\(4\\left(-8\\right)\\) <span class=\"token\">\u24d3<\/span> \\(-27\u00f73.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831872478\"><p id=\"fs-id1167834063137\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}-100\u00f7\\left(-4\\right)\\hfill \\\\ \\text{Divide, with signs that are}\\hfill &amp; &amp; &amp; \\\\ \\text{the same the quotient is positive.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}25\\hfill \\end{array}\\)<p id=\"fs-id1167834191278\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}7\u00b76\\hfill \\\\ \\text{Multiply, with same signs.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}42\\hfill \\end{array}\\)<p id=\"fs-id1167834422857\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4.5em}{0ex}}4\\left(-8\\right)\\hfill \\\\ \\text{Multiply, with different signs.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4.5em}{0ex}}-32\\hfill \\end{array}\\)<p id=\"fs-id1167832075898\"><span class=\"token\">\u24d3<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4.5em}{0ex}}-27\u00f73\\hfill \\\\ \\text{Divide, with different signs,}\\hfill &amp; &amp; &amp; \\\\ \\text{the quotient is negative.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4.5em}{0ex}}-9\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834428912\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834345685\"><div data-type=\"problem\" id=\"fs-id1167834345687\"><p id=\"fs-id1167834345689\">Multiply or divide: <span class=\"token\">\u24d0<\/span> \\(-115\u00f7\\left(-5\\right)\\) <span class=\"token\">\u24d1<\/span> \\(5\u00b712\\) <span class=\"token\">\u24d2<\/span> \\(9\\left(-7\\right)\\) <span class=\"token\">\u24d3<\/span>\\(-63\u00f77.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834422894\"><p id=\"fs-id1167834422896\"><span class=\"token\">\u24d0<\/span> 23 <span class=\"token\">\u24d1<\/span> 60 <span class=\"token\">\u24d2<\/span> \\(-63\\) <span class=\"token\">\u24d3<\/span> \\(-9\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826804652\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826804656\"><div data-type=\"problem\" id=\"fs-id1167826804658\"><p id=\"fs-id1167834539469\">Multiply or divide: <span class=\"token\">\u24d0<\/span> \\(-117\u00f7\\left(-3\\right)\\) <span class=\"token\">\u24d1<\/span> \\(3\u00b713\\) <span class=\"token\">\u24d2<\/span> \\(7\\left(-4\\right)\\) <span class=\"token\">\u24d3<\/span>\\(-42\u00f76.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831880276\"><p id=\"fs-id1167831882554\"><span class=\"token\">\u24d0<\/span> 39 <span class=\"token\">\u24d1<\/span> 39 <span class=\"token\">\u24d2<\/span> \u221228 <span class=\"token\">\u24d3<\/span> \u22127<\/p><\/div><\/div><\/div><p id=\"fs-id1167834228840\">When we multiply a number by 1, the result is the same number. Each time we multiply a number by \\(-1,\\) we get its opposite!<\/p><div data-type=\"note\" id=\"fs-id1167834537368\"><div data-type=\"title\">Multiplication by \\(-1\\) <\/div><div data-type=\"equation\" id=\"fs-id1167835575838\" class=\"unnumbered\" data-label=\"\">\\(-1a=\\text{\u2212}a\\)<\/div><p id=\"fs-id1167831880476\">Multiplying a number by \\(-1\\) gives its opposite.<\/p><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832053258\"><h3 data-type=\"title\">Simplify Expressions with Integers<\/h3><p id=\"fs-id1167832053263\">What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember Please Excuse My Dear Aunt Sally?<\/p><p id=\"fs-id1167834335086\">Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.<\/p><div data-type=\"example\" id=\"fs-id1167834536158\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834536160\"><div data-type=\"problem\" id=\"fs-id1167834536162\"><p id=\"fs-id1167834536165\">Simplify: <span class=\"token\">\u24d0<\/span> \\({\\left(-2\\right)}^{4}\\) <span class=\"token\">\u24d1<\/span> \\(\\text{\u2212}{2}^{4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832066494\"><p id=\"fs-id1167832066496\">Notice the difference in parts (a) and (b). In part (a), the exponent means to raise what is in the parentheses, the \\(-2\\) to the 4<sup>th<\/sup> power. In part (b), the exponent means to raise just the 2 to the 4<sup>th<\/sup> power and then take the opposite.<\/p><p id=\"fs-id1167834062721\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(-2\\right)}^{4}\\hfill \\\\ \\text{Write in expanded form.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(-2\\right)\\left(-2\\right)\\left(-2\\right)\\left(-2\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}4\\left(-2\\right)\\left(-2\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}-8\\left(-2\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}16\\hfill \\end{array}\\)<p id=\"fs-id1167831239202\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4.6em}{0ex}}\\text{\u2212}{2}^{4}\\hfill \\\\ \\text{Write in expanded form.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4.6em}{0ex}}\\text{\u2212}\\left(2\u00b72\u00b72\u00b72\\right)\\hfill \\\\ \\text{We are asked to find}\\hfill &amp; &amp; &amp; \\\\ \\text{the opposite of}\\phantom{\\rule{0.2em}{0ex}}{2}^{4}\\text{.}\\hfill &amp; &amp; &amp; \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4.6em}{0ex}}\\text{\u2212}\\left(4\u00b72\u00b72\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4.6em}{0ex}}\\text{\u2212}\\left(8\u00b72\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4.6em}{0ex}}-16\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832042791\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834505582\"><div data-type=\"problem\" id=\"fs-id1167834505584\"><p id=\"fs-id1167834505586\">Simplify: <span class=\"token\">\u24d0<\/span> \\({\\left(-3\\right)}^{4}\\) <span class=\"token\">\u24d1<\/span> \\(\\text{\u2212}{3}^{4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831826507\"><p id=\"fs-id1167831826509\"><span class=\"token\">\u24d0<\/span> 81 <span class=\"token\">\u24d1<\/span> \\(-81\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834228426\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834536588\"><div data-type=\"problem\" id=\"fs-id1167834536591\"><p id=\"fs-id1167834536593\">Simplify: <span class=\"token\">\u24d0<\/span> \\({\\left(-7\\right)}^{2}\\) <span class=\"token\">\u24d1<\/span> \\(\\text{\u2212}{7}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831871457\"><p id=\"fs-id1167831871460\"><span class=\"token\">\u24d0<\/span> 49 <span class=\"token\">\u24d1<\/span> \\(-49\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167832053420\">The last example showed us the difference between \\({\\left(-2\\right)}^{4}\\) and \\(\\text{\u2212}{2}^{4}.\\) This distinction is important to prevent future errors. The next example reminds us to multiply and divide in order left to right.<\/p><div data-type=\"example\" id=\"fs-id1167834556092\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834556094\"><div data-type=\"problem\" id=\"fs-id1167834183406\"><p id=\"fs-id1167834183408\">Simplify: <span class=\"token\">\u24d0<\/span> \\(8\\left(-9\\right)\u00f7{\\left(-2\\right)}^{3}\\) <span class=\"token\">\u24d1<\/span> \\(-30\u00f72+\\left(-3\\right)\\left(-7\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835378126\"><p id=\"fs-id1167835378128\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{8em}{0ex}}8\\left(-9\\right)\u00f7{\\left(-2\\right)}^{3}\\hfill \\\\ \\text{Exponents first.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{8em}{0ex}}8\\left(-9\\right)\u00f7\\left(-8\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{8em}{0ex}}-72\u00f7\\left(-8\\right)\\hfill \\\\ \\text{Divide.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{8em}{0ex}}9\\hfill \\end{array}\\)<p id=\"fs-id1167831880168\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}-30\u00f72+\\left(-3\\right)\\left(-7\\right)\\hfill \\\\ \\text{Multiply and divide}\\hfill &amp; &amp; &amp; \\\\ \\text{left to right, so divide first.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}-15+\\left(-3\\right)\\left(-7\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}-15+21\\hfill \\\\ \\text{Add.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}6\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832053084\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832053087\"><div data-type=\"problem\" id=\"fs-id1167832053089\"><p id=\"fs-id1167832053091\">Simplify: <span class=\"token\">\u24d0<\/span> \\(12\\left(-9\\right)\u00f7{\\left(-3\\right)}^{3}\\) <span class=\"token\">\u24d1<\/span> \\(-27\u00f73+\\left(-5\\right)\\left(-6\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826880228\"><p id=\"fs-id1167831826512\"><span class=\"token\">\u24d0<\/span> 4 <span class=\"token\">\u24d1<\/span> 21<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834536337\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834536342\"><div data-type=\"problem\" id=\"fs-id1167834536344\"><p id=\"fs-id1167834536346\">Simplify: <span class=\"token\">\u24d0<\/span> \\(18\\left(-4\\right)\u00f7{\\left(-2\\right)}^{3}\\) <span class=\"token\">\u24d1<\/span> \\(-32\u00f74+\\left(-2\\right)\\left(-7\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835575709\"><p id=\"fs-id1167835575711\"><span class=\"token\">\u24d0<\/span> 9 <span class=\"token\">\u24d1<\/span> 6<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832053123\"><h3 data-type=\"title\">Evaluate Variable Expressions with Integers<\/h3><p id=\"fs-id1167832053128\">Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.<\/p><div data-type=\"example\" id=\"fs-id1167832053133\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832053135\"><div data-type=\"problem\" id=\"fs-id1167832053137\"><p id=\"fs-id1167831880328\">Evaluate \\(4{x}^{2}-2xy+3{y}^{2}\\) when \\(x=2,y=-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834238705\"><table id=\"fs-id1167834238707\" summary=\"The expression is 4 x squared minus 2 x y plus 3 y squared. Substitute x with 2 and y with minus 1. Using parentheses to show multiplication, we get 4 open parentheses 2 close parentheses squared minus 2 open parentheses 2 close parentheses open parentheses minus 1 close parentheses plus 3 open parentheses minus 1 close parentheses squared. Simplifying exponents, we get 4 times 4 minus open parentheses minus 4 close parentheses plus 3 times 1. We multiply to get 16 minus open parentheses minus 4 close parentheses plus 3. We subtract to get 20 plus 3. We add to get 23.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr valign=\"top\"><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834539432\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834535493\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{4em}{0ex}}\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831871648\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify exponents.<\/td><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831880151\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831880503\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract.<\/td><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832076130\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add.<\/td><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832053446\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834422813\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834422817\"><div data-type=\"problem\" id=\"fs-id1167834422819\"><p id=\"fs-id1167834422821\">Evaluate: \\(3{x}^{2}-2xy+6{y}^{2}\\) when \\(x=1,y=-2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835514354\"><p id=\"fs-id1167835514356\">31<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834536416\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834536420\"><div data-type=\"problem\" id=\"fs-id1167834536422\"><p id=\"fs-id1167834536424\">Evaluate: \\(4{x}^{2}-xy+5{y}^{2}\\) when \\(x=-2,y=3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831910923\"><p id=\"fs-id1167831910925\">67<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831910933\"><h3 data-type=\"title\">Translate Phrases to Expressions with Integers<\/h3><p id=\"fs-id1167832066700\">Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.<\/p><div data-type=\"example\" id=\"fs-id1167832066704\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832066706\"><div data-type=\"problem\" id=\"fs-id1167832066708\"><p id=\"fs-id1167832066711\">Translate and simplify: the sum of 8 and \\(-12,\\) increased by \\(3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834539696\"><p id=\"fs-id1167834539698\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}\\text{the}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{sum}}\\phantom{\\rule{0.2em}{0ex}}\\underset{\\text{\u2013}}{\\text{of}}\\phantom{\\rule{0.2em}{0ex}}8\\phantom{\\rule{0.2em}{0ex}}\\underset{\\text{\u2013\u2013\u2013}}{\\text{and}}\\phantom{\\rule{0.2em}{0ex}}-12\\phantom{\\rule{0.2em}{0ex}}\\text{increased by 3}\\hfill \\\\ \\text{Translate.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}\\left[8+\\left(-12\\right)\\right]+3\\hfill \\\\ \\text{Simplify. Be careful not to confuse the}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}\\left(-4\\right)+3\\hfill \\\\ \\text{brackets with an absolute value sign.}\\hfill &amp; &amp; &amp; \\\\ \\text{Add.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}-1\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834539459\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835352729\"><div data-type=\"problem\" id=\"fs-id1167835352731\"><p id=\"fs-id1167835352733\">Translate and simplify the sum of 9 and \\(-16,\\) increased by 4.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835352745\"><p id=\"fs-id1167835352747\">\\(\\left(9+\\left(-16\\right)\\right)+4;-3\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834535728\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834535732\"><div data-type=\"problem\" id=\"fs-id1167834535735\"><p id=\"fs-id1167834537376\">Translate and simplify the sum of \\(-8\\) and \\(-12,\\) increased by 7.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834537393\"><p id=\"fs-id1167834537395\">\\(\\left(-8+\\left(-12\\right)\\right)+7;-13\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832076190\"><h3 data-type=\"title\">Use Integers in Applications<\/h3><p id=\"fs-id1167832076195\">We\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.<\/p><div data-type=\"example\" id=\"fs-id1167834537844\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve Application Problems Using Integers<\/div><div data-type=\"exercise\" id=\"fs-id1167834537846\"><div data-type=\"problem\" id=\"fs-id1167834537848\"><p id=\"fs-id1167834537850\">The temperature in Kendallville, Indiana one morning was 11 degrees. By mid-afternoon, the temperature had dropped to \\(-9\\) degrees. What was the difference in the morning and afternoon temperatures?<\/p><span data-type=\"media\" id=\"fs-id1167832076198\" data-alt=\"Figure shows a glass thermometer, with temperature markings ranging from minus 10 to 30. Two markings are highlighted, minus 9 degrees C and 11 degrees C.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_016_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a glass thermometer, with temperature markings ranging from minus 10 to 30. Two markings are highlighted, minus 9 degrees C and 11 degrees C.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167834182486\"><span data-type=\"media\" id=\"fs-id1167834182489\" data-alt=\"Step 1 is to read the problem and make sure all the words and ideas are understood.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to read the problem and make sure all the words and ideas are understood.\"><\/span><span data-type=\"media\" id=\"fs-id1167831881150\" data-alt=\"Step 2 is to identify what we are asked to find. Here, we need to find the difference of the morning and afternoon temperatures.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to identify what we are asked to find. Here, we need to find the difference of the morning and afternoon temperatures.\"><\/span><span data-type=\"media\" id=\"fs-id1167831881162\" data-alt=\"Step 3 is to write a phrase that gives the information to find it. In this case, the phrase is the difference of 11 and minus 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to write a phrase that gives the information to find it. In this case, the phrase is the difference of 11 and minus 9.\"><\/span><span data-type=\"media\" id=\"fs-id1167835511995\" data-alt=\"Step 4 is to translate the phrase to an expression. Here, we write 11 minus open parentheses minus 9 close parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to translate the phrase to an expression. Here, we write 11 minus open parentheses minus 9 close parentheses.\"><\/span><span data-type=\"media\" id=\"fs-id1167835512007\" data-alt=\"In step 5, we simplify the expression to get 20.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017e_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 5, we simplify the expression to get 20.\"><\/span><span data-type=\"media\" id=\"fs-id1167832068492\" data-alt=\"Step 6 is to answer the question with a complete sentence: The difference in temperatures was 20 degrees.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017f_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to answer the question with a complete sentence: The difference in temperatures was 20 degrees.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832068508\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832068512\"><div data-type=\"problem\" id=\"fs-id1167835187799\"><p id=\"fs-id1167835187801\">The temperature in Anchorage, Alaska one morning was \\(15\\) degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835187812\"><p id=\"fs-id1167835187814\">The difference in temperatures was 45 degrees Fahrenheit.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835187821\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835287446\"><div data-type=\"problem\" id=\"fs-id1167835287448\"><p id=\"fs-id1167835287450\">The temperature in Denver was \\(-6\\) degrees at lunchtime. By sunset the temperature had dropped to \\(-15\\) degrees. What was the difference in the lunchtime and sunset temperatures?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835287465\"><p id=\"fs-id1167835287467\">The difference in temperatures was 9 degrees.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834538046\" class=\"howto\"><div data-type=\"title\">Use Integers in Applications.<\/div><ol id=\"fs-id1167834538053\" type=\"1\" class=\"stepwise\"><li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li><li><strong data-effect=\"bold\">Identify<\/strong> what we are asked to find.<\/li><li><strong data-effect=\"bold\">Write a phrase<\/strong> that gives the information to find it.<\/li><li><strong data-effect=\"bold\">Translate<\/strong> the phrase to an expression.<\/li><li><strong data-effect=\"bold\">Simplify<\/strong> the expression.<\/li><li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li><\/ol><\/div><div data-type=\"note\" id=\"fs-id1167834533182\" class=\"media-2\"><p id=\"fs-id1167834533186\">Access this online resource for additional instruction and practice with integers.<\/p><ul id=\"fs-id1167834533189\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/25subintegers\">Subtracting Integers with Counters<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167834535516\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167836310060\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Opposite Notation<\/strong><div data-type=\"newline\"><br><\/div><div data-type=\"equation\" id=\"fs-id1167834535530\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\text{\u2212}a\\phantom{\\rule{0.2em}{0ex}}\\text{means the opposite of the number}\\phantom{\\rule{0.2em}{0ex}}a\\hfill \\\\ \\text{The notation}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2212}a\\phantom{\\rule{0.2em}{0ex}}\\text{is read as \u201cthe opposite of}\\phantom{\\rule{0.2em}{0ex}}a\\text{.\u201d}\\hfill \\end{array}\\)<\/div><\/li><li><strong data-effect=\"bold\">Absolute Value<\/strong><div data-type=\"newline\"><br><\/div> The absolute value of a number is its distance from 0 on the number line.<div data-type=\"newline\"><br><\/div> The absolute value of a number <em data-effect=\"italics\">n<\/em> is written as \\(|n|\\) and \\(|n|\\ge 0\\) for all numbers.<div data-type=\"newline\"><br><\/div> Absolute values are always greater than or equal to zero.<\/li><li><strong data-effect=\"bold\">Grouping Symbols<\/strong><div data-type=\"newline\"><br><\/div><div data-type=\"equation\" id=\"fs-id1167834537897\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccccccc}\\text{Parentheses}\\hfill &amp; &amp; &amp; \\left(\\phantom{\\rule{0.2em}{0ex}}\\right)\\hfill &amp; &amp; &amp; \\text{Braces}\\hfill &amp; &amp; &amp; \\left\\{\\phantom{\\rule{0.2em}{0ex}}\\right\\}\\hfill \\\\ \\text{Brackets}\\hfill &amp; &amp; &amp; \\left[\\phantom{\\rule{0.2em}{0ex}}\\right]\\hfill &amp; &amp; &amp; \\text{Absolute value}\\hfill &amp; &amp; &amp; \\phantom{\\rule{0.2em}{0ex}}|\\phantom{\\rule{0.2em}{0ex}}|\\hfill \\end{array}\\)<\/div><\/li><li><strong data-effect=\"bold\">Subtraction Property<\/strong><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{2em}{0ex}}a-b=a+\\left(\\text{\u2212}b\\right)\\)<div data-type=\"newline\"><br><\/div> Subtracting a number is the same as adding its opposite.<\/li><li><strong data-effect=\"bold\">Multiplication and Division of Signed Numbers<\/strong><div data-type=\"newline\"><br><\/div> For multiplication and division of two signed numbers:<div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167834533537\" summary=\".\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><u data-effect=\"underline\">Same signs<\/u><\/strong><\/th><th data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><u data-effect=\"underline\">Result<\/u><\/strong><\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Two positives<\/td><td data-valign=\"top\" data-align=\"left\">Positive<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Two negatives<\/td><td data-valign=\"top\" data-align=\"left\">Positive<\/td><\/tr><\/tbody><\/table><div data-type=\"newline\"><br><\/div> \\(\\phantom{\\rule{12.5em}{0ex}}\\)If the signs are the same, the result is positive.<div data-type=\"newline\"><br><\/div> <table id=\"fs-id1167834526182\" summary=\".\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><u data-effect=\"underline\">Different signs<\/u><\/strong><\/th><th data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><u data-effect=\"underline\">Result<\/u><\/strong><\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Positive and negative<\/td><td data-valign=\"top\" data-align=\"left\">Negative<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Negative and positive<\/td><td data-valign=\"top\" data-align=\"left\">Negative<\/td><\/tr><\/tbody><\/table><div data-type=\"newline\"><br><\/div> \\(\\phantom{\\rule{12.2em}{0ex}}\\)If the signs are different, the result is negative.<\/li><li><strong data-effect=\"bold\">Multiplication by<\/strong>\\(-1\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{2em}{0ex}}-1a=\\text{\u2212}a\\)<div data-type=\"newline\"><br><\/div> Multiplying a number by \\(-1\\) gives its opposite.<\/li><li><strong data-effect=\"bold\">How to Use Integers in Applications.<\/strong><div data-type=\"newline\"><br><\/div><ol id=\"fs-id1167834539249\" type=\"1\" class=\"stepwise\"><li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood<\/li><li><strong data-effect=\"bold\">Identify<\/strong> what we are asked to find.<\/li><li><strong data-effect=\"bold\">Write a phrase<\/strong> that gives the information to find it.<\/li><li><strong data-effect=\"bold\">Translate<\/strong> the phrase to an expression.<\/li><li><strong data-effect=\"bold\">Simplify<\/strong> the expression.<\/li><li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831826351\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167831826355\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167831826362\"><strong data-effect=\"bold\">Simplify Expressions with Absolute Value<\/strong><\/p><p id=\"fs-id1167831826368\">In the following exercises, fill in \\(&lt;,&gt;,\\) or \\(=\\) for each of the following pairs of numbers.<\/p><div data-type=\"exercise\" id=\"fs-id1167834238749\"><div data-type=\"problem\" id=\"fs-id1167834238751\"><p id=\"fs-id1167834238753\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(|-7|___-|-7|\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(6___-|-6|\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(|-11|___-11\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\text{\u2212}\\left(-13\\right)___-|-13|\\)<\/div><div data-type=\"solution\" id=\"fs-id1167831882179\"><p id=\"fs-id1167831882181\"><span class=\"token\">\u24d0<\/span>\\(&gt;\\)<span class=\"token\">\u24d1<\/span>\\(&gt;\\)<span class=\"token\">\u24d2<\/span>\\(&gt;\\)<span class=\"token\">\u24d3<\/span>\\(&gt;\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834534488\"><div data-type=\"problem\" id=\"fs-id1167834244227\"><p id=\"fs-id1167834244229\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\text{\u2212}|-9|___|-9|\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-8___|-8|\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(|-1|___-1\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\text{\u2212}\\left(-14\\right)___-|-14|\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826978155\"><div data-type=\"problem\" id=\"fs-id1167826978157\"><p id=\"fs-id1167832086878\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\text{\u2212}|2|___-|-2|\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-12___-|-12|\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(|-3|___-3\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(|-19|___-\\left(-19\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834538202\"><p id=\"fs-id1167834538205\"><span class=\"token\">\u24d0<\/span>\\(=\\)<span class=\"token\">\u24d1<\/span>\\(=\\)<span class=\"token\">\u24d2<\/span>\\(&gt;\\)<span class=\"token\">\u24d3<\/span>\\(=\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831826418\"><div data-type=\"problem\" id=\"fs-id1167831826420\"><p id=\"fs-id1167831826422\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\text{\u2212}|-4|___-|4|\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(5___-|-5|\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\text{\u2212}|-10|___-10\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(\\text{\u2212}|-0|___-\\left(-0\\right)\\)<\/div><\/div><p id=\"fs-id1167835511702\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167835511705\"><div data-type=\"problem\" id=\"fs-id1167835511707\"><p id=\"fs-id1167835511709\">\\(|15-7|-|14-6|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835511742\"><p id=\"fs-id1167835511744\">0<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835511750\"><div data-type=\"problem\" id=\"fs-id1167834536356\"><p id=\"fs-id1167834536358\">\\(|17-8|-|13-4|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834536399\"><div data-type=\"problem\" id=\"fs-id1167834536401\"><p id=\"fs-id1167834536403\">\\(18-|2\\left(8-3\\right)|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831912722\"><p id=\"fs-id1167831912724\">8<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831912729\"><div data-type=\"problem\" id=\"fs-id1167831912731\"><p id=\"fs-id1167831912733\">\\(15-|3\\left(8-5\\right)|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835379691\"><div data-type=\"problem\" id=\"fs-id1167835379693\"><p id=\"fs-id1167835379695\">\\(18-|12-4\\left(4-1\\right)+3|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835510052\"><p id=\"fs-id1167835510054\">15<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835510059\"><div data-type=\"problem\" id=\"fs-id1167835510061\"><p id=\"fs-id1167835510064\">\\(27-|19+4\\left(3-1\\right)-7|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834531328\"><div data-type=\"problem\" id=\"fs-id1167834531330\"><p id=\"fs-id1167834531332\">\\(10-3|9-3\\left(3-1\\right)|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834531370\"><p id=\"fs-id1167834531372\">1<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834372402\"><div data-type=\"problem\" id=\"fs-id1167834372404\"><p id=\"fs-id1167834372406\">\\(13-2|11-2\\left(5-2\\right)|\\)<\/p><\/div><\/div><p id=\"fs-id1167834372451\"><strong data-effect=\"bold\">Add and Subtract Integers<\/strong><\/p><p id=\"fs-id1167834372456\">In the following exercises, simplify each expression.<\/p><div data-type=\"exercise\" id=\"fs-id1167834372460\"><div data-type=\"problem\" id=\"fs-id1167834372462\"><p id=\"fs-id1167834372464\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-7+\\left(-4\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-7+4\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(7+\\left(-4\\right).\\)<\/div><div data-type=\"solution\" id=\"fs-id1167832042872\"><p id=\"fs-id1167832042874\"><span class=\"token\">\u24d0<\/span>\\(-11\\)<span class=\"token\">\u24d1<\/span>\\(-3\\)<span class=\"token\">\u24d2<\/span>\\(3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834372359\"><div data-type=\"problem\" id=\"fs-id1167834372361\"><p id=\"fs-id1167834372363\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-5+\\left(-9\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-5+9\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(5+\\left(-9\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834533260\"><div data-type=\"problem\" id=\"fs-id1167834533262\"><p id=\"fs-id1167834533264\">\\(48+\\left(-16\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835622299\"><p id=\"fs-id1167835622302\">32<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835622307\"><div data-type=\"problem\" id=\"fs-id1167835622309\"><p id=\"fs-id1167835622311\">\\(34+\\left(-19\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835622337\"><div data-type=\"problem\" id=\"fs-id1167835622339\"><p id=\"fs-id1167835622341\">\\(-14+\\left(-12\\right)+4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834535752\"><p id=\"fs-id1167834535754\">\\(-22\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535763\"><div data-type=\"problem\" id=\"fs-id1167834535765\"><p id=\"fs-id1167834535767\">\\(-17+\\left(-18\\right)+6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535800\"><div data-type=\"problem\" id=\"fs-id1167834535802\"><p id=\"fs-id1167834535804\">\\(19+2\\left(-3+8\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831880853\"><p id=\"fs-id1167831880855\">\\(29\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831880864\"><div data-type=\"problem\" id=\"fs-id1167831880866\"><p id=\"fs-id1167831880868\">\\(24+3\\left(-5+9\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831880900\"><div data-type=\"problem\" id=\"fs-id1167831880902\"><p id=\"fs-id1167831880904\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(13-7\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-13-\\left(-7\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(-13-7\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(13-\\left(-7\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834538296\"><p id=\"fs-id1167834538298\"><span class=\"token\">\u24d0<\/span> 6 <span class=\"token\">\u24d1<\/span> \\(-6\\) <span class=\"token\">\u24d2<\/span> \\(-20\\) <span class=\"token\">\u24d3<\/span> \\(20\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835575382\"><div data-type=\"problem\" id=\"fs-id1167835575384\"><p id=\"fs-id1167835575386\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(15-8\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-15-\\left(-8\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(-15-8\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(15-\\left(-8\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835622273\"><div data-type=\"problem\" id=\"fs-id1167834536446\"><p id=\"fs-id1167834536448\">\\(-17-42\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834536459\"><p id=\"fs-id1167834536462\">\\(-59\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834536470\"><div data-type=\"problem\" id=\"fs-id1167834536472\"><p id=\"fs-id1167834536474\">\\(-58-\\left(-67\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834536502\"><div data-type=\"problem\" id=\"fs-id1167834536504\"><p id=\"fs-id1167834536506\">\\(-14-\\left(-27\\right)+9\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835479492\"><p id=\"fs-id1167835479494\">22<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835479499\"><div data-type=\"problem\" id=\"fs-id1167835479501\"><p id=\"fs-id1167835479504\">\\(64+\\left(-17\\right)-9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835479533\"><div data-type=\"problem\" id=\"fs-id1167835479536\"><p id=\"fs-id1167835479538\"><span class=\"token\">\u24d0<\/span>\\(44-28\\)<span class=\"token\">\u24d1<\/span>\\(44+\\left(-28\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835319147\"><p id=\"fs-id1167835319149\"><span class=\"token\">\u24d0<\/span> 16 <span class=\"token\">\u24d1<\/span>16<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835319163\"><div data-type=\"problem\" id=\"fs-id1167835319166\"><p id=\"fs-id1167835319168\"><span class=\"token\">\u24d0<\/span>\\(35-16\\)<span class=\"token\">\u24d1<\/span>\\(35+\\left(-16\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835575747\"><div data-type=\"problem\" id=\"fs-id1167835575749\"><p id=\"fs-id1167835575751\"><span class=\"token\">\u24d0<\/span>\\(27-\\left(-18\\right)\\)<span class=\"token\">\u24d1<\/span>\\(27+18\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835575789\"><p id=\"fs-id1167835575792\"><span class=\"token\">\u24d0<\/span> 45 <span class=\"token\">\u24d1<\/span>45<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835575806\"><div data-type=\"problem\" id=\"fs-id1167835575808\"><p id=\"fs-id1167835575810\"><span class=\"token\">\u24d0<\/span>\\(46-\\left(-37\\right)\\)<span class=\"token\">\u24d1<\/span>\\(46+37\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826808779\"><div data-type=\"problem\" id=\"fs-id1167826808781\"><p id=\"fs-id1167826808783\">\\(\\left(2-7\\right)-\\left(3-8\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826808816\"><p id=\"fs-id1167826808818\">0<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826808824\"><div data-type=\"problem\" id=\"fs-id1167835575614\"><p id=\"fs-id1167835575616\">\\(\\left(1-8\\right)-\\left(2-9\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835575657\"><div data-type=\"problem\" id=\"fs-id1167835575659\"><p id=\"fs-id1167835575661\">\\(\\text{\u2212}\\left(6-8\\right)-\\left(2-4\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835575841\"><p id=\"fs-id1167835575843\">4<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835575848\"><div data-type=\"problem\" id=\"fs-id1167835575850\"><p id=\"fs-id1167835575853\">\\(\\text{\u2212}\\left(4-5\\right)-\\left(7-8\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835575895\"><div data-type=\"problem\" id=\"fs-id1167835575897\"><p id=\"fs-id1167835575900\">\\(25-\\left[10-\\left(3-12\\right)\\right]\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831912593\"><p id=\"fs-id1167831912595\">6<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831912600\"><div data-type=\"problem\" id=\"fs-id1167831912602\"><p id=\"fs-id1167831912604\">\\(32-\\left[5-\\left(15-20\\right)\\right]\\)<\/p><\/div><\/div><p id=\"fs-id1167831912645\"><strong data-effect=\"bold\">Multiply and Divide Integers<\/strong><\/p><p id=\"fs-id1167831912651\">In the following exercises, multiply or divide.<\/p><div data-type=\"exercise\" id=\"fs-id1167831912654\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831912656\"><p id=\"fs-id1167831912659\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-4\u00b78\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(13\\left(-5\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(-24\u00f76\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(-52\u00f7\\left(-4\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834182419\"><p id=\"fs-id1167834182421\"><span class=\"token\">\u24d0<\/span>\\(-32\\)<span class=\"token\">\u24d1<\/span>\\(-65\\)<span class=\"token\">\u24d2<\/span>\\(-4\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(13\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834538486\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834538488\"><p id=\"fs-id1167834538490\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-3\u00b79\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(9\\left(-7\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(35\u00f7\\left(-7\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(-84\u00f7\\left(-6\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835575498\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835575500\"><p id=\"fs-id1167835575502\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-28\u00f77\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-180\u00f715\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(3\\left(-13\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(-1\\left(-14\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167831881018\"><p id=\"fs-id1167831881020\"><span class=\"token\">\u24d0<\/span>\\(-4\\)<span class=\"token\">\u24d1<\/span>\\(-12\\)<span class=\"token\">\u24d2<\/span>\\(-39\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(14\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831881067\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831881069\"><p id=\"fs-id1167831881071\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-36\u00f74\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-192\u00f712\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(9\\left(-7\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span>\\(-1\\left(-19\\right)\\)<\/div><\/div><p id=\"fs-id1167834449021\"><strong data-effect=\"bold\">Simplify and Evaluate Expressions with Integers<\/strong><\/p><p id=\"fs-id1167834449027\">In the following exercises, simplify each expression.<\/p><div data-type=\"exercise\" id=\"fs-id1167834449030\"><div data-type=\"problem\" id=\"fs-id1167834449032\"><p id=\"fs-id1167834449035\"><span class=\"token\">\u24d0<\/span>\\({\\left(-2\\right)}^{6}\\)<span class=\"token\">\u24d1<\/span>\\(\\text{\u2212}{2}^{6}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832053585\"><p id=\"fs-id1167832053587\"><span class=\"token\">\u24d0<\/span>\\(64\\)<span class=\"token\">\u24d1<\/span>\\(-64\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832053611\"><div data-type=\"problem\" id=\"fs-id1167832053613\"><p id=\"fs-id1167832053616\"><span class=\"token\">\u24d0<\/span>\\({\\left(-3\\right)}^{5}\\)<span class=\"token\">\u24d1<\/span>\\(\\text{\u2212}{3}^{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834533290\"><div data-type=\"problem\" id=\"fs-id1167834533292\"><p id=\"fs-id1167834533294\">\\(5\\left(-6\\right)+7\\left(-2\\right)-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834533327\"><p id=\"fs-id1167834533329\">\\(-47\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834533338\"><div data-type=\"problem\" id=\"fs-id1167834533340\"><p id=\"fs-id1167834533342\">\\(8\\left(-4\\right)+5\\left(-4\\right)-6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834533386\"><div data-type=\"problem\" id=\"fs-id1167834533388\"><p id=\"fs-id1167834533390\">\\(-3\\left(-5\\right)\\left(6\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831826200\"><p id=\"fs-id1167831826202\">\\(90\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831826211\"><div data-type=\"problem\" id=\"fs-id1167831826213\"><p id=\"fs-id1167831826215\">\\(-4\\left(-6\\right)\\left(3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831826249\"><div data-type=\"problem\" id=\"fs-id1167831826252\"><p id=\"fs-id1167831826254\">\\(\\left(8-11\\right)\\left(9-12\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831826285\"><p id=\"fs-id1167831826287\">\\(9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831826294\"><div data-type=\"problem\" id=\"fs-id1167831826296\"><p id=\"fs-id1167831826298\">\\(\\left(6-11\\right)\\left(8-13\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834541913\"><div data-type=\"problem\" id=\"fs-id1167834541915\"><p id=\"fs-id1167834541917\">\\(26-3\\left(2-7\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834541941\"><p id=\"fs-id1167834541944\">\\(41\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834541952\"><div data-type=\"problem\" id=\"fs-id1167834541954\"><p id=\"fs-id1167834541956\">\\(23-2\\left(4-6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834541992\"><div data-type=\"problem\" id=\"fs-id1167834541994\"><p id=\"fs-id1167834541996\">\\(65\u00f7\\left(-5\\right)+\\left(-28\\right)\u00f7\\left(-7\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834542035\"><p id=\"fs-id1167834542037\">\\(-9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834542046\"><div data-type=\"problem\" id=\"fs-id1167834542048\"><p id=\"fs-id1167834542050\">\\(52\u00f7\\left(-4\\right)+\\left(-32\\right)\u00f7\\left(-8\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835534365\"><div data-type=\"problem\" id=\"fs-id1167835534368\"><p id=\"fs-id1167835534370\">\\(9-2\\left[3-8\\left(-2\\right)\\right]\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835534403\"><p id=\"fs-id1167835534405\">\\(-29\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835534414\"><div data-type=\"problem\" id=\"fs-id1167835534416\"><p id=\"fs-id1167835534418\">\\(11-3\\left[7-4\\left(-2\\right)\\right]\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835534462\"><div data-type=\"problem\" id=\"fs-id1167835534465\"><p id=\"fs-id1167835534467\">\\(8-|2-4\\left(4-1\\right)+3|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834448778\"><p id=\"fs-id1167834448780\">\\(1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834448787\"><div data-type=\"problem\" id=\"fs-id1167834448789\"><p id=\"fs-id1167834448792\">\\(7-|5-3\\left(4-1\\right)-6|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834448842\"><div data-type=\"problem\" id=\"fs-id1167834448844\"><p id=\"fs-id1167834448846\">\\(9-3|2\\left(2-6\\right)-\\left(3-7\\right)|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834448894\"><p id=\"fs-id1167834448896\">\\(-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834448905\"><div data-type=\"problem\" id=\"fs-id1167834448907\"><p id=\"fs-id1167834448909\">\\(5-2|2\\left(1-4\\right)-\\left(2-5\\right)|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826987727\"><div data-type=\"problem\" id=\"fs-id1167826987729\"><p id=\"fs-id1167826987731\">\\({\\left(-3\\right)}^{2}-24\u00f7\\left(8-2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826987769\"><p id=\"fs-id1167826987771\">\\(5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826987778\"><div data-type=\"problem\" id=\"fs-id1167826987781\"><p id=\"fs-id1167826987783\">\\({\\left(-4\\right)}^{2}-32\u00f7\\left(12-4\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167826987831\">In the following exercises, evaluate each expression.<\/p><div data-type=\"exercise\" id=\"fs-id1167826987834\"><div data-type=\"problem\" id=\"fs-id1167826987836\"><p id=\"fs-id1167826987838\">\\(y+\\left(-14\\right)\\) when<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(y=-33\\) <span class=\"token\">\u24d1<\/span> \\(y=30\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835216331\"><p id=\"fs-id1167835216333\"><span class=\"token\">\u24d0<\/span>\\(-47\\)<span class=\"token\">\u24d1<\/span>\\(16\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835216358\"><div data-type=\"problem\" id=\"fs-id1167835216360\"><p id=\"fs-id1167835216362\">\\(x+\\left(-21\\right)\\) when<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(x=-27\\) <span class=\"token\">\u24d1<\/span> \\(x=44\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835216437\"><div data-type=\"problem\" id=\"fs-id1167835216439\"><p id=\"fs-id1167835216441\">\\({\\left(x+y\\right)}^{2}\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(x=-3,y=14\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835216482\"><p id=\"fs-id1167835216484\">\\(121\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835216492\"><div data-type=\"problem\" id=\"fs-id1167835216495\"><p id=\"fs-id1167835216497\">\\({\\left(y+z\\right)}^{2}\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(y=-3,z=15\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834534772\"><div data-type=\"problem\" id=\"fs-id1167834534774\"><p id=\"fs-id1167834534776\">\\(9a-2b-8\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(a=-6\\) and \\(b=-3\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834534816\"><p id=\"fs-id1167834534818\">\\(-56\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834534827\"><div data-type=\"problem\" id=\"fs-id1167834534829\"><p id=\"fs-id1167834534831\">\\(7m-4n-2\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(m=-4\\) and \\(n=-9\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834534881\"><div data-type=\"problem\" id=\"fs-id1167834534883\"><p id=\"fs-id1167834534885\">\\(3{x}^{2}-4xy+2{y}^{2}\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(x=-2,y=-3\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834534935\"><p id=\"fs-id1167834534937\">\\(6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834534944\"><div data-type=\"problem\" id=\"fs-id1167834534946\"><p id=\"fs-id1167834534949\">\\(4{x}^{2}-xy+3{y}^{2}\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(x=-3,y=-2\\)<\/div><\/div><p id=\"fs-id1167834535007\"><strong data-effect=\"bold\">Translate English Phrases to Algebraic Expressions<\/strong><\/p><p id=\"fs-id1167835518988\">In the following exercises, translate to an algebraic expression and simplify if possible.<\/p><div data-type=\"exercise\" id=\"fs-id1167835518992\"><div data-type=\"problem\" id=\"fs-id1167835518994\"><p id=\"fs-id1167835518996\">the sum of 3 and \\(-15,\\) increased by 7<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835519008\"><p id=\"fs-id1167835519010\">\\(\\left(3+\\left(-15\\right)\\right)+7;-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835519047\"><div data-type=\"problem\" id=\"fs-id1167835519049\"><p id=\"fs-id1167835519051\">the sum of \\(-8\\) and \\(-9,\\) increased by \\(23\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835519110\"><div data-type=\"problem\" id=\"fs-id1167835519112\"><p id=\"fs-id1167835519114\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> the difference of \\(10\\) and \\(-18\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> subtract \\(11\\) from \\(-25\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835519149\"><p id=\"fs-id1167835519151\"><span class=\"token\">\u24d0<\/span>\\(10-\\left(-18\\right);28\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-25-11;-36\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835519200\"><div data-type=\"problem\" id=\"fs-id1167835519202\"><p id=\"fs-id1167835519205\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> the difference of \\(-5\\) and \\(-30\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> subtract \\(-6\\) from \\(-13\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535035\"><div data-type=\"problem\" id=\"fs-id1167834535037\"><p id=\"fs-id1167834535039\">the quotient of \\(-6\\) and the sum of \\(a\\) and \\(b\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834535057\"><p id=\"fs-id1167834535059\">\\(\\frac{-6}{a+b}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535078\"><div data-type=\"problem\" id=\"fs-id1167834535080\"><p id=\"fs-id1167834535082\">the product of \\(-13\\) and the difference of \\(c\\) and \\(d\\)<\/p><\/div><\/div><p id=\"fs-id1167834535123\"><strong data-effect=\"bold\">Use Integers in Applications<\/strong><\/p><p id=\"fs-id1167834535130\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167834535133\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834535135\"><p id=\"fs-id1167834535137\"><strong data-effect=\"bold\">Temperature<\/strong> On January 15, the high temperature in Anaheim, California, was 84\u00b0. That same day, the high temperature in Embarrass, Minnesota, was \\(\\text{\u2212}12\\text{\u00b0}.\\) What was the difference between the temperature in Anaheim and the temperature in Embarrass?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834535162\"><p id=\"fs-id1167834535163\">\\(96\\text{\u00b0}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535176\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834535178\"><p id=\"fs-id1167834535180\"><strong data-effect=\"bold\">Temperature<\/strong> On January 21, the high temperature in Palm Springs, California, was \\(89\\text{\u00b0},\\) and the high temperature in Whitefield, New Hampshire, was \\(\\text{\u2212}31\\text{\u00b0}.\\) What was the difference between the temperature in Palm Springs and the temperature in Whitefield?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535230\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834535232\"><p id=\"fs-id1167834535234\"><strong data-effect=\"bold\">Football<\/strong> On the first down, the Chargers had the ball on their 25-yard line. On the next three downs, they lost 6 yards, gained 10 yards, and lost 8 yards. What was the yard line at the end of the fourth down?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834535244\"><p id=\"fs-id1167834535246\">21<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535252\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834535254\"><p id=\"fs-id1167834535256\"><strong data-effect=\"bold\">Football<\/strong> On the first down, the Steelers had the ball on their 30-yard line. On the next three downs, they gained 9 yards, lost 14 yards, and lost 2 yards. What was the yard line at the end of the fourth down?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535281\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834535283\"><p id=\"fs-id1167834535285\"><strong data-effect=\"bold\">Checking Account<\/strong> Mayra has ?124 in her checking account. She writes a check for ?152. What is the new balance in her checking account?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834535295\"><p id=\"fs-id1167834535297\">\\(\\text{\u2212}\\text{?}28\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535310\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834535312\"><p id=\"fs-id1167834535314\"><strong data-effect=\"bold\">Checking Account<\/strong> Reymonte has a balance of \\(\\text{\u2212}\\text{?}49\\) in his checking account. He deposits ?281 to the account. What is the new balance?<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834537505\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167834537512\"><div data-type=\"problem\" id=\"fs-id1167834537515\"><p id=\"fs-id1167834537517\">Explain why the sum of \\(-8\\) and 2 is negative, but the sum of 8 and \\(-2\\) is positive.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834537532\"><p id=\"fs-id1167834537534\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834537539\"><div data-type=\"problem\" id=\"fs-id1167834537541\"><p id=\"fs-id1167834537543\">Give an example from your life experience of adding two negative numbers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834537554\"><div data-type=\"problem\" id=\"fs-id1167834537556\"><p id=\"fs-id1167834537559\">In your own words, state the rules for multiplying and dividing integers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834537563\"><p id=\"fs-id1167834537564\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834537570\"><div data-type=\"problem\" id=\"fs-id1167834537572\"><p id=\"fs-id1167834537574\">Why is \\(\\text{\u2212}{4}^{3}={\\left(-4\\right)}^{3}?\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834537614\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167834537619\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167834537635\" data-alt=\"This table has 4 columns, 6 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: simplify expressions with absolute value, add and subtract integers, multiply and divide integers, simplify and evaluate expressions with integers, translate English phrases to algebraic expressions, use integers in applications. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 6 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: simplify expressions with absolute value, add and subtract integers, multiply and divide integers, simplify and evaluate expressions with integers, translate English phrases to algebraic expressions, use integers in applications. The remaining columns are blank.\"><\/span><p id=\"fs-id1167834537643\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167834537656\"><dt>absolute value<\/dt><dd id=\"fs-id1167834537661\">The absolute value of a number is its distance from \\(0\\) on the number line.<\/dd><\/dl><dl id=\"fs-id1167834537670\"><dt>integers<\/dt><dd id=\"fs-id1167834537674\">The whole numbers and their opposites are called the integers.<\/dd><\/dl><dl id=\"fs-id1167834537678\"><dt>negative numbers<\/dt><dd id=\"fs-id1167834537683\">Numbers less than \\(0\\) are negative numbers.<\/dd><\/dl><dl id=\"fs-id1167834537692\"><dt>opposite<\/dt><dd id=\"fs-id1167834537696\">The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Simplify expressions with absolute value<\/li>\n<li>Add and subtract integers<\/li>\n<li>Multiply and divide integers<\/li>\n<li>Simplify expressions with integers<\/li>\n<li>Evaluate variable expressions with integers<\/li>\n<li>Translate phrases to expressions with integers<\/li>\n<li>Use integers in applications<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835364461\" class=\"be-prepared\">\n<p id=\"fs-id1167835343769\">A more thorough introduction to the topics covered in this section can be found in the <em data-effect=\"italics\">Elementary Algebra<\/em> chapter, Foundations.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832051980\">\n<h3 data-type=\"title\">Simplify Expressions with Absolute Value<\/h3>\n<p id=\"fs-id1167835321040\">A <span data-type=\"term\">negative numbers<\/span> is a number less than 0. The negative numbers are to the left of zero on the number line. See <a href=\"#CNX_IntAlg_Figure_01_02_001\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_02_001\">\n<div class=\"bc-figcaption figcaption\">The number line shows the location of positive and negative numbers.<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835325238\" data-alt=\"Figure shows a horizontal line marked with numbers at equal distances. At the center of the line is 0. To the right of this, starting from the number closest to 0 are 1, 2, 3 and 4. These are labeled positive numbers. To the left of 0, starting from the number closest to 0 are minus 1, minus 2, minus 3 and minus 4. These are labeled negative numbers.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a horizontal line marked with numbers at equal distances. At the center of the line is 0. To the right of this, starting from the number closest to 0 are 1, 2, 3 and 4. These are labeled positive numbers. To the left of 0, starting from the number closest to 0 are minus 1, minus 2, minus 3 and minus 4. These are labeled negative numbers.\" \/><\/span><\/div>\n<p id=\"fs-id1167834062128\">You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> are the same distance from zero, each one is called the <span data-type=\"term\">opposite<\/span> of the other. The opposite of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3bd060f32a71334fb5cdf65d10fc75c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> and the opposite of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc9f98af08ef42304971ee2f5b39b7e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167835326232\">\n<div data-type=\"title\">Opposite<\/div>\n<p id=\"fs-id1167834186346\">The <strong data-effect=\"bold\">opposite<\/strong> of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.<\/p>\n<\/div>\n<p id=\"fs-id1167835368578\"><a href=\"#CNX_IntAlg_Figure_01_02_002\" class=\"autogenerated-content\">(Figure)<\/a> illustrates the definition.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_02_002\">\n<div class=\"bc-figcaption figcaption\">The opposite of 3 is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834195463\" data-alt=\"Figure shows a number line with the numbers 3 and minus 3 highlighted. These are equidistant from 0, both being 3 numbers away from 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_002_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line with the numbers 3 and minus 3 highlighted. These are equidistant from 0, both being 3 numbers away from 0.\" \/><\/span><\/div>\n<div data-type=\"note\" id=\"fs-id1167835596363\">\n<div data-type=\"title\">Opposite Notation<\/div>\n<div data-type=\"equation\" id=\"fs-id1167835347808\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02c66db09b467c4d2db92647ba8abda0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#101;&#97;&#110;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#111;&#116;&#97;&#116;&#105;&#111;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#114;&#101;&#97;&#100;&#32;&#97;&#115;&#32;&#96;&#96;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"345\" style=\"vertical-align: -36px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835307051\">We saw that numbers such as 3 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the <span data-type=\"term\">absolute value<\/span> of that number.<\/p>\n<div data-type=\"note\" id=\"fs-id1167831871666\">\n<div data-type=\"title\">Absolute Value<\/div>\n<p id=\"fs-id1167835304200\">The <strong data-effect=\"bold\">absolute value<\/strong> of a number is its distance from 0 on the number line.<\/p>\n<p id=\"fs-id1167834138250\">The absolute value of a number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b38a586dc76a31dfa818cbaca2da26f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#110;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"17\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6cd98f4266f7f491a976c3270ec303d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#110;&#124;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> for all numbers.<\/p>\n<p id=\"fs-id1167834345757\">Absolute values are always greater than or equal to zero.<\/p>\n<\/div>\n<p id=\"fs-id1167835325056\">For example,<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834063148\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af8c21add0262e90c0516e4f62b800c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#53;&#32;&#117;&#110;&#105;&#116;&#115;&#32;&#97;&#119;&#97;&#121;&#32;&#102;&#114;&#111;&#109;&#32;&#48;&#44;&#32;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#45;&#53;&#124;&#61;&#53;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#32;&#105;&#115;&#32;&#53;&#32;&#117;&#110;&#105;&#116;&#115;&#32;&#97;&#119;&#97;&#121;&#32;&#102;&#114;&#111;&#109;&#32;&#48;&#44;&#32;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#53;&#124;&#61;&#53;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"306\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"fs-id1167834464240\"><a href=\"#CNX_IntAlg_Figure_01_02_003\" class=\"autogenerated-content\">(Figure)<\/a> illustrates this idea.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_02_003\">\n<div class=\"bc-figcaption figcaption\">The numbers 5 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/> are 5 units away from 0.<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834439195\" data-alt=\"Figure shows a number line showing the numbers 0, 5 and minus 5. 5 and minus 5 are equidistant from 0, both being 5 units away from 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_003_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a number line showing the numbers 0, 5 and minus 5. 5 and minus 5 are equidistant from 0, both being 5 units away from 0.\" \/><\/span><\/div>\n<p id=\"fs-id1167832031212\">The absolute value of a number is never negative because distance cannot be negative. The only number with absolute value equal to zero is the number zero itself because the distance from 0 to 0 on the number line is zero units.<\/p>\n<p id=\"fs-id1167834063255\">In the next example, we\u2019ll order expressions with absolute values.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835595046\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835362334\">\n<div data-type=\"problem\" id=\"fs-id1167834219221\">\n<p id=\"fs-id1167835217963\">Fill in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7afd5b67d1411fd169472ab43b36654a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;&#44;&#62;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/> for each of the following pairs of numbers:<\/p>\n<p id=\"fs-id1167831883212\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3358a5350aa17e1fdfd1c43869f2614_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#53;&#124;&#95;&#95;&#45;&#124;&#45;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"74\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf89fec2427019ed9214ca6a6d6e3466_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#95;&#95;&#45;&#124;&#45;&#56;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"44\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f8b20e8a99fbb923978d559ff0d3d3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#95;&#95;&#45;&#124;&#45;&#57;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80c396230638327f5ea86bba8392e8f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#95;&#95;&#124;&#45;&#49;&#54;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"81\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835301570\">\n<p id=\"fs-id1167835380308\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d1e08bd8a257c997cc127cf7344972e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#45;&#53;&#124;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#95;&#95;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#124;&#45;&#53;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#95;&#95;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#114;&#100;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#62;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#45;&#53;&#124;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#62;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#124;&#45;&#53;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"585\" width=\"95\" style=\"vertical-align: -363px;\" \/><\/p>\n<p id=\"fs-id1167835342749\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b988fabb1c9a23c4b51e056a61635cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#95;&#95;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#124;&#45;&#56;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#95;&#95;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#114;&#100;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#62;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#62;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#124;&#45;&#56;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"316\" style=\"vertical-align: -37px;\" \/><\/p>\n<p id=\"fs-id1167831880002\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b054eb9a3059b7184584485c60f0682_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#57;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#95;&#95;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#124;&#45;&#57;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#57;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#95;&#95;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#114;&#100;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#57;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#57;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#124;&#45;&#57;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"327\" style=\"vertical-align: -37px;\" \/><\/p>\n<p id=\"fs-id1167835333080\"><span class=\"token\">\u24d3<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be4e1c2ceb70e7ab96b5361bca5567f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#95;&#95;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#124;&#45;&#49;&#54;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#54;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#95;&#95;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#114;&#100;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#54;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#124;&#45;&#49;&#54;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"308\" style=\"vertical-align: -37px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835365755\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835370336\">\n<div data-type=\"problem\" id=\"fs-id1167834195431\">\n<p id=\"fs-id1167831928877\">Fill in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7afd5b67d1411fd169472ab43b36654a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;&#44;&#62;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/> for each of the following pairs of numbers:<\/p>\n<p id=\"fs-id1167832056480\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f8b20e8a99fbb923978d559ff0d3d3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#95;&#95;&#45;&#124;&#45;&#57;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df43589c7bacf46b7609ad6fb24c892a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#95;&#95;&#45;&#124;&#45;&#50;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"44\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-400bd8695a005256961e40e7fcb5f20f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#95;&#95;&#124;&#45;&#56;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"47\" style=\"vertical-align: -8px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1c1121950be36606fb271bf5c8175f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#95;&#95;&#124;&#45;&#57;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"65\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835381034\">\n<p id=\"fs-id1167832052135\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5fae947beab7b30f45d5b6603772f41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835225722\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835355749\">\n<div data-type=\"problem\" id=\"fs-id1167835417963\">\n<p id=\"fs-id1167831076591\">Fill in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7afd5b67d1411fd169472ab43b36654a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;&#44;&#62;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/> for each of the following pairs of numbers:<\/p>\n<p id=\"fs-id1167835321729\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7a8c43cbf7f05fd6e745e17eeda7475_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#95;&#95;&#45;&#124;&#45;&#55;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"44\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff4b4c5ca0aed8ab754917cf3958cdae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#95;&#95;&#124;&#45;&#49;&#48;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"77\" style=\"vertical-align: -10px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-246db8e329c6458c9a6e86d7d0047dac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#52;&#124;&#95;&#95;&#45;&#124;&#45;&#52;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"74\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb4683f0b17e3e02dc7e76f92de1fd24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#95;&#95;&#124;&#45;&#49;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"51\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835198777\">\n<p id=\"fs-id1167835367362\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5fae947beab7b30f45d5b6603772f41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832052677\">We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834433550\">\n<div data-type=\"title\">Grouping Symbols<\/div>\n<div data-type=\"equation\" id=\"fs-id1167835321488\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6b2434bc0550a9dbbd3e25fca8de02f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#114;&#101;&#110;&#116;&#104;&#101;&#115;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#97;&#99;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#97;&#99;&#107;&#101;&#116;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#98;&#115;&#111;&#108;&#117;&#116;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"340\" style=\"vertical-align: -16px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835166543\">In the next example, we simplify the expressions inside absolute value bars first just like we do with parentheses.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835319324\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834180019\">\n<div data-type=\"problem\" id=\"fs-id1167834430895\">\n<p id=\"fs-id1167832058609\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ec6c01a8311817433afab9509af55ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#45;&#124;&#49;&#57;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"161\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306143\">\n<p id=\"fs-id1167832054257\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16f72f0acf5637da261d381329f0a9ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#52;&#45;&#124;&#49;&#57;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#111;&#114;&#107;&#32;&#105;&#110;&#115;&#105;&#100;&#101;&#32;&#112;&#97;&#114;&#101;&#110;&#116;&#104;&#101;&#115;&#101;&#115;&#32;&#102;&#105;&#114;&#115;&#116;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#32;&#50;&#32;&#102;&#114;&#111;&#109;&#32;&#54;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#52;&#45;&#124;&#49;&#57;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#51;&#40;&#52;&#41;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#52;&#45;&#124;&#49;&#57;&#45;&#49;&#50;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#32;&#105;&#110;&#115;&#105;&#100;&#101;&#32;&#116;&#104;&#101;&#32;&#97;&#98;&#115;&#111;&#108;&#117;&#116;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#32;&#98;&#97;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#52;&#45;&#124;&#55;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#97;&#107;&#101;&#32;&#116;&#104;&#101;&#32;&#97;&#98;&#115;&#111;&#108;&#117;&#116;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#52;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"147\" width=\"581\" style=\"vertical-align: -67px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835206644\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835180537\">\n<div data-type=\"problem\" id=\"fs-id1167832055359\">\n<p id=\"fs-id1167834124820\">Simplify:<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c93f2536250ca3e9faa2a6fcd8aedce1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#45;&#124;&#49;&#49;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"160\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835259043\">\n<p id=\"fs-id1167835258413\">16<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835164568\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835609424\">\n<div data-type=\"problem\" id=\"fs-id1167826880456\">\n<p id=\"fs-id1167835346341\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c98daaffdcbfae614dfc9df99be6a11d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#45;&#124;&#56;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835365656\">\n<p id=\"fs-id1167835262356\">9<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835511674\">\n<h3 data-type=\"title\">Add and Subtract Integers<\/h3>\n<p id=\"fs-id1167831908724\">So far, we have only used the counting numbers and the whole numbers.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835355753\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a787f938b567dd4352c2db195d2162cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#117;&#110;&#116;&#105;&#110;&#103;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#49;&#44;&#50;&#44;&#51;&#92;&#100;&#111;&#116;&#115;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#111;&#108;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#100;&#111;&#116;&#115;&#32;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"281\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"fs-id1167835335608\">Our work with opposites gives us a way to define the <span data-type=\"term\">integers<\/span>. The whole numbers and their opposites are called the integers. The integers are the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d09c237ff601972dca27dfb548908db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#111;&#116;&#115;&#32;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"178\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167835307207\">\n<div data-type=\"title\">Integers<\/div>\n<p id=\"fs-id1167835232918\">The whole numbers and their opposites are called the <strong data-effect=\"bold\">integers<\/strong>.<\/p>\n<p id=\"fs-id1167835304330\">The integers are the numbers<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834192221\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5980ec992558d70651b5f8d3ef87360_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#111;&#116;&#115;&#32;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"182\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835302203\">Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.<\/p>\n<p id=\"fs-id1167835213624\">We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.<\/p>\n<p id=\"fs-id1167835311335\">We let one color (blue) represent positive. The other color (red) will represent the negatives.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835309360\" data-alt=\"Figure show two circles labeled positive blue and negative red.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_004_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure show two circles labeled positive blue and negative red.\" \/><\/span><\/p>\n<p id=\"fs-id1167835309409\">If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835309392\" data-alt=\"Figure shows a blue circle and a red circle encircled in a larger shape. This is labeled 1 plus minus 1 equals 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_005_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a blue circle and a red circle encircled in a larger shape. This is labeled 1 plus minus 1 equals 0.\" \/><\/span><\/p>\n<p id=\"fs-id1167835350247\">We will use the counters to show how to add:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167832076598\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-392b3957f3f78c80aabdb4a062ddf276_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#53;&#43;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"418\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167835326907\">The first example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5acc9b4ad63073fb1b26e058edc2a8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> adds 5 positives and 3 positives\u2014both positives.<\/p>\n<p id=\"fs-id1167835303389\">The second example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9dbdbd2c5eb27ab38d13bce69555a04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -4px;\" \/> adds 5 negatives and 3 negatives\u2014both negatives.<\/p>\n<p id=\"fs-id1167835191059\">When the signs are the same, the counters are all the same color, and so we add them. In each case we get 8\u2014either 8 positives or 8 negatives.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835190851\" data-alt=\"Figure on the left is labeled 5 plus 3. It shows 8 blue circles. 5 plus 3 equals 8. Figure on the right is labeled minus 5 plus open parentheses minus 3 close parentheses. It shows 8 blue circles labeled 8 negatives. Minus 5 plus open parentheses minus 3 close parentheses equals minus 8.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_006_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure on the left is labeled 5 plus 3. It shows 8 blue circles. 5 plus 3 equals 8. Figure on the right is labeled minus 5 plus open parentheses minus 3 close parentheses. It shows 8 blue circles labeled 8 negatives. Minus 5 plus open parentheses minus 3 close parentheses equals minus 8.\" \/><\/span><\/p>\n<p id=\"fs-id1167831934032\">So what happens when the signs are different? Let\u2019s add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9db94c4517228eb349cd4426ad66e71b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba8a2eacb81b1b362eb246f849de3b99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835346415\">When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835346024\" data-alt=\"Figure on the left is labeled minus 5 plus 3. It has 5 red circles and 3 blue circles. Three pairs of red and blue circles are formed. More negatives means the sum is negative. The figure on the right is labeled 5 plus minus 3. It has 5 blue and 3 red circles. Three pairs of red and blue circles are formed. More positives means the sum is positive.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_007_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure on the left is labeled minus 5 plus 3. It has 5 red circles and 3 blue circles. Three pairs of red and blue circles are formed. More negatives means the sum is negative. The figure on the right is labeled 5 plus minus 3. It has 5 blue and 3 red circles. Three pairs of red and blue circles are formed. More positives means the sum is positive.\" \/><\/span><\/p>\n<div data-type=\"example\" id=\"fs-id1167835365552\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834190186\">\n<div data-type=\"problem\" id=\"fs-id1167835214274\">\n<p id=\"fs-id1167835345490\">Add: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-929de2c2af9a64f22297516f77f37aa1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1211b5f02078795082d5e423b624cad8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3870a01a9f922967a9c9f3f208229f8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835214138\">\n<p id=\"fs-id1167832055795\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835166183\" summary=\"Figure shows five red circles. It is labeled minus 1 minus 4. 1 negative plus 4 negatives is 5 negatives. The answer is minus 5.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834195654\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_008b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835355963\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_008a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">1 negative plus 4 negatives is 5 negatives<\/td>\n<td><\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834121084\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_008c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835232646\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835254685\" summary=\"Figure is labeled minus 1 plus 5. It has 5 blue circles and 1 red circle. 1 pair of red and blue circles is formed. There are more positives, so the sum is positive. The answer is 4.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834183995\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_009a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835214048\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_009b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">There are more positives, so the sum is positive.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228881\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_009c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835167257\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835329511\" summary=\"The figure is labeled 1 plus minus 5. There is 1 blue circle and 5 red circles. 1 pair of blue and red circles is formed. There are more negatives, so the sum is negative. The answer is minus 4.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835354908\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_010a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834274063\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_010b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">There are more negatives, so the sum is negative.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835421621\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_010c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835303770\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835420812\">\n<div data-type=\"problem\" id=\"fs-id1167835319023\">\n<p id=\"fs-id1167835418075\">Add: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0210f277e32ab025744cce7d267b17c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a4dcc55080d931ad32225aa3a323aca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"52\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-863baab61efe01e3388b7d8cb842c159_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835334779\">\n<p id=\"fs-id1167835360011\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span> 2 <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826814134\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835380916\">\n<div data-type=\"problem\" id=\"fs-id1167834186295\">\n<p id=\"fs-id1167826977942\">Add: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0c2d36ef70e9e10fd645e5b1e254fa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d37ddf9df3c0b876ebeed0ce08e85bca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9689938c85494aa24ee9f8118f5153ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835269016\">\n<p id=\"fs-id1167835283690\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00c6d30c5f7439a21caf981437f64be1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span> 3 <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835361703\">We will continue to use counters to model the subtraction. Perhaps when you were younger, you read <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-740f2c70f1192d5a960cc80e92c14e05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#96;&#96;&#53;&#45;&#51;&#39;&#39;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"54\" style=\"vertical-align: 0px;\" \/> as \u201c5 take away 3.\u201d When you use counters, you can think of subtraction the same way!<\/p>\n<p id=\"fs-id1167835189607\">We will use the counters to show to subtract:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835478988\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b6ef0cfad69fba0f8d92f69a54026ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#53;&#45;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"418\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167835304878\">The first example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a738d157afc3555c06ab0724f3127ead_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> we subtract 3 positives from 5 positives and end up with 2 positives.<\/p>\n<p id=\"fs-id1167832053392\">In the second example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95c17916478e824f1f8fa70dd6576fe4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -4px;\" \/> we subtract 3 negatives from 5 negatives and end up with 2 negatives.<\/p>\n<p id=\"fs-id1167834536601\">Each example used counters of only one color, and the \u201ctake away\u201d model of subtraction was easy to apply.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834282821\" data-alt=\"Figure on the left is labeled 5 minus 3 equals 2. There are 5 blue circles. Three of these are encircled and an arrow indicates that they are taken away. The figure on the right is labeled minus 5 minus open parentheses minus 3 close parentheses equals minus 2. There are 5 red circles. Three of these are encircled and an arrow indicates that they are taken away.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_011_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure on the left is labeled 5 minus 3 equals 2. There are 5 blue circles. Three of these are encircled and an arrow indicates that they are taken away. The figure on the right is labeled minus 5 minus open parentheses minus 3 close parentheses equals minus 2. There are 5 red circles. Three of these are encircled and an arrow indicates that they are taken away.\" \/><\/span><\/p>\n<p id=\"fs-id1167835348640\">What happens when we have to subtract one positive and one negative number? We\u2019ll need to use both blue and red counters as well as some neutral pairs. If we don\u2019t have the number of counters needed to take away, we add neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels\u2014the value is the same, but it looks different.<\/p>\n<p id=\"fs-id1167835370956\">Let\u2019s look at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a029da03c39ec48a0d5bebd17696a97b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d15ebbe0d1c869b2a752c69b85e842f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<table id=\"fs-id1167834536310\" summary=\"Calculations of two expressions are shown. The expression on the left is minus 5 minus 3. The expression on the right is 5 minus open parentheses minus 3 close parentheses. We first model the first number. So, we have 5 red counters on the left and 5 white counters on the right. We now add three neutral pairs for both expressions. So we have 8 red and 3 white counters on the left and 8 white and 3 red counters on the right. We now remove the number of counters modeled by the second number. Three white counters are removed from the left and three red counters are removed from the right. We now have eight red counters on the left and 8 white counters on the right. Hence, minus 5 minus 3, on the left, is minus 8, whereas 5 minus minus 3, on the right, is 8.\" class=\"unnumbered unstyled can-break\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831911244\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835621605\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012h_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Model the first number.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831883380\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835306654\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012i_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We now add the needed neutral pairs.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835370213\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835353953\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012j_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We remove the number of counters modeled by the second number.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835418298\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832055009\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012k_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Count what is left.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835358812\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834314791\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012l_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834064550\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834183933\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012m_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831872542\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834505122\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_012n_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\" id=\"fs-id1167835514389\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835514391\">\n<div data-type=\"problem\" id=\"fs-id1167835187188\">\n<p id=\"fs-id1167835187190\">Subtract: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8ca17d1166d111b870604af4706a28f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffb4eb491bf7eadb06fb9defa6f7f02c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1d1c3eed482bbd31339b691e61cf6ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdedf418cebe7dd8b8238593cb1eed06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834227489\">\n<p id=\"fs-id1167831882978\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835479134\" summary=\"The expression is 3 minus 1. Three white counters are shown. An arrow indicates that 1 is taken away from these, and 2 remain. The answer is 2.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d3dbc4c42841c8465c416ec545ec4c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><span data-type=\"media\" id=\"fs-id1167835510839\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_018a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835358825\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_018b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Take 1 positive from 3 positives and get 2 positives.<\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835511311\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_018c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167834535740\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167832076029\" summary=\"The expression is minus 3 minus minus 1. Three red counters are shown. An arrow indicates that 1 is taken away from these, and 2 remain. The answer is minus 2.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826978713\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_019a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834539540\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_019b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Take 1 positive from 3 negatives and get 2 negatives.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832066812\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_019c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167834120450\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834280273\" summary=\"The expression is minus 3 minus 1. 4 red and 1 blue counters are shown. An arrow indicates that 1 blue counter is taken away from these. 4 red counters remain. The answer is minus 4.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94978be6cbbb429e976666f6d7e2b76e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><span data-type=\"media\" id=\"fs-id1167831138153\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_013a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834422963\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_013c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Take 1 positive from the one added neutral pair.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167835301849\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_013b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831239549\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_013d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835367421\"><span class=\"token\">\u24d3<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834539525\" summary=\"The expression is 3 minus minus 1. 4 blue and 1 red counters are shown. An arrow indicates that 1 red counter is taken away from these. 4 blue counters remain. The answer is 4.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6dbcf36993d713ed395c145314d02327_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><span data-type=\"media\" id=\"fs-id1167834539309\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_014a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834538618\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_014c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Take 1 negative from the one added neutral pair.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167832054944\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_014b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834190252\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_014d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834536304\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831882604\">\n<div data-type=\"problem\" id=\"fs-id1167831882606\">\n<p id=\"fs-id1167835340804\">Subtract: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5998bc2d047a935f18b7f8b1bead811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-717239a47bf9731de552df2098d9058f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6302a9df0c3462c19601388a3e35158_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d5077e21e5995699af9bebefb148d0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834183611\">\n<p id=\"fs-id1167834539501\"><span class=\"token\">\u24d0<\/span> 2 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d3<\/span> 10<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831882121\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835346029\">\n<div data-type=\"problem\" id=\"fs-id1167835346032\">\n<p id=\"fs-id1167835346034\">Subtract: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad08e565aed640d068767233478786cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d54412b4ab04674969c56370bac23fe3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4fefb3a4e0684f1bdfa5cf519f4e71f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"52\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d202489ec98d1fe8b203ba77f4ba363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835367941\">\n<p id=\"fs-id1167835367944\"><span class=\"token\">\u24d0<\/span> 3 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2981688f5b26fd15fa08c897afa741ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d3<\/span> 11<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834464452\">Have you noticed that <em data-effect=\"italics\">subtraction of signed numbers can be done by adding the opposite<\/em>? In the last example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1d1c3eed482bbd31339b691e61cf6ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/> is the same as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75db80a14417df8d60e8ec6d03bb6f06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5b23456808ccbcc7fc4f81a2a5bac91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/> is the same as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2dba75755ade966cf92cc900595a7b94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"44\" style=\"vertical-align: -2px;\" \/> You will often see this idea, the Subtraction Property, written as follows:<\/p>\n<div data-type=\"note\" id=\"fs-id1167835322024\">\n<div data-type=\"title\">Subtraction Property<\/div>\n<div data-type=\"equation\" id=\"fs-id1167835322029\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2094b7b77332eac21bdccd1d0b2ad1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#98;&#61;&#97;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167835281540\">Subtracting a number is the same as adding its opposite.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834539392\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834539394\">\n<div data-type=\"problem\" id=\"fs-id1167834182501\">\n<p id=\"fs-id1167834182503\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-943e7adbeeef4dc05755b90ce9c5b2b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2d8f34ecb522efb4afe43fc9df2a9b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f472d3b8c713d824953c1df6887c61f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#55;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f5defab49e5e2d67bc530493611b02d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#55;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd833b6f4419d5b777515813efd64512_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e7c6f979c580aadf3daff79575ff86d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"48\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d54412b4ab04674969c56370bac23fe3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d03f7332847ca5f6fa628fb70db32ff4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835378105\">\n<p id=\"fs-id1167835307054\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ac7f8b2b101f73c9ac23909235019c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#51;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"415\" style=\"vertical-align: -12px;\" \/><\/p>\n<p id=\"fs-id1167835254052\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e75af39f5b2ef06ae99256808bde3d8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#55;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#55;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"438\" style=\"vertical-align: -12px;\" \/><\/p>\n<p id=\"fs-id1167834346258\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92d80b5a3728bdffb9be6d070ff14d2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#43;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"390\" style=\"vertical-align: -12px;\" \/><\/p>\n<p id=\"fs-id1167834555060\"><span class=\"token\">\u24d3<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c76666d70691bc01633ab8717be1ffb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#55;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"407\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832075833\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834252708\">\n<div data-type=\"problem\" id=\"fs-id1167834252710\">\n<p id=\"fs-id1167835369538\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8d1db3b80dc35b7b60d36b8b0f75e81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#49;&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40d01dceba05f67e282cc39583a98104_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1bfa277e8ad3a65864851171edca3c6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f575a5be244e8ae7126a8516466fdfe8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b41f96b708c4cfc76755ff21ecdf264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ba9ad1085e91165970318aae353eb86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#43;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-697ebdd76454722d1ddf2380fd4f5084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15763bae1f0950ec258e1053368aad2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834506085\">\n<p id=\"fs-id1167834506087\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5e1fdb76078334103e6fc44e07f8b2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#44;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b5c104554d29a3849819e2ab426cfd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb9e52ddecc045b16dbf509b1c89f11b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61df1b0d2c108c12a0d1179240d82ab2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#44;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"43\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-868764e4e7f6822206e639607822a724_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835379147\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835380208\">\n<div data-type=\"problem\" id=\"fs-id1167835380210\">\n<p id=\"fs-id1167834538449\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44a108bd66b3875a41f393bf24c495f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"47\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b0576267e3a19ae5f4481d2457da7cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6798c1ffabed81d68e9b74dda10cf202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90212d6f23a201afff9d58ad62bc3458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-922ae15bd0c3c3939f7ee8be178a9342_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cafcd1114e77b18b2c610269afde9c86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#43;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a36cddef0e4e628976a53a80f1a1ed36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b189b9cf571cd229667a8a856f7eccd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#43;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834505550\">\n<p id=\"fs-id1167835379157\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5e1fdb76078334103e6fc44e07f8b2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#44;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47d8e24cb5e94e0ad2c16c4cb8b212bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-851674cf8d214f2dc493930ee7e6ee73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25080588314095004f9496a2564db5b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#51;&#44;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"44\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1cd431044023aa1fa5cc325dd846184_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#44;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834539673\">What happens when there are more than three integers? We just use the order of operations as usual.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835416831\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835416833\">\n<div data-type=\"problem\" id=\"fs-id1167835416835\">\n<p id=\"fs-id1167834555195\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2474111bd1f1233419d120f292a12361_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831880380\">\n<p id=\"fs-id1167831887723\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca31e16779828afd69afed6cd76d08b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#105;&#110;&#115;&#105;&#100;&#101;&#32;&#116;&#104;&#101;&#32;&#112;&#97;&#114;&#101;&#110;&#116;&#104;&#101;&#115;&#101;&#115;&#32;&#102;&#105;&#114;&#115;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#32;&#108;&#101;&#102;&#116;&#32;&#116;&#111;&#32;&#114;&#105;&#103;&#104;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#52;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"529\" style=\"vertical-align: -34px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831908823\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831908827\">\n<div data-type=\"problem\" id=\"fs-id1167835216901\">\n<p id=\"fs-id1167835216903\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7276e6d5da26f6a1d0fa52cc80026f80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831881377\">\n<p id=\"fs-id1167831881379\">3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832067665\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835349053\">\n<div data-type=\"problem\" id=\"fs-id1167835349055\">\n<p id=\"fs-id1167835349057\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18d516a282453ddd9361a8f7c0e462f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306347\">\n<p id=\"fs-id1167835306349\">13<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834094558\">\n<h3 data-type=\"title\">Multiply and Divide Integers<\/h3>\n<p id=\"fs-id1167826804678\">Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we are using the model just to help us discover the pattern.<\/p>\n<p id=\"fs-id1167834505365\">We remember that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9ae6fbdaf87158bf2d304feeeab2e27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&middot;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/> means <strong data-effect=\"bold\">add<\/strong> <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em> <strong data-effect=\"bold\">times<\/strong>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834222458\" data-alt=\"The figure on the left is labeled 5 dot 3. Here, we need to add 5, 3 times. Three rows of five blue counters each are shown. This makes 15 positives. Hence, 5 times 3 is 15. The figure on the right is labeled minus 5 open parentheses 3 close parentheses. Here we need to add minus 5, 3 times. Three rows of five red counters each are shown. This makes 15 negatives. Hence, minus 5 times 3 is minus 15.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_020_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure on the left is labeled 5 dot 3. Here, we need to add 5, 3 times. Three rows of five blue counters each are shown. This makes 15 positives. Hence, 5 times 3 is 15. The figure on the right is labeled minus 5 open parentheses 3 close parentheses. Here we need to add minus 5, 3 times. Three rows of five red counters each are shown. This makes 15 negatives. Hence, minus 5 times 3 is minus 15.\" \/><\/span><\/p>\n<p id=\"fs-id1167834183356\">The next two examples are more interesting. What does it mean to multiply 5 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80cc5c6a8396c9c51a989af972cef479_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/> It means subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bec62d514bb6966b610f4a8e2ac51641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#44;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"26\" style=\"vertical-align: -4px;\" \/> times. Looking at subtraction as \u201ctaking away\u201d, it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831881942\" data-alt=\"The figure on the left is labeled 5 open parentheses minus 3 close parentheses. We need to take away 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 negatives. Hence, 5 times minus 3 is minus 15. The figure on the right is labeled open parentheses minus 5 close parentheses open parentheses minus 3 close parentheses. We need to take away minus 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 positives. Hence, minus 5 times minus 3 is 15.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_021_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure on the left is labeled 5 open parentheses minus 3 close parentheses. We need to take away 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 negatives. Hence, 5 times minus 3 is minus 15. The figure on the right is labeled open parentheses minus 5 close parentheses open parentheses minus 3 close parentheses. We need to take away minus 5, three times. Three rows of five positive counters each and three rows of five negative counters each are shown. What is left is 15 positives. Hence, minus 5 times minus 3 is 15.\" \/><\/span><\/p>\n<p id=\"fs-id1167834556774\">In summary:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834556777\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3cafbe252fa39706b7f3f12c9a54f8d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#53;&middot;&#51;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"269\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"fs-id1167835304702\">Notice that for multiplication of two signed numbers, when the<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835304705\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23a4efaee55780407313f998eda21db1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#103;&#110;&#115;&#32;&#97;&#114;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#97;&#109;&#101;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#103;&#110;&#115;&#32;&#97;&#114;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"327\" style=\"vertical-align: -14px;\" \/><\/div>\n<p id=\"fs-id1167831910778\">What about division? Division is the inverse operation of multiplication. So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50b7fa13ae1876c1613d434e8648a230_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&divide;&#51;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -1px;\" \/> because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a7dbdb756bdf669325de4d27edf867a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&middot;&#51;&#61;&#49;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"63\" style=\"vertical-align: -1px;\" \/> In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834183503\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ded8127c19393f7c39c276cb3f71591e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#53;&middot;&#51;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#53;&divide;&#51;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#53;&divide;&#51;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#53;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#53;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"572\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"fs-id1167832076237\">Division follows the same rules as multiplication with regard to signs.<\/p>\n<div data-type=\"note\" id=\"fs-id1167832076240\">\n<div data-type=\"title\">Multiplication and Division of Signed Numbers<\/div>\n<p id=\"fs-id1167835468553\">For multiplication and division of two signed numbers:<\/p>\n<table id=\"fs-id1167835468556\" summary=\".\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Same signs<\/th>\n<th data-valign=\"top\" data-align=\"left\">Result<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Two positives<\/td>\n<td data-valign=\"top\" data-align=\"left\">Positive<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Two negatives<\/td>\n<td data-valign=\"top\" data-align=\"left\">Positive<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167834505247\">If the signs are the same, the result is positive.<\/p>\n<table id=\"fs-id1167834505250\" summary=\".\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Different signs<\/th>\n<th data-valign=\"top\" data-align=\"left\">Result<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Positive and negative<\/td>\n<td data-valign=\"top\" data-align=\"left\">Negative<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Negative and positive<\/td>\n<td data-valign=\"top\" data-align=\"left\">Negative<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835232461\">If the signs are different, the result is negative.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834536325\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834536327\">\n<div data-type=\"problem\" id=\"fs-id1167834536329\">\n<p id=\"fs-id1167834535837\">Multiply or divide: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5739a56510647f1a1d2d43ad47070cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;&#48;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcd35d513f0ed51a8087556d45a89249_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&middot;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70eabcb64a7d9494be9d191cbf23d11d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39afd9905f08036fa6436f244f5582d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#55;&divide;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831872478\">\n<p id=\"fs-id1167834063137\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-013c0b081c25afa5fbbc2e89cc91773a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#48;&#48;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#44;&#32;&#119;&#105;&#116;&#104;&#32;&#115;&#105;&#103;&#110;&#115;&#32;&#116;&#104;&#97;&#116;&#32;&#97;&#114;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#105;&#115;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"429\" style=\"vertical-align: -25px;\" \/><\/p>\n<p id=\"fs-id1167834191278\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6238e93eba04e5990762d55dca46ee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#55;&middot;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#44;&#32;&#119;&#105;&#116;&#104;&#32;&#115;&#97;&#109;&#101;&#32;&#115;&#105;&#103;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"386\" style=\"vertical-align: -14px;\" \/><\/p>\n<p id=\"fs-id1167834422857\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-018036058484030ecd54f6a88d880eca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#44;&#32;&#119;&#105;&#116;&#104;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#32;&#115;&#105;&#103;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"405\" style=\"vertical-align: -14px;\" \/><\/p>\n<p id=\"fs-id1167832075898\"><span class=\"token\">\u24d3<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a58fdf3f173cfe1e95bf0eb45f932ca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#55;&divide;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#44;&#32;&#119;&#105;&#116;&#104;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#32;&#115;&#105;&#103;&#110;&#115;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#105;&#115;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"391\" style=\"vertical-align: -25px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834428912\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834345685\">\n<div data-type=\"problem\" id=\"fs-id1167834345687\">\n<p id=\"fs-id1167834345689\">Multiply or divide: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-395c978b72371bcb6e6e2f17c6b966f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#53;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-641c03d44d003183153a6b94f0fa704e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&middot;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"26\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b361129d452a3d90c0bec2bb004052be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd85d5a2a294686836b8d987b2f6577c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#51;&divide;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834422894\">\n<p id=\"fs-id1167834422896\"><span class=\"token\">\u24d0<\/span> 23 <span class=\"token\">\u24d1<\/span> 60 <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c081a4d1588de9f9de1dd7d5282b4c9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826804652\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826804656\">\n<div data-type=\"problem\" id=\"fs-id1167826804658\">\n<p id=\"fs-id1167834539469\">Multiply or divide: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-854f402314ae428b7170accd800f0218_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#55;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e973c8e2c25726a6ca584d2b2fd1fe1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&middot;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e99ff919ab15281b2038d914222f91d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60edddbef30b32b592cf2f25e76087b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#50;&divide;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831880276\">\n<p id=\"fs-id1167831882554\"><span class=\"token\">\u24d0<\/span> 39 <span class=\"token\">\u24d1<\/span> 39 <span class=\"token\">\u24d2<\/span> \u221228 <span class=\"token\">\u24d3<\/span> \u22127<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834228840\">When we multiply a number by 1, the result is the same number. Each time we multiply a number by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df802eaa8e08eb73e8cd4d30d23d4b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> we get its opposite!<\/p>\n<div data-type=\"note\" id=\"fs-id1167834537368\">\n<div data-type=\"title\">Multiplication by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> <\/div>\n<div data-type=\"equation\" id=\"fs-id1167835575838\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2023743c7964e496a9d8d09a6a9b28b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#97;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"64\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"fs-id1167831880476\">Multiplying a number by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> gives its opposite.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832053258\">\n<h3 data-type=\"title\">Simplify Expressions with Integers<\/h3>\n<p id=\"fs-id1167832053263\">What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember Please Excuse My Dear Aunt Sally?<\/p>\n<p id=\"fs-id1167834335086\">Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834536158\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834536160\">\n<div data-type=\"problem\" id=\"fs-id1167834536162\">\n<p id=\"fs-id1167834536165\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39e546b1083a74884e4981f6405b861e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4f94e1e59f3a742952cb98a7106e8c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832066494\">\n<p id=\"fs-id1167832066496\">Notice the difference in parts (a) and (b). In part (a), the exponent means to raise what is in the parentheses, the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> to the 4<sup>th<\/sup> power. In part (b), the exponent means to raise just the 2 to the 4<sup>th<\/sup> power and then take the opposite.<\/p>\n<p id=\"fs-id1167834062721\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3f63f6affa3dec19bc29da4b08655b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#105;&#110;&#32;&#101;&#120;&#112;&#97;&#110;&#100;&#101;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"109\" width=\"467\" style=\"vertical-align: -48px;\" \/><\/p>\n<p id=\"fs-id1167831239202\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed8df7a5bcdf0a178157c366391efda1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#105;&#110;&#32;&#101;&#120;&#112;&#97;&#110;&#100;&#101;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&middot;&#50;&middot;&#50;&middot;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#97;&#114;&#101;&#32;&#97;&#115;&#107;&#101;&#100;&#32;&#116;&#111;&#32;&#102;&#105;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&middot;&#50;&middot;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&middot;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"151\" width=\"372\" style=\"vertical-align: -69px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832042791\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834505582\">\n<div data-type=\"problem\" id=\"fs-id1167834505584\">\n<p id=\"fs-id1167834505586\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f030f29170d3e23cfadee1ce2e2d79f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20115a7348be13aff2fe09fcb234ecb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#51;&#125;&#94;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831826507\">\n<p id=\"fs-id1167831826509\"><span class=\"token\">\u24d0<\/span> 81 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e9ef1c6bd7a9b6e2b87026a76cda03b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834228426\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834536588\">\n<div data-type=\"problem\" id=\"fs-id1167834536591\">\n<p id=\"fs-id1167834536593\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd5486faed5cb7a4ed65fdd9632f24fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3472017cbc2006be951120eba3b35cde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#55;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831871457\">\n<p id=\"fs-id1167831871460\"><span class=\"token\">\u24d0<\/span> 49 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87b45b657a37f37cb779527d2a666770_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832053420\">The last example showed us the difference between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39e546b1083a74884e4981f6405b861e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4f94e1e59f3a742952cb98a7106e8c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: 0px;\" \/> This distinction is important to prevent future errors. The next example reminds us to multiply and divide in order left to right.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834556092\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834556094\">\n<div data-type=\"problem\" id=\"fs-id1167834183406\">\n<p id=\"fs-id1167834183408\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e01597f7c7a3d4cd7c7bda391922815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"95\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19ace3cb74d68af311996bf2eba7b23b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#48;&divide;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835378126\">\n<p id=\"fs-id1167835378128\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6dea49dc550f0005c28d0119f05e7f02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#115;&#32;&#102;&#105;&#114;&#115;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#55;&#50;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"408\" style=\"vertical-align: -34px;\" \/><\/p>\n<p id=\"fs-id1167831880168\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68851fb0dca29ed6f89377673baed722_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;&#48;&divide;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#97;&#110;&#100;&#32;&#100;&#105;&#118;&#105;&#100;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#101;&#102;&#116;&#32;&#116;&#111;&#32;&#114;&#105;&#103;&#104;&#116;&#44;&#32;&#115;&#111;&#32;&#100;&#105;&#118;&#105;&#100;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#53;&#43;&#50;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"103\" width=\"440\" style=\"vertical-align: -45px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832053084\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832053087\">\n<div data-type=\"problem\" id=\"fs-id1167832053089\">\n<p id=\"fs-id1167832053091\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12418c6cf21fe3b5305ef45db5234146_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"103\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-074a90978897a16846560a64400049d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#55;&divide;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826880228\">\n<p id=\"fs-id1167831826512\"><span class=\"token\">\u24d0<\/span> 4 <span class=\"token\">\u24d1<\/span> 21<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834536337\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834536342\">\n<div data-type=\"problem\" id=\"fs-id1167834536344\">\n<p id=\"fs-id1167834536346\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72ea0facc417a9afcb7cf77ec951ef5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"103\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53543953f55c8ad1676fdf22c8ba534d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#50;&divide;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835575709\">\n<p id=\"fs-id1167835575711\"><span class=\"token\">\u24d0<\/span> 9 <span class=\"token\">\u24d1<\/span> 6<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832053123\">\n<h3 data-type=\"title\">Evaluate Variable Expressions with Integers<\/h3>\n<p id=\"fs-id1167832053128\">Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832053133\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832053135\">\n<div data-type=\"problem\" id=\"fs-id1167832053137\">\n<p id=\"fs-id1167831880328\">Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d36069afd091ee940d5182ad9461281a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#121;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0721123fdfaf6dd25e31954507f7c56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;&#121;&#61;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834238705\">\n<table id=\"fs-id1167834238707\" summary=\"The expression is 4 x squared minus 2 x y plus 3 y squared. Substitute x with 2 and y with minus 1. Using parentheses to show multiplication, we get 4 open parentheses 2 close parentheses squared minus 2 open parentheses 2 close parentheses open parentheses minus 1 close parentheses plus 3 open parentheses minus 1 close parentheses squared. Simplifying exponents, we get 4 times 4 minus open parentheses minus 4 close parentheses plus 3 times 1. We multiply to get 16 minus open parentheses minus 4 close parentheses plus 3. We subtract to get 20 plus 3. We add to get 23.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834539432\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834535493\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9ab22ad991fe7c719067fb4633bd8e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831871648\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify exponents.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831880151\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831880503\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832076130\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832053446\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_015g_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834422813\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834422817\">\n<div data-type=\"problem\" id=\"fs-id1167834422819\">\n<p id=\"fs-id1167834422821\">Evaluate: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6daf298d6991eb3c7b90d03d3d5f1ebc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#121;&#43;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39c491afc3f59ce77ea95fa66c4dc7bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#44;&#121;&#61;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835514354\">\n<p id=\"fs-id1167835514356\">31<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834536416\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834536420\">\n<div data-type=\"problem\" id=\"fs-id1167834536422\">\n<p id=\"fs-id1167834536424\">Evaluate: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a57cad1bd60410c91dee4ef6db6e2ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#121;&#43;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28afa06d9d0d66795ff31c6b62be578e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#44;&#121;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831910923\">\n<p id=\"fs-id1167831910925\">67<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831910933\">\n<h3 data-type=\"title\">Translate Phrases to Expressions with Integers<\/h3>\n<p id=\"fs-id1167832066700\">Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832066704\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832066706\">\n<div data-type=\"problem\" id=\"fs-id1167832066708\">\n<p id=\"fs-id1167832066711\">Translate and simplify: the sum of 8 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffa5d3c4f2839e2c1a583aeb49aaf520_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -4px;\" \/> increased by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a650b6362973a0817ee43187a6682e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834539696\">\n<p id=\"fs-id1167834539698\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1970ba0432f20fdbc0289253702db73e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#109;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#102;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#45;&#45;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#99;&#114;&#101;&#97;&#115;&#101;&#100;&#32;&#98;&#121;&#32;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#91;&#56;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#32;&#66;&#101;&#32;&#99;&#97;&#114;&#101;&#102;&#117;&#108;&#32;&#110;&#111;&#116;&#32;&#116;&#111;&#32;&#99;&#111;&#110;&#102;&#117;&#115;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#114;&#97;&#99;&#107;&#101;&#116;&#115;&#32;&#119;&#105;&#116;&#104;&#32;&#97;&#110;&#32;&#97;&#98;&#115;&#111;&#108;&#117;&#116;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#32;&#115;&#105;&#103;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"681\" style=\"vertical-align: -48px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834539459\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835352729\">\n<div data-type=\"problem\" id=\"fs-id1167835352731\">\n<p id=\"fs-id1167835352733\">Translate and simplify the sum of 9 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa33003053879d040ac71240c5b97afb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -4px;\" \/> increased by 4.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835352745\">\n<p id=\"fs-id1167835352747\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8e716ae233ec878fec938c9bdef0de6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#59;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834535728\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834535732\">\n<div data-type=\"problem\" id=\"fs-id1167834535735\">\n<p id=\"fs-id1167834537376\">Translate and simplify the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffa5d3c4f2839e2c1a583aeb49aaf520_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -4px;\" \/> increased by 7.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537393\">\n<p id=\"fs-id1167834537395\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a04e5c9a619148e4098f13be5c9f296f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#55;&#59;&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"173\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832076190\">\n<h3 data-type=\"title\">Use Integers in Applications<\/h3>\n<p id=\"fs-id1167832076195\">We\u2019ll outline a plan to solve applications. It\u2019s hard to find something if we don\u2019t know what we\u2019re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we\u2019ll write a phrase that gives the information to find it. We\u2019ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834537844\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve Application Problems Using Integers<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834537846\">\n<div data-type=\"problem\" id=\"fs-id1167834537848\">\n<p id=\"fs-id1167834537850\">The temperature in Kendallville, Indiana one morning was 11 degrees. By mid-afternoon, the temperature had dropped to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> degrees. What was the difference in the morning and afternoon temperatures?<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832076198\" data-alt=\"Figure shows a glass thermometer, with temperature markings ranging from minus 10 to 30. Two markings are highlighted, minus 9 degrees C and 11 degrees C.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_016_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a glass thermometer, with temperature markings ranging from minus 10 to 30. Two markings are highlighted, minus 9 degrees C and 11 degrees C.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834182486\"><span data-type=\"media\" id=\"fs-id1167834182489\" data-alt=\"Step 1 is to read the problem and make sure all the words and ideas are understood.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to read the problem and make sure all the words and ideas are understood.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167831881150\" data-alt=\"Step 2 is to identify what we are asked to find. Here, we need to find the difference of the morning and afternoon temperatures.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to identify what we are asked to find. Here, we need to find the difference of the morning and afternoon temperatures.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167831881162\" data-alt=\"Step 3 is to write a phrase that gives the information to find it. In this case, the phrase is the difference of 11 and minus 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to write a phrase that gives the information to find it. In this case, the phrase is the difference of 11 and minus 9.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835511995\" data-alt=\"Step 4 is to translate the phrase to an expression. Here, we write 11 minus open parentheses minus 9 close parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017d_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to translate the phrase to an expression. Here, we write 11 minus open parentheses minus 9 close parentheses.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835512007\" data-alt=\"In step 5, we simplify the expression to get 20.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017e_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 5, we simplify the expression to get 20.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167832068492\" data-alt=\"Step 6 is to answer the question with a complete sentence: The difference in temperatures was 20 degrees.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_017f_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to answer the question with a complete sentence: The difference in temperatures was 20 degrees.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832068508\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832068512\">\n<div data-type=\"problem\" id=\"fs-id1167835187799\">\n<p id=\"fs-id1167835187801\">The temperature in Anchorage, Alaska one morning was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b59223d2c162a0fdd163af8597f0845e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/> degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835187812\">\n<p id=\"fs-id1167835187814\">The difference in temperatures was 45 degrees Fahrenheit.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835187821\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835287446\">\n<div data-type=\"problem\" id=\"fs-id1167835287448\">\n<p id=\"fs-id1167835287450\">The temperature in Denver was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> degrees at lunchtime. By sunset the temperature had dropped to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0329d19b932005c996d83dbb9e09ffed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/> degrees. What was the difference in the lunchtime and sunset temperatures?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835287465\">\n<p id=\"fs-id1167835287467\">The difference in temperatures was 9 degrees.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834538046\" class=\"howto\">\n<div data-type=\"title\">Use Integers in Applications.<\/div>\n<ol id=\"fs-id1167834538053\" type=\"1\" class=\"stepwise\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are asked to find.<\/li>\n<li><strong data-effect=\"bold\">Write a phrase<\/strong> that gives the information to find it.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> the phrase to an expression.<\/li>\n<li><strong data-effect=\"bold\">Simplify<\/strong> the expression.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834533182\" class=\"media-2\">\n<p id=\"fs-id1167834533186\">Access this online resource for additional instruction and practice with integers.<\/p>\n<ul id=\"fs-id1167834533189\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/25subintegers\">Subtracting Integers with Counters<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167834535516\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836310060\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Opposite Notation<\/strong>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167834535530\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e131802d30c4d28d032a059404c67e81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#101;&#97;&#110;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#111;&#116;&#97;&#116;&#105;&#111;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#114;&#101;&#97;&#100;&#32;&#97;&#115;&#32;&#96;&#96;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#39;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"346\" style=\"vertical-align: -14px;\" \/><\/div>\n<\/li>\n<li><strong data-effect=\"bold\">Absolute Value<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> The absolute value of a number is its distance from 0 on the number line.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> The absolute value of a number <em data-effect=\"italics\">n<\/em> is written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b38a586dc76a31dfa818cbaca2da26f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#110;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"17\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6cd98f4266f7f491a976c3270ec303d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#110;&#124;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> for all numbers.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Absolute values are always greater than or equal to zero.<\/li>\n<li><strong data-effect=\"bold\">Grouping Symbols<\/strong>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167834537897\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81af12df88d0c2702421283057319935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#114;&#101;&#110;&#116;&#104;&#101;&#115;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#97;&#99;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#97;&#99;&#107;&#101;&#116;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#98;&#115;&#111;&#108;&#117;&#116;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"389\" style=\"vertical-align: -16px;\" \/><\/div>\n<\/li>\n<li><strong data-effect=\"bold\">Subtraction Property<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36945374811302ce9cf2b8714c4d88df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#45;&#98;&#61;&#97;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Subtracting a number is the same as adding its opposite.<\/li>\n<li><strong data-effect=\"bold\">Multiplication and Division of Signed Numbers<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> For multiplication and division of two signed numbers:<\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834533537\" summary=\".\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><u data-effect=\"underline\">Same signs<\/u><\/strong><\/th>\n<th data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><u data-effect=\"underline\">Result<\/u><\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Two positives<\/td>\n<td data-valign=\"top\" data-align=\"left\">Positive<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Two negatives<\/td>\n<td data-valign=\"top\" data-align=\"left\">Positive<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2081e142d6cecaf6d79164a805e7335a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>If the signs are the same, the result is positive.<\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834526182\" summary=\".\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><u data-effect=\"underline\">Different signs<\/u><\/strong><\/th>\n<th data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\"><u data-effect=\"underline\">Result<\/u><\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Positive and negative<\/td>\n<td data-valign=\"top\" data-align=\"left\">Negative<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2022&nbsp;&nbsp;Negative and positive<\/td>\n<td data-valign=\"top\" data-align=\"left\">Negative<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2f315efc7f650504b9439c202a2b8f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#50;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>If the signs are different, the result is negative.<\/li>\n<li><strong data-effect=\"bold\">Multiplication by<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a37076105fe1d151efba711abd2c5a22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#97;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"68\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Multiplying a number by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> gives its opposite.<\/li>\n<li><strong data-effect=\"bold\">How to Use Integers in Applications.<\/strong>\n<div data-type=\"newline\"><\/div>\n<ol id=\"fs-id1167834539249\" type=\"1\" class=\"stepwise\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are asked to find.<\/li>\n<li><strong data-effect=\"bold\">Write a phrase<\/strong> that gives the information to find it.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> the phrase to an expression.<\/li>\n<li><strong data-effect=\"bold\">Simplify<\/strong> the expression.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831826351\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167831826355\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167831826362\"><strong data-effect=\"bold\">Simplify Expressions with Absolute Value<\/strong><\/p>\n<p id=\"fs-id1167831826368\">In the following exercises, fill in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7afd5b67d1411fd169472ab43b36654a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#60;&#44;&#62;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"39\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/> for each of the following pairs of numbers.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834238749\">\n<div data-type=\"problem\" id=\"fs-id1167834238751\">\n<p id=\"fs-id1167834238753\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-866980087b983b7897916f7605ca3939_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#55;&#124;&#95;&#95;&#95;&#45;&#124;&#45;&#55;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"71\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6330033158ce4a18f05fc5a980b8f25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#95;&#95;&#95;&#45;&#124;&#45;&#54;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"41\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc0c215d75107d69c1024cad756471bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#49;&#49;&#124;&#95;&#95;&#95;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"68\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-936cf47a9bf17d055ca6add53cc7be56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#95;&#95;&#95;&#45;&#124;&#45;&#49;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"82\" style=\"vertical-align: -10px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167831882179\">\n<p id=\"fs-id1167831882181\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834534488\">\n<div data-type=\"problem\" id=\"fs-id1167834244227\">\n<p id=\"fs-id1167834244229\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4da017312d3c88fee2105ba8436c4091_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#124;&#45;&#57;&#124;&#95;&#95;&#95;&#124;&#45;&#57;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"61\" style=\"vertical-align: -11px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a672a32e7caf11d4468c8cce459ff550_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#95;&#95;&#95;&#124;&#45;&#56;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"44\" style=\"vertical-align: -11px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a7fa488395dd2efe8dd9b1b49c6d1ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#49;&#124;&#95;&#95;&#95;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"54\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cc2f0e1e564955b2c59f56ce7a589bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#95;&#95;&#95;&#45;&#124;&#45;&#49;&#52;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"82\" style=\"vertical-align: -10px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826978155\">\n<div data-type=\"problem\" id=\"fs-id1167826978157\">\n<p id=\"fs-id1167832086878\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7b4c3bf7e8e00869e98443895cb4b8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#124;&#50;&#124;&#95;&#95;&#95;&#45;&#124;&#45;&#50;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"49\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31d2bf34fecfb81cfc45c124f46604a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#95;&#95;&#95;&#45;&#124;&#45;&#49;&#50;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7468a496056ec4fa6433ff4886de6328_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#51;&#124;&#95;&#95;&#95;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-600c5596d7d503ec0dfff25bdd80b7b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#49;&#57;&#124;&#95;&#95;&#95;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"87\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834538202\">\n<p id=\"fs-id1167834538205\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a5598f6c52dfad4d548aabcf09fbca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"12\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831826418\">\n<div data-type=\"problem\" id=\"fs-id1167831826420\">\n<p id=\"fs-id1167831826422\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7134a051dc45b64fe71de1dc7757634_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#124;&#45;&#52;&#124;&#95;&#95;&#95;&#45;&#124;&#52;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"62\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-733d2bbbf3ceaf0539cbed9e675482d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#95;&#95;&#95;&#45;&#124;&#45;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"41\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc4ad432d0fef914ed12b5d664eb31df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#124;&#45;&#49;&#48;&#124;&#95;&#95;&#95;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-015db81065dcfb238198447fb0f37e1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#124;&#45;&#48;&#124;&#95;&#95;&#95;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835511702\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835511705\">\n<div data-type=\"problem\" id=\"fs-id1167835511707\">\n<p id=\"fs-id1167835511709\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c025cb6f15bed386886aa63a6e7ab57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#49;&#53;&#45;&#55;&#124;&#45;&#124;&#49;&#52;&#45;&#54;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835511742\">\n<p id=\"fs-id1167835511744\">0<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835511750\">\n<div data-type=\"problem\" id=\"fs-id1167834536356\">\n<p id=\"fs-id1167834536358\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b1375f94792dd48ba0f73c476b92186_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#49;&#55;&#45;&#56;&#124;&#45;&#124;&#49;&#51;&#45;&#52;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834536399\">\n<div data-type=\"problem\" id=\"fs-id1167834536401\">\n<p id=\"fs-id1167834536403\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7817b31b5d79341c225c1a320bddd44a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#45;&#124;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831912722\">\n<p id=\"fs-id1167831912724\">8<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831912729\">\n<div data-type=\"problem\" id=\"fs-id1167831912731\">\n<p id=\"fs-id1167831912733\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8ffffc5c377006e13f2def455b45349_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#45;&#124;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835379691\">\n<div data-type=\"problem\" id=\"fs-id1167835379693\">\n<p id=\"fs-id1167835379695\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-481515b44dda41f6fdcc2ec0b0c3c710_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#45;&#124;&#49;&#50;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"181\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835510052\">\n<p id=\"fs-id1167835510054\">15<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835510059\">\n<div data-type=\"problem\" id=\"fs-id1167835510061\">\n<p id=\"fs-id1167835510064\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-658e832f8013b170da665249fc80ee71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#45;&#124;&#49;&#57;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#55;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"182\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834531328\">\n<div data-type=\"problem\" id=\"fs-id1167834531330\">\n<p id=\"fs-id1167834531332\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82f77b02f827e7594693c9f64cbf97eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#45;&#51;&#124;&#57;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834531370\">\n<p id=\"fs-id1167834531372\">1<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834372402\">\n<div data-type=\"problem\" id=\"fs-id1167834372404\">\n<p id=\"fs-id1167834372406\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d92663d87070ee469f022a092d8e943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#45;&#50;&#124;&#49;&#49;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834372451\"><strong data-effect=\"bold\">Add and Subtract Integers<\/strong><\/p>\n<p id=\"fs-id1167834372456\">In the following exercises, simplify each expression.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834372460\">\n<div data-type=\"problem\" id=\"fs-id1167834372462\">\n<p id=\"fs-id1167834372464\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48cbf3de904136f43596ae0f10bbe0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc1c6c1bd765502fa4c0854c57316b51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0d22e963b79a10d1198508aba8bc275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167832042872\">\n<p id=\"fs-id1167832042874\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2981688f5b26fd15fa08c897afa741ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834372359\">\n<div data-type=\"problem\" id=\"fs-id1167834372361\">\n<p id=\"fs-id1167834372363\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ef9e6596d7ddba7b9e28c72b6d4a490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b765ce693ae985dab9c90893a73490c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd05131a3388e29ab254e7337daf8e80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834533260\">\n<div data-type=\"problem\" id=\"fs-id1167834533262\">\n<p id=\"fs-id1167834533264\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5347e78e1388867c78f84257e82214bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835622299\">\n<p id=\"fs-id1167835622302\">32<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835622307\">\n<div data-type=\"problem\" id=\"fs-id1167835622309\">\n<p id=\"fs-id1167835622311\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-288c931cb2f8a6e3d2228e3c03dc1af8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835622337\">\n<div data-type=\"problem\" id=\"fs-id1167835622339\">\n<p id=\"fs-id1167835622341\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-070ef83fdc77ea566edbd4518f4c8c3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834535752\">\n<p id=\"fs-id1167834535754\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-851674cf8d214f2dc493930ee7e6ee73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535763\">\n<div data-type=\"problem\" id=\"fs-id1167834535765\">\n<p id=\"fs-id1167834535767\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07755c6ff6fa8706c39591961a371300_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#55;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535800\">\n<div data-type=\"problem\" id=\"fs-id1167834535802\">\n<p id=\"fs-id1167834535804\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-918e4de58e3bd439d356305182c48d8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831880853\">\n<p id=\"fs-id1167831880855\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da60de369e4a2bdb1a6166bdb4b6f1c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831880864\">\n<div data-type=\"problem\" id=\"fs-id1167831880866\">\n<p id=\"fs-id1167831880868\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ba44c0b49b4c239fdec18f97acb9339_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831880900\">\n<div data-type=\"problem\" id=\"fs-id1167831880902\">\n<p id=\"fs-id1167831880904\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da1eae46d26089099068f7337bf60e0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"47\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edb304a3696cd950f75def4b61b0d5f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d72aa622d0bbc04ef9062e03b1928f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e55ee0d2060df01a87d8a0491ef0402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834538296\">\n<p id=\"fs-id1167834538298\"><span class=\"token\">\u24d0<\/span> 6 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b93660fe4bfeaf9a69e38dd0cba4d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-930d01a5f622cee5952bab4752e56063_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835575382\">\n<div data-type=\"problem\" id=\"fs-id1167835575384\">\n<p id=\"fs-id1167835575386\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38f53881d17ad69d85523142d63b8f85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"47\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0bc75e0fa69dce5753011020448c083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8e7f3b6b7123d50e5450bfc3e7df8bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27f64549631257f903356f7284372af8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835622273\">\n<div data-type=\"problem\" id=\"fs-id1167834536446\">\n<p id=\"fs-id1167834536448\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edd48401ecbeced99097cef4de7a30e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#55;&#45;&#52;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"69\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834536459\">\n<p id=\"fs-id1167834536462\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8592fdae394223d029fcc21317133ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834536470\">\n<div data-type=\"problem\" id=\"fs-id1167834536472\">\n<p id=\"fs-id1167834536474\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49a167f3520739b34343d60162587522_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#56;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834536502\">\n<div data-type=\"problem\" id=\"fs-id1167834536504\">\n<p id=\"fs-id1167834536506\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18410c1f656e226b5e7a70abf2d5196c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835479492\">\n<p id=\"fs-id1167835479494\">22<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835479499\">\n<div data-type=\"problem\" id=\"fs-id1167835479501\">\n<p id=\"fs-id1167835479504\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c9f40df2aca63b40512242cccfc97b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835479533\">\n<div data-type=\"problem\" id=\"fs-id1167835479536\">\n<p id=\"fs-id1167835479538\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f6e73a75721e926ae49e3abb7b13a4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#52;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2c343faa3d44761a7060b26e307afcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835319147\">\n<p id=\"fs-id1167835319149\"><span class=\"token\">\u24d0<\/span> 16 <span class=\"token\">\u24d1<\/span>16<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835319163\">\n<div data-type=\"problem\" id=\"fs-id1167835319166\">\n<p id=\"fs-id1167835319168\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3081f7139091f0f4177c8720d811cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acf31906464b025c7c1e4c67b8ed6766_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835575747\">\n<div data-type=\"problem\" id=\"fs-id1167835575749\">\n<p id=\"fs-id1167835575751\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c731405c20b6850ef82dba43e0fc598b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c25edd25eccda1d968816153366e08f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#43;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835575789\">\n<p id=\"fs-id1167835575792\"><span class=\"token\">\u24d0<\/span> 45 <span class=\"token\">\u24d1<\/span>45<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835575806\">\n<div data-type=\"problem\" id=\"fs-id1167835575808\">\n<p id=\"fs-id1167835575810\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ba3739ae38961e4faadab8a591ad110_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#54;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3309d8c2eff4737b32b60b60659ccdb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#54;&#43;&#51;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826808779\">\n<div data-type=\"problem\" id=\"fs-id1167826808781\">\n<p id=\"fs-id1167826808783\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bbb5fa7c49285663cf98f2b43cb7f6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826808816\">\n<p id=\"fs-id1167826808818\">0<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826808824\">\n<div data-type=\"problem\" id=\"fs-id1167835575614\">\n<p id=\"fs-id1167835575616\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7693ebcc77a2316b7f5b785bd496073e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835575657\">\n<div data-type=\"problem\" id=\"fs-id1167835575659\">\n<p id=\"fs-id1167835575661\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3096804c2de3830c0694e94bd29e049b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835575841\">\n<p id=\"fs-id1167835575843\">4<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835575848\">\n<div data-type=\"problem\" id=\"fs-id1167835575850\">\n<p id=\"fs-id1167835575853\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed9eefcbaa2f184ece8a65f3a4238b8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835575895\">\n<div data-type=\"problem\" id=\"fs-id1167835575897\">\n<p id=\"fs-id1167835575900\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93bff9c2baf995720da4030aee245f22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#91;&#49;&#48;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831912593\">\n<p id=\"fs-id1167831912595\">6<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831912600\">\n<div data-type=\"problem\" id=\"fs-id1167831912602\">\n<p id=\"fs-id1167831912604\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb825f53789b236f424ae01b5da9a187_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#50;&#45;&#92;&#108;&#101;&#102;&#116;&#91;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#45;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831912645\"><strong data-effect=\"bold\">Multiply and Divide Integers<\/strong><\/p>\n<p id=\"fs-id1167831912651\">In the following exercises, multiply or divide.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831912654\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831912656\">\n<p id=\"fs-id1167831912659\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60335547639ba5b446fbb1dbd0f61854_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&middot;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e26d28a6d9c5af3feb6f409cca797f2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d702e9b7915f85c122ea0c782f7002a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#52;&divide;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28d8a7f02cb807d81013eb64b477dc8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#50;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834182419\">\n<p id=\"fs-id1167834182421\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fce2979511e6a27a1bd61ca970c655b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-166222493e0c05c592c1e8e54c275158_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-837d0d6b05c238eddf106b68ec9dcfd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834538486\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834538488\">\n<p id=\"fs-id1167834538490\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e0f4424762fb677174e72f3d4a33138_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&middot;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b361129d452a3d90c0bec2bb004052be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1252589e3700b2aed653595d9828b47a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-343de80fd3ee87b695a068aab0729b9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#52;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835575498\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835575500\">\n<p id=\"fs-id1167835575502\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca492837f9f6efa8c318e568c5b4c117_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#56;&divide;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4696a3bd719050055f1f13135c3f509_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#56;&#48;&divide;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab13b5515e80b4c88edf828114a3c963_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d398b90f253a04cd8d61419909697cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167831881018\">\n<p id=\"fs-id1167831881020\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9eae543c53f9301fa411ef541a1f4433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf64b03624682b0a337c10836b136ca1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38826cadfc5dbf2d92abbbfd6d370410_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831881067\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831881069\">\n<p id=\"fs-id1167831881071\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fca168a7994c064ddf32cd38420ec464_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#54;&divide;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7a4079b0e15bb27282126d7b4a87290_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#57;&#50;&divide;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b361129d452a3d90c0bec2bb004052be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af4d3e14c40317fc6cb99953de35ec8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167834449021\"><strong data-effect=\"bold\">Simplify and Evaluate Expressions with Integers<\/strong><\/p>\n<p id=\"fs-id1167834449027\">In the following exercises, simplify each expression.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834449030\">\n<div data-type=\"problem\" id=\"fs-id1167834449032\">\n<p id=\"fs-id1167834449035\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3d91d19a200676911902f50f3ee4e43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34c97249f43053adaf939b2e7568ac7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#50;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832053585\">\n<p id=\"fs-id1167832053587\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de25eb06760758d2c59eb63fca270413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d47325be40c0ee5b346187292f647b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832053611\">\n<div data-type=\"problem\" id=\"fs-id1167832053613\">\n<p id=\"fs-id1167832053616\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e00bf55759c5b8bf6723f02ba00afdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ecef8b630ea8cdc552b9bdc8c4d25c7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#51;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834533290\">\n<div data-type=\"problem\" id=\"fs-id1167834533292\">\n<p id=\"fs-id1167834533294\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85f0f85521a080391f767cabf8d51180_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834533327\">\n<p id=\"fs-id1167834533329\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6193f2be927539d70e972d662785fd1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834533338\">\n<div data-type=\"problem\" id=\"fs-id1167834533340\">\n<p id=\"fs-id1167834533342\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42d324e2b5ce01d2839a7dd66ec820c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834533386\">\n<div data-type=\"problem\" id=\"fs-id1167834533388\">\n<p id=\"fs-id1167834533390\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa75b0df6dbc1e81f7fd2a16258eb2f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831826200\">\n<p id=\"fs-id1167831826202\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b5f1c3a0e1ca514e4e63519e4d0c016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831826211\">\n<div data-type=\"problem\" id=\"fs-id1167831826213\">\n<p id=\"fs-id1167831826215\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92ee7e0dfb4eff0145c51b2c590533bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831826249\">\n<div data-type=\"problem\" id=\"fs-id1167831826252\">\n<p id=\"fs-id1167831826254\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0394b023c4f67d69d7a99987647f7f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831826285\">\n<p id=\"fs-id1167831826287\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831826294\">\n<div data-type=\"problem\" id=\"fs-id1167831826296\">\n<p id=\"fs-id1167831826298\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b74138c11841b38808f35613eea0d9c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834541913\">\n<div data-type=\"problem\" id=\"fs-id1167834541915\">\n<p id=\"fs-id1167834541917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-653f4ea6625efc3317803561a1dcf962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#54;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834541941\">\n<p id=\"fs-id1167834541944\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20bf590fc81a78a705705c8aad69df9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834541952\">\n<div data-type=\"problem\" id=\"fs-id1167834541954\">\n<p id=\"fs-id1167834541956\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de8aa2f9c3c0b7b1426fbf05a26db1ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#51;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834541992\">\n<div data-type=\"problem\" id=\"fs-id1167834541994\">\n<p id=\"fs-id1167834541996\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b843b0ce465977a53eaa4249643cff01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#53;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834542035\">\n<p id=\"fs-id1167834542037\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834542046\">\n<div data-type=\"problem\" id=\"fs-id1167834542048\">\n<p id=\"fs-id1167834542050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e1294eabab0cc5373c28d2d3c440a61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#50;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835534365\">\n<div data-type=\"problem\" id=\"fs-id1167835534368\">\n<p id=\"fs-id1167835534370\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d53067a8da43befde6f153cea9ad046d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#91;&#51;&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835534403\">\n<p id=\"fs-id1167835534405\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94535250d2f2556f9c0cb93ff04db6e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835534414\">\n<div data-type=\"problem\" id=\"fs-id1167835534416\">\n<p id=\"fs-id1167835534418\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-736f06f6d3a1259461be721afeff8c7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#91;&#55;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835534462\">\n<div data-type=\"problem\" id=\"fs-id1167835534465\">\n<p id=\"fs-id1167835534467\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5306d0d2c79ce7d081b9ef5937d6ea16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#124;&#50;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"165\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834448778\">\n<p id=\"fs-id1167834448780\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834448787\">\n<div data-type=\"problem\" id=\"fs-id1167834448789\">\n<p id=\"fs-id1167834448792\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-374e928cbc4fcec237471b3ced787c20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#45;&#124;&#53;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#54;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"165\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834448842\">\n<div data-type=\"problem\" id=\"fs-id1167834448844\">\n<p id=\"fs-id1167834448846\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88ec7c3ee507fdc61f424da2a1ebac38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#45;&#51;&#124;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"190\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834448894\">\n<p id=\"fs-id1167834448896\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834448905\">\n<div data-type=\"problem\" id=\"fs-id1167834448907\">\n<p id=\"fs-id1167834448909\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-163795a51ab6de40c570e404aa192d10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#50;&#124;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"190\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826987727\">\n<div data-type=\"problem\" id=\"fs-id1167826987729\">\n<p id=\"fs-id1167826987731\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a85564ebc792b94edbad0ec191087bb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826987769\">\n<p id=\"fs-id1167826987771\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826987778\">\n<div data-type=\"problem\" id=\"fs-id1167826987781\">\n<p id=\"fs-id1167826987783\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71b8a2c84bdd8213bc7d80417de1a881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#50;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826987831\">In the following exercises, evaluate each expression.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826987834\">\n<div data-type=\"problem\" id=\"fs-id1167826987836\">\n<p id=\"fs-id1167826987838\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d68be4b037a45e488d7b547084f3bd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"76\" style=\"vertical-align: -4px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72122a4c8ec17c289eec6cc0fba169c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"65\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02d2d43e4042eb1d7c4946f1691acc28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835216331\">\n<p id=\"fs-id1167835216333\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6193f2be927539d70e972d662785fd1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8dde8d4cade60a4f3ca0e779512b974c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835216358\">\n<div data-type=\"problem\" id=\"fs-id1167835216360\">\n<p id=\"fs-id1167835216362\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d2604c0dc262cf6cd9bc6e0cd7644f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -4px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d50b89d15ab8c25c12da93a6cef1268b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"66\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a0c2fd9a040adbffa2b489ee1ed09ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835216437\">\n<div data-type=\"problem\" id=\"fs-id1167835216439\">\n<p id=\"fs-id1167835216441\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a469f70946862971ba45ee6cb67facc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"61\" style=\"vertical-align: -4px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f19ccce3cbd3c2f724bc70eab550496e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#44;&#121;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835216482\">\n<p id=\"fs-id1167835216484\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c71eeef0468cd1464a1c1d6ade7e3f1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835216492\">\n<div data-type=\"problem\" id=\"fs-id1167835216495\">\n<p id=\"fs-id1167835216497\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66cbf4f9745ec400cb2d1ada5b287f62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07e18f1bc07fb0f5cd162592d51d40be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#44;&#122;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834534772\">\n<div data-type=\"problem\" id=\"fs-id1167834534774\">\n<p id=\"fs-id1167834534776\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fed435a1f6e46141f785b1aeee66dece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#97;&#45;&#50;&#98;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"87\" style=\"vertical-align: 0px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09c7ecdc865f5ca84f19934ec6f2c6bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4df6e82e2e030684f8d30ec89dda7639_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: 0px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834534816\">\n<p id=\"fs-id1167834534818\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eeacf7895c3fd138514ad215aa482af4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834534827\">\n<div data-type=\"problem\" id=\"fs-id1167834534829\">\n<p id=\"fs-id1167834534831\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50fa975d04c832a8447a0f621f724e27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#109;&#45;&#52;&#110;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"95\" style=\"vertical-align: -1px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-754283866fe7f24eadf016df34fe5cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"62\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eba9cf21d60d2e805dd964c0e855d596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834534881\">\n<div data-type=\"problem\" id=\"fs-id1167834534883\">\n<p id=\"fs-id1167834534885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11427e890f20a69145b04fa766ca2ec4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#121;&#43;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-543e03572659af7e21eff0637e72bbcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#44;&#121;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834534935\">\n<p id=\"fs-id1167834534937\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834534944\">\n<div data-type=\"problem\" id=\"fs-id1167834534946\">\n<p id=\"fs-id1167834534949\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5db52fa353eab373555b4e34bbf21e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#121;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad1f6de67c9a7af6a71473d5d5e39c41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#44;&#121;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167834535007\"><strong data-effect=\"bold\">Translate English Phrases to Algebraic Expressions<\/strong><\/p>\n<p id=\"fs-id1167835518988\">In the following exercises, translate to an algebraic expression and simplify if possible.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835518992\">\n<div data-type=\"problem\" id=\"fs-id1167835518994\">\n<p id=\"fs-id1167835518996\">the sum of 3 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-328c7ef88b11dd44f16dc983336c4b0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"35\" style=\"vertical-align: -4px;\" \/> increased by 7<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835519008\">\n<p id=\"fs-id1167835519010\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04b4413671d3707a89c0e1f126f09c5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#55;&#59;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835519047\">\n<div data-type=\"problem\" id=\"fs-id1167835519049\">\n<p id=\"fs-id1167835519051\">the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d33a2f5760bb02cc8868fb05cfa50c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> increased by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e554ca92166de4bd98acfbef8f678cd2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835519110\">\n<div data-type=\"problem\" id=\"fs-id1167835519112\">\n<p id=\"fs-id1167835519114\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5450913cc453faf132dc56e5965ca797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb9e52ddecc045b16dbf509b1c89f11b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cba90047eec6a05e400b16dfb86d7f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" style=\"vertical-align: -1px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-170b493658855d7211748be1a7029d86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: 0px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835519149\">\n<p id=\"fs-id1167835519151\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09017fbbaa0316e319972d53dd1367c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#59;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3242c61270a55090c7b71433a476f332_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#53;&#45;&#49;&#49;&#59;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"110\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835519200\">\n<div data-type=\"problem\" id=\"fs-id1167835519202\">\n<p id=\"fs-id1167835519205\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec1795c5af248d148a298ec6e4785280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee7bea24371f97fb5673002404153494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535035\">\n<div data-type=\"problem\" id=\"fs-id1167834535037\">\n<p id=\"fs-id1167834535039\">the quotient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834535057\">\n<p id=\"fs-id1167834535059\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-107bf84796215066792d64ae12f51759_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#125;&#123;&#97;&#43;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"24\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535078\">\n<div data-type=\"problem\" id=\"fs-id1167834535080\">\n<p id=\"fs-id1167834535082\">the product of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee7bea24371f97fb5673002404153494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> and the difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e8716946f6a868f015e0d62f28bc540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834535123\"><strong data-effect=\"bold\">Use Integers in Applications<\/strong><\/p>\n<p id=\"fs-id1167834535130\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834535133\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834535135\">\n<p id=\"fs-id1167834535137\"><strong data-effect=\"bold\">Temperature<\/strong> On January 15, the high temperature in Anaheim, California, was 84\u00b0. That same day, the high temperature in Embarrass, Minnesota, was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-881ce7159f93e129159506063c274f85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> What was the difference between the temperature in Anaheim and the temperature in Embarrass?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834535162\">\n<p id=\"fs-id1167834535163\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44ba2bd2b1855f3d633162e2f779ed6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535176\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834535178\">\n<p id=\"fs-id1167834535180\"><strong data-effect=\"bold\">Temperature<\/strong> On January 21, the high temperature in Palm Springs, California, was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb663423819d74fdb9c4cbe83bf8a094_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"22\" style=\"vertical-align: -4px;\" \/> and the high temperature in Whitefield, New Hampshire, was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce76ebeb4e699c77cff69eff5d6a2e05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#51;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> What was the difference between the temperature in Palm Springs and the temperature in Whitefield?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535230\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834535232\">\n<p id=\"fs-id1167834535234\"><strong data-effect=\"bold\">Football<\/strong> On the first down, the Chargers had the ball on their 25-yard line. On the next three downs, they lost 6 yards, gained 10 yards, and lost 8 yards. What was the yard line at the end of the fourth down?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834535244\">\n<p id=\"fs-id1167834535246\">21<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535252\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834535254\">\n<p id=\"fs-id1167834535256\"><strong data-effect=\"bold\">Football<\/strong> On the first down, the Steelers had the ball on their 30-yard line. On the next three downs, they gained 9 yards, lost 14 yards, and lost 2 yards. What was the yard line at the end of the fourth down?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535281\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834535283\">\n<p id=\"fs-id1167834535285\"><strong data-effect=\"bold\">Checking Account<\/strong> Mayra has ?124 in her checking account. She writes a check for ?152. What is the new balance in her checking account?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834535295\">\n<p id=\"fs-id1167834535297\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95b77e15422d99d4c80f7dc251eaa3fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535310\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834535312\">\n<p id=\"fs-id1167834535314\"><strong data-effect=\"bold\">Checking Account<\/strong> Reymonte has a balance of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfb54cc5452ccaeb6cbd9f1df736b373_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> in his checking account. He deposits ?281 to the account. What is the new balance?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834537505\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167834537512\">\n<div data-type=\"problem\" id=\"fs-id1167834537515\">\n<p id=\"fs-id1167834537517\">Explain why the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and 2 is negative, but the sum of 8 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> is positive.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537532\">\n<p id=\"fs-id1167834537534\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834537539\">\n<div data-type=\"problem\" id=\"fs-id1167834537541\">\n<p id=\"fs-id1167834537543\">Give an example from your life experience of adding two negative numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834537554\">\n<div data-type=\"problem\" id=\"fs-id1167834537556\">\n<p id=\"fs-id1167834537559\">In your own words, state the rules for multiplying and dividing integers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537563\">\n<p id=\"fs-id1167834537564\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834537570\">\n<div data-type=\"problem\" id=\"fs-id1167834537572\">\n<p id=\"fs-id1167834537574\">Why is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dec065851d4c5a475548b5a6f029cc08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#52;&#125;&#94;&#123;&#51;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834537614\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167834537619\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834537635\" data-alt=\"This table has 4 columns, 6 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: simplify expressions with absolute value, add and subtract integers, multiply and divide integers, simplify and evaluate expressions with integers, translate English phrases to algebraic expressions, use integers in applications. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_02_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 6 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: simplify expressions with absolute value, add and subtract integers, multiply and divide integers, simplify and evaluate expressions with integers, translate English phrases to algebraic expressions, use integers in applications. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167834537643\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167834537656\">\n<dt>absolute value<\/dt>\n<dd id=\"fs-id1167834537661\">The absolute value of a number is its distance from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> on the number line.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834537670\">\n<dt>integers<\/dt>\n<dd id=\"fs-id1167834537674\">The whole numbers and their opposites are called the integers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834537678\">\n<dt>negative numbers<\/dt>\n<dd id=\"fs-id1167834537683\">Numbers less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> are negative numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834537692\">\n<dt>opposite<\/dt>\n<dd id=\"fs-id1167834537696\">The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-881","chapter","type-chapter","status-publish","hentry"],"part":762,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/881","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/881\/revisions"}],"predecessor-version":[{"id":14620,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/881\/revisions\/14620"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/762"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/881\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=881"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=881"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=881"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=881"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}