{"id":15231,"date":"2019-05-22T18:25:55","date_gmt":"2019-05-22T22:25:55","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-absolute-value-inequalities\/"},"modified":"2019-05-22T18:26:36","modified_gmt":"2019-05-22T22:26:36","slug":"solve-absolute-value-inequalities","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-absolute-value-inequalities\/","title":{"raw":"Solve Absolute Value Inequalities","rendered":"Solve Absolute Value Inequalities"},"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>Solve absolute value equations<\/li><li>Solve absolute value inequalities with \u201cless than\u201d<\/li><li>Solve absolute value inequalities with \u201cgreater than\u201d<\/li><li>Solve applications with absolute value<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167831894154\" class=\"be-prepared\"><p id=\"fs-id1167834111770\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167831890429\" type=\"1\"><li>Evaluate: \\(\\text{\u2212}|7|.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167835595046\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Fill in \\(\\text{&lt;},\\text{&gt;},\\) or \\(=\\) for each of the following pairs of numbers.<div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d0<\/span> \\(|-8|\\text{___}-|-8|\\) <span class=\"token\">\u24d1<\/span> \\(12\\text{___}-|-12|\\) <span class=\"token\">\u24d2<\/span> \\(|-6|\\text{___}-6\\) <span class=\"token\">\u24d3<\/span> \\(\\text{\u2212}\\left(-15\\right)\\text{___}-|-15|\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167835595046\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(14-2|8-3\\left(4-1\\right)|.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167835319324\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834065781\"><h3 data-type=\"title\">Solve Absolute Value Equations<\/h3><p id=\"fs-id1167835333127\">As we prepare to solve absolute value equations, we review our definition of <span data-type=\"term\">absolute value<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167830962404\"><div data-type=\"title\">Absolute Value<\/div><p>The absolute value of a number is its distance from zero on the number line.<\/p><p>The absolute value of a number <em data-effect=\"italics\">n<\/em> is written as \\(|n|\\) and \\(|n|\\ge 0\\) for all numbers.<\/p><p id=\"fs-id1167834120880\">Absolute values are always greater than or equal to zero.<\/p><\/div><p id=\"fs-id1167831923735\">We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:<\/p><p id=\"fs-id1167835338742\">\\(\\phantom{\\rule{3em}{0ex}}-5\\) is 5 units away from 0, so \\(|-5|=5.\\)<\/p><p>\\(\\phantom{\\rule{3.65em}{0ex}}5\\) is 5 units away from 0, so \\(|5|=5.\\)<\/p><p id=\"fs-id1167835339214\"><a href=\"#CNX_IntAlg_Figure_02_07_001\" class=\"autogenerated-content\">(Figure)<\/a> illustrates this idea.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_001\"><div class=\"bc-figcaption figcaption\">The numbers 5 and \\(-5\\) are both five units away from zero.<\/div><span data-type=\"media\" data-alt=\"The figure is a number line with tick marks at negative 5, 0, and 5. The distance between negative 5 and 0 is given as 5 units, so the absolute value of negative 5 is 5. The distance between 5 and 0 is 5 units, so the absolute value of 5 is 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with tick marks at negative 5, 0, and 5. The distance between negative 5 and 0 is given as 5 units, so the absolute value of negative 5 is 5. The distance between 5 and 0 is 5 units, so the absolute value of 5 is 5.\"><\/span><\/div><p id=\"fs-id1167835357520\">For the equation \\(|x|=5,\\) we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and \\(-5\\) are five units from zero on the number line. They are the solutions to the equation.<\/p><div data-type=\"equation\" id=\"fs-id1167835215234\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\text{If}\\hfill &amp; &amp; &amp; &amp; &amp; |x|=5\\hfill \\\\ \\text{then}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.3em}{0ex}}x=-5\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}x=5\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167834300003\">The solution can be simplified to a single statement by writing \\(x=\\text{\u00b1}5.\\) This is read, \u201c<em data-effect=\"italics\">x<\/em> is equal to positive or negative 5\u201d.<\/p><p id=\"fs-id1167830837184\">We can generalize this to the following property for absolute value equations.<\/p><div data-type=\"note\" id=\"fs-id1167834299926\"><div data-type=\"title\">Absolute Value Equations<\/div><p id=\"fs-id1167832041537\">For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<\/p><div data-type=\"equation\" id=\"fs-id1167835284995\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\text{if}\\hfill &amp; &amp; &amp; &amp; &amp; |u|=a\\hfill \\\\ \\text{then}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.3em}{0ex}}u=\\text{\u2212}a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u=a\\hfill \\end{array}\\)<\/div><p>Remember that an absolute value cannot be a negative number.<\/p><\/div><div data-type=\"example\" id=\"fs-id1167831112021\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835363618\"><p id=\"fs-id1167834495232\">Solve: <span class=\"token\">\u24d0<\/span> \\(|x|=8\\) <span class=\"token\">\u24d1<\/span> \\(|y|=-6\\) <span class=\"token\">\u24d2<\/span> \\(|z|=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834124409\"><p id=\"fs-id1167835400249\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; |x|=8\\hfill \\\\ \\text{Write the equivalent equations.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.45em}{0ex}}x=-8\\phantom{\\rule{0.2em}{0ex}}\\text{or}\\phantom{\\rule{0.4em}{0ex}}x=8\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.45em}{0ex}}x=\\text{\u00b1}8\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{12.6em}{0ex}}|y|=-6\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{12.6em}{0ex}}\\text{No solution}\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div> Since an absolute value is always positive, there are no solutions to this equation.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d2<\/span><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; |z|=0\\hfill \\\\ \\text{Write the equivalent equations.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.45em}{0ex}}z=-0\\phantom{\\rule{0.2em}{0ex}}\\text{or}\\phantom{\\rule{0.2em}{0ex}}z=0\\hfill \\\\ \\text{Since}\\phantom{\\rule{0.2em}{0ex}}-0=0,\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.45em}{0ex}}z=0\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div> Both equations tell us that \\(z=0\\) and so there is only one solution.<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831106822\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167831922992\">Solve: <span class=\"token\">\u24d0<\/span> \\(|x|=2\\) <span class=\"token\">\u24d1<\/span> \\(|y|=-4\\) <span class=\"token\">\u24d2<\/span> \\(|z|=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835336858\"><p><span class=\"token\">\u24d0<\/span>\\(\\text{\u00b1}2\\)<span class=\"token\">\u24d1<\/span> no solution <span class=\"token\">\u24d2<\/span> 0<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835595443\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835348491\"><div data-type=\"problem\" id=\"fs-id1167834430172\"><p id=\"fs-id1167834133052\">Solve: <span class=\"token\">\u24d0<\/span> \\(|x|=11\\) <span class=\"token\">\u24d1<\/span> \\(|y|=-5\\) <span class=\"token\">\u24d2<\/span> \\(|z|=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830894511\"><p id=\"fs-id1167834528036\"><span class=\"token\">\u24d0<\/span>\\(\\text{\u00b1}11\\)<span class=\"token\">\u24d1<\/span> no solution <span class=\"token\">\u24d2<\/span> 0<\/p><\/div><\/div><\/div><p>To solve an <span data-type=\"term\" class=\"no-emphasis\">absolute value equation<\/span>, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.<\/p><div data-type=\"example\" id=\"fs-id1167834339985\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve Absolute Value Equations<\/div><div data-type=\"exercise\" id=\"fs-id1167826801729\"><div data-type=\"problem\" id=\"fs-id1167835339566\"><p id=\"fs-id1167831910493\">Solve \\(|5x-4|-3=8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835258843\"><span data-type=\"media\" id=\"fs-id1167835288367\" data-alt=\"Step 1 is to isolate the absolute value expression. The difference between the absolute value of the quantity 5 x minus 4 and 3 is equal to 8. Add 3 to both sides. The result is the absolute value of the quantity 5 x minus 4 is equal to 11.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to isolate the absolute value expression. The difference between the absolute value of the quantity 5 x minus 4 and 3 is equal to 8. Add 3 to both sides. The result is the absolute value of the quantity 5 x minus 4 is equal to 11.\"><\/span><span data-type=\"media\" id=\"fs-id1167826995876\" data-alt=\"Step 2 is to write the equivalent equations, 5 x minus 4 is equal to negative 11 and 5 x minus 4 is equal to 11.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to write the equivalent equations, 5 x minus 4 is equal to negative 11 and 5 x minus 4 is equal to 11.\"><\/span><span data-type=\"media\" id=\"fs-id1167835362546\" data-alt=\"Step 3 is to solve each equation. Add 4 to each side. 5 x is equal to negative 7 or 5 x is equal to 15. Divide each side by 5. The result is x is equal to negative seven-fifths or x is equal to 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve each equation. Add 4 to each side. 5 x is equal to negative 7 or 5 x is equal to 15. Divide each side by 5. The result is x is equal to negative seven-fifths or x is equal to 3.\"><\/span><span data-type=\"media\" id=\"fs-id1167835264946\" data-alt=\"Step 4 is to check each solution. Substitute 3 and negative seven-fifths into the original equation, the difference between the absolute value of the quantity 5 x minus 4 and 3 is equal to 8. Substitute 3 for x. Is the difference between the absolute value of the quantity 5 times 3 minus 4 and 3 equal to 8? Is the difference between the absolute value of the quantity 15 minus 4 and 3 equal to 8? Is the difference between the absolute value of the 11 and 3 equal to 8? Is 11 minus 3 equal to 8? 8 is equal to 8, so the solution x is equal to 3 checks. Substitute negative seven-fifths for x. Is the difference between the absolute value of the quantity 5 times negative seven-fifths minus 4 and 3 equal to 8? Is the difference between the absolute value of the quantity negative 7 minus 4 and 3 equal to 8? Is the difference between the absolute value of the negative 11 and 3 equal to 8? Is 11 minus 3 equal to 8? 8 is equal to 8, so the solution x is equal to negative seven-fifths checks.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check each solution. Substitute 3 and negative seven-fifths into the original equation, the difference between the absolute value of the quantity 5 x minus 4 and 3 is equal to 8. Substitute 3 for x. Is the difference between the absolute value of the quantity 5 times 3 minus 4 and 3 equal to 8? Is the difference between the absolute value of the quantity 15 minus 4 and 3 equal to 8? Is the difference between the absolute value of the 11 and 3 equal to 8? Is 11 minus 3 equal to 8? 8 is equal to 8, so the solution x is equal to 3 checks. Substitute negative seven-fifths for x. Is the difference between the absolute value of the quantity 5 times negative seven-fifths minus 4 and 3 equal to 8? Is the difference between the absolute value of the quantity negative 7 minus 4 and 3 equal to 8? Is the difference between the absolute value of the negative 11 and 3 equal to 8? Is 11 minus 3 equal to 8? 8 is equal to 8, so the solution x is equal to negative seven-fifths checks.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835304704\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167835511680\">Solve: \\(|3x-5|-1=6.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835596157\"><p id=\"fs-id1167835327087\">\\(x=4,x=-\\frac{2}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835320274\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835353084\"><div data-type=\"problem\" id=\"fs-id1167831024455\"><p>Solve: \\(|4x-3|-5=2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835304970\"><p id=\"fs-id1167831923395\">\\(x=-1,x=\\frac{5}{2}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835338375\">The steps for solving an absolute value equation are summarized here.<\/p><div data-type=\"note\" class=\"howto\"><div data-type=\"title\">Solve absolute value equations.<\/div><ol id=\"fs-id1167835262974\" type=\"1\" class=\"stepwise\"><li>Isolate the absolute value expression.<\/li><li>Write the equivalent equations.<\/li><li>Solve each equation.<\/li><li>Check each solution.<\/li><\/ol><\/div><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835364757\"><div data-type=\"problem\" id=\"fs-id1167832043509\"><p id=\"fs-id1167832153416\">Solve \\(2|x-7|+5=9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171790373440\"><table id=\"fs-id1167835335371\" class=\"unnumbered unstyled can-break\" summary=\"The sum of 2 times the absolute value of the quantity x minus 7 and 5 is equal to 9. Isolate the absolute value expression. 2 times the absolute value of the quantity x minus 7 is equal to 4 simplifies to the absolute value of the quantity x minus 7 is equal to 2. Write the equivalent equations. They are x minus 7 is equal to negative 2 or x minus 7 is equal to 2. Solve each equation. The solutions x is equal to 5 or x is equal to 9. Check using the original equation, the sum of 2 times the absolute value of the quantity x minus 7 and 5 is equal to 9. Is the sum of 2 times the absolute value of the quantity 5 minus 7 and 5 equal to 9? Is the sum of 2 times the absolute value of negative 2 and 5 equal to 9? Is 2 times 2 plus 5 equal to 9? Is 4 plus 5 equal to 9? 9 is equal to 9, so the solution x is equal to 5 checks. Is the sum of 2 times the absolute value of the quantity 9 minus 7 and 5 equal to 9? Is the sum of 2 times the absolute value of 2 and 5 equal to 9? Is 2 times 2 plus 5 equal to 9? Is 4 plus 5 equal to 9? 9 is equal to 9, so the solution x is equal to 9 checks.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(2|x-7|+5=9\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Isolate the absolute value expression.\u2003\u2003\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"right\">\\(2|x-7|=4\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(|x-7|=2\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the equivalent equations.<\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{0.6em}{0ex}}x-7=\\text{\u2212}2\\) or \\(x-7=2\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve each equation.<\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{2.25em}{0ex}}x=5\\phantom{\\rule{0.7em}{0ex}}\\) or \\(\\phantom{\\rule{1.6em}{0ex}}x=9\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835333456\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835198774\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835371191\"><div data-type=\"problem\" id=\"fs-id1167835419823\"><p id=\"fs-id1167835368941\">Solve: \\(3|x-4|-4=8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834308115\"><p id=\"fs-id1167834228781\">\\(x=8,x=0\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835181971\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835349582\"><div data-type=\"problem\" id=\"fs-id1167830962365\"><p id=\"fs-id1167826880268\">Solve: \\(2|x-5|+3=9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834228481\"><p id=\"fs-id1167835370272\">\\(x=8,x=2\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1166400945810\">Remember, an absolute value is always positive!<\/p><div data-type=\"example\" id=\"fs-id1167835281948\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834539505\"><div data-type=\"problem\" id=\"fs-id1167834274143\"><p id=\"fs-id1167834279299\">Solve: \\(|\\frac{2}{3}x-4|+11=3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835529802\"><p id=\"fs-id1167835183544\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; |\\frac{2}{3}x-4|+11=3\\hfill \\\\ \\text{Isolate the absolute value term.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{2.2em}{0ex}}|\\frac{2}{3}x-4|=-8\\hfill \\\\ \\text{An absolute value cannot be negative.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{1.2em}{0ex}}\\text{No solution}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831025415\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834192376\"><div data-type=\"problem\" id=\"fs-id1167835274654\"><p id=\"fs-id1167831823492\">Solve: \\(|\\frac{3}{4}x-5|+9=4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835236935\"><p id=\"fs-id1167826994048\">No solution<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831923617\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834489662\"><div data-type=\"problem\"><p id=\"fs-id1167832060267\">Solve: \\(|\\frac{5}{6}x+3|+8=6.\\)<\/p><\/div><div data-type=\"solution\"><p>No solution<\/p><\/div><\/div><\/div><p id=\"fs-id1167830960678\">Some of our absolute value equations could be of the form \\(|u|=|v|\\) where <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em> are algebraic expressions. For example, \\(|x-3|=|2x+1|.\\)<\/p><p id=\"fs-id1167835361268\">How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, <em data-effect=\"italics\">u<\/em>, and a positive real number, <em data-effect=\"italics\">a<\/em>, if \\(|u|=a,\\) then \\(u=\\text{\u2212}a\\) or \\(u=a.\\)<\/p><p id=\"fs-id1167835339561\">This tell us that<\/p><div data-type=\"equation\" id=\"fs-id1167835213198\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccccccccc}\\text{if}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.3em}{0ex}}|u|=|v|\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\\\ \\text{then}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.3em}{0ex}}u=\\text{\u2212}v\\hfill &amp; \\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u=v&amp; &amp; \\text{or}\\hfill &amp; &amp; &amp; u=v\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835288119\">This leads us to the following property for equations with two absolute values.<\/p><div data-type=\"note\"><div data-type=\"title\">Equations with Two Absolute Values<\/div><p id=\"fs-id1167834557281\">For any algebraic expressions, <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em>,<\/p><div data-type=\"equation\" id=\"fs-id1167834523020\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\text{if}\\hfill &amp; &amp; &amp; &amp; &amp; |u|=|v|\\hfill \\\\ \\text{then}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.3em}{0ex}}u=\\text{\u2212}v\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u=v\\hfill \\end{array}\\)<\/div><\/div><p id=\"fs-id1166400208360\">When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835269461\"><div data-type=\"problem\" id=\"fs-id1167835186322\"><p id=\"fs-id1167835326417\">Solve: \\(|5x-1|=|2x+3|.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1171792882089\"><div data-type=\"equation\" id=\"fs-id1167836729601\" class=\"unnumbered\" data-label=\"\">\\(\\phantom{\\rule{3em}{0ex}}|5x-1|=|2x+3|\\)<\/div><p id=\"fs-id1167834189093\">\\(\\begin{array}{cccccccccc}\\text{Write the equivalent equations.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}5x-1&amp; =\\hfill &amp; \\text{\u2212}\\left(2x+3\\right)\\hfill &amp; \\phantom{\\rule{1em}{0ex}}\\text{or}\\phantom{\\rule{1em}{0ex}}\\hfill &amp; \\hfill 5x-1&amp; =\\hfill &amp; 2x+3\\hfill \\\\ \\text{Solve each equation.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}5x-1&amp; =\\hfill &amp; -2x-3\\hfill &amp; \\phantom{\\rule{1em}{0ex}}\\text{or}\\phantom{\\rule{1em}{0ex}}\\hfill &amp; 3x-1\\hfill &amp; =\\hfill &amp; 3\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}7x-1&amp; =\\hfill &amp; -3\\hfill &amp; &amp; \\hfill 3x&amp; =\\hfill &amp; 4\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}7x&amp; =\\hfill &amp; -2\\hfill &amp; &amp; \\hfill x&amp; =\\hfill &amp; \\frac{4}{3}\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}x&amp; =\\hfill &amp; -\\frac{2}{7}\\hfill &amp; \\phantom{\\rule{1em}{0ex}}\\text{or}\\phantom{\\rule{1em}{0ex}}\\hfill &amp; \\hfill x&amp; =\\hfill &amp; \\frac{4}{3}\\hfill \\\\ \\text{Check.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\\\ \\text{We leave the check to you.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835361950\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835310674\"><div data-type=\"problem\" id=\"fs-id1167835194778\"><p id=\"fs-id1167834228660\">Solve: \\(|7x-3|=|3x+7|.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831040602\"><p id=\"fs-id1167835360737\">\\(x=-\\frac{2}{5},\\)\\(x=\\frac{5}{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831891635\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167826857424\"><p id=\"fs-id1167835370427\">Solve: \\(|6x-5|=|3x+4|.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835610057\">\\(x=3,\\)\\(x=\\frac{1}{9}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Solve Absolute Value Inequalities with \u201cLess Than\u201d<\/h3><p>Let\u2019s look now at what happens when we have an <span data-type=\"term\" class=\"no-emphasis\">absolute value inequality<\/span>. Everything we\u2019ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.<\/p><p id=\"fs-id1167835378734\">Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line. For the equation \\(|x|=5,\\) we saw that both 5 and \\(-5\\) are five units from zero on the number line. They are the solutions to the equation.<\/p><div data-type=\"equation\" id=\"fs-id1167835283055\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill |x|=5\\hfill \\\\ \\hfill x=-5\\phantom{\\rule{4em}{0ex}}\\text{or}\\phantom{\\rule{4em}{0ex}}x=5\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835365479\">What about the inequality \\(|x|\\le 5?\\) Where are the numbers whose distance is less than or equal to 5? We know \\(-5\\) and 5 are both five units from zero. All the numbers between \\(-5\\) and 5 are less than five units from zero. See <a href=\"#CNX_IntAlg_Figure_02_07_004\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_004\"><span data-type=\"media\" id=\"fs-id1167835333448\" data-alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a left bracket at negative 5 and a right bracket at 5. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x which is less than or equal to 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a left bracket at negative 5 and a right bracket at 5. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x which is less than or equal to 5.\"><\/span><\/div><p id=\"fs-id1167835421120\">In a more general way, we can see that if \\(|u|\\le a,\\) then \\(\\text{\u2212}a\\le u\\le a.\\) See <a href=\"#CNX_IntAlg_Figure_02_07_005\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_005\"><span data-type=\"media\" id=\"fs-id1167834432227\" data-alt=\"The figure is a number line with negative a 0, and a displayed. There is a left bracket at negative a and a right bracket at a. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is less than or equal to a, then negative a is less than or equal to u which is less than or equal to a.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative a 0, and a displayed. There is a left bracket at negative a and a right bracket at a. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is less than or equal to a, then negative a is less than or equal to u which is less than or equal to a.\"><\/span><\/div><p id=\"fs-id1167835301566\">This result is summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167835331002\"><div data-type=\"title\">Absolute Value Inequalities with \\(&lt;\\) or \\(\\le \\)<\/div><p id=\"fs-id1167835381129\">For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<\/p><div data-type=\"equation\" id=\"fs-id1167834526452\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccccccc}\\text{if}\\hfill &amp; &amp; &amp; &amp; |u|&lt;a,\\hfill &amp; &amp; &amp; &amp; &amp; \\text{then}\\phantom{\\rule{0.5em}{0ex}}\\text{\u2212}a&lt;u&lt;a\\hfill \\\\ \\text{if}\\hfill &amp; &amp; &amp; &amp; |u|\\le a,\\hfill &amp; &amp; &amp; &amp; &amp; \\text{then}\\phantom{\\rule{0.5em}{0ex}}\\text{\u2212}a\\le u\\le a\\hfill \\end{array}\\)<\/div><\/div><p id=\"fs-id1167835349608\">After solving an inequality, it is often helpful to check some points to see if the solution makes sense. The graph of the solution divides the number line into three sections. Choose a value in each section and substitute it in the original inequality to see if it makes the inequality true or not. While this is not a complete check, it often helps verify the solution.<\/p><div data-type=\"example\" id=\"fs-id1167835352166\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167826993390\"><p id=\"fs-id1167826779359\">Solve \\(|x|&lt;7.\\) Graph the solution and write the solution in interval notation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835354119\"><table id=\"fs-id1167835344288\" class=\"unnumbered unstyled\" summary=\"The absolute value of x is less than 7. Write the equivalent inequality. It is negative 7 is less than x which is less than 7. Graph the solution. It is a number line with a left parenthesis at negative 7, a right parenthesis at 7 and shading between the parentheses. Write the solution using interval notation. It is negative 7 to 7 within parentheses.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834228084\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the equivalent inequality.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835343024\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the solution.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the solution using interval notation.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831238937\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835483823\">Check:<\/p><p id=\"fs-id1167835304135\">To verify, check a value in each section of the number line showing the solution. Choose numbers such as \\(-8,\\) 1, and 9.<\/p><span data-type=\"media\" id=\"fs-id1167832066987\" data-alt=\"The figure is a number line with a left parenthesis at negative 7, a right parenthesis at 7 and shading between the parentheses. The values negative 8, 1, and 9 are marked with points. The absolute value of negative 8 is less than 7 is false. It does not satisfy the absolute value of x is less than 7. The absolute value of 1 is less than 7 is true. It does satisfy the absolute value of x is less than 7. The absolute value of 9 is less than 7 is false. It does not satisfy the absolute value of x is less than 7.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with a left parenthesis at negative 7, a right parenthesis at 7 and shading between the parentheses. The values negative 8, 1, and 9 are marked with points. The absolute value of negative 8 is less than 7 is false. It does not satisfy the absolute value of x is less than 7. The absolute value of 1 is less than 7 is true. It does satisfy the absolute value of x is less than 7. The absolute value of 9 is less than 7 is false. It does not satisfy the absolute value of x is less than 7.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835375988\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832066840\"><div data-type=\"problem\" id=\"fs-id1167834141948\"><p id=\"fs-id1167831107059\">Graph the solution and write the solution in interval notation: \\(|x|&lt;9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835225764\"><span data-type=\"media\" id=\"fs-id1167826967468\" data-alt=\"The solution is negative 9 is less than x which is less than 9. The number line shows open circles at negative 9 and 9 with shading in between the circles. The interval notation is negative 9 to 9 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_301_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 9 is less than x which is less than 9. The number line shows open circles at negative 9 and 9 with shading in between the circles. The interval notation is negative 9 to 9 within parentheses.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835358565\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835350323\"><div data-type=\"problem\" id=\"fs-id1167835341270\"><p id=\"fs-id1167834535568\">Graph the solution and write the solution in interval notation: \\(|x|&lt;1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835307947\"><span data-type=\"media\" id=\"fs-id1167831923510\" data-alt=\"The solution is negative 1 is less than x which is less than 1. The number line shows open circles at negative 1 and 1 with shading in between the circles. The interval notation is negative 1 to 1 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than 1. The number line shows open circles at negative 1 and 1 with shading in between the circles. The interval notation is negative 1 to 1 within parentheses.\"><\/span><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167835363985\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167826967314\"><div data-type=\"problem\" id=\"fs-id1167834132141\"><p id=\"fs-id1167834097907\">Solve \\(|5x-6|\\le 4.\\) Graph the solution and write the solution in interval notation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835364122\"><table id=\"fs-id1167835366818\" class=\"unnumbered unstyled can-break\" summary=\"Step 1 is to isolate the absolute value expression, the absolute value of the quantity 5 x minus 6 is less than or equal to 4. It is isolated. Step 2 is to write the equivalent compound inequality. Negative 4 is less than or equal to 5 x minus 6 which is less than 4. Step 3 is to solve the compound inequality. 2 is less than or equal to 5 x which is less than or equal to 10. Step 4 is to graph the solution. The graph showed closed points at two-fifths and 2 with shading between the circles. Step 5 is to write the solution using interval notation. It is two-fifths to 2 within brackets. The check is left to you.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Isolate the absolute value expression.<div data-type=\"newline\"><br><\/div>It is isolated.<\/td><td data-valign=\"top\" data-align=\"center\">\\(|5x-6|\\le 4\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Write the equivalent compound inequality.<\/td><td data-valign=\"top\" data-align=\"center\">\\(-4\\le 5x-6\\le 4\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Solve the compound inequality.<\/td><td data-valign=\"top\" data-align=\"center\">\\(2\\le 5x\\le 10\\)<div data-type=\"newline\"><br><\/div>\\(\\frac{2}{5}\\le x\\le 2\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4.<\/strong> Graph the solution.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835416933\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5.<\/strong> Write the solution using interval notation.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left[\\frac{2}{5},2\\right]\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div>The check is left to you.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834189360\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835358273\"><div data-type=\"problem\" id=\"fs-id1167834238700\"><p id=\"fs-id1167835596723\">Solve \\(|2x-1|\\le 5.\\) Graph the solution and write the solution in interval notation:<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834238687\"><span data-type=\"media\" id=\"fs-id1167835327750\" data-alt=\"The solution is negative 2 is less than or equal to x which is less than or equal to 3. The number line shows closed circles at negative 2 and 3 with shading between the circles. The interval notation is negative 2 to 3 within brackets.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than or equal to x which is less than or equal to 3. The number line shows closed circles at negative 2 and 3 with shading between the circles. The interval notation is negative 2 to 3 within brackets.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832051824\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826782420\"><div data-type=\"problem\" id=\"fs-id1167835513688\"><p id=\"fs-id1167834094549\">Solve \\(|4x-5|\\le 3.\\) Graph the solution and write the solution in interval notation:<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834432147\"><span data-type=\"media\" id=\"fs-id1167834593016\" data-alt=\"The solution is one-half is less than or equal to x which is less than or equal to 2. The number line shows closed circles at one-half and 2 with shading between the circles. The interval notation is one-half to 2 within brackets.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is one-half is less than or equal to x which is less than or equal to 2. The number line shows closed circles at one-half and 2 with shading between the circles. The interval notation is one-half to 2 within brackets.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834246739\" class=\"howto\"><div data-type=\"title\">Solve absolute value inequalities with &lt; or \u2264.<\/div><ol id=\"fs-id1167832067486\" type=\"1\" class=\"stepwise\"><li>Isolate the absolute value expression.<\/li><li>Write the equivalent compound inequality.<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167835419866\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccccccccccc}|u|&lt;a\\hfill &amp; &amp; &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{\u2212}a&lt;u&lt;a\\hfill \\\\ |u|\\le a\\hfill &amp; &amp; &amp; &amp; &amp; \\text{is equivalent to}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{\u2212}a\\le u\\le a\\hfill \\end{array}\\)<\/div><\/li><li>Solve the compound inequality.<\/li><li>Graph the solution<\/li><li>Write the solution using interval notation.<\/li><\/ol><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834308172\"><h3 data-type=\"title\">Solve Absolute Value Inequalities with \u201cGreater Than\u201d<\/h3><p id=\"fs-id1167832060088\">What happens for absolute value inequalities that have \u201cgreater than\u201d? Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.<\/p><p id=\"fs-id1167835337199\">We started with the inequality \\(|x|\\le 5.\\) We saw that the numbers whose distance is less than or equal to five from zero on the number line were \\(-5\\) and 5 and all the numbers between \\(-5\\) and 5. See <a href=\"#CNX_IntAlg_Figure_02_07_009\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_009\"><span data-type=\"media\" id=\"fs-id1167835320124\" data-alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its right and a right bracket at 5 with shading to its left. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x is less than or equal to 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its right and a right bracket at 5 with shading to its left. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x is less than or equal to 5.\"><\/span><\/div><p id=\"fs-id1167835304662\">Now we want to look at the inequality \\(|x|\\ge 5.\\) Where are the numbers whose distance from zero is greater than or equal to five?<\/p><p id=\"fs-id1167826994597\">Again both \\(-5\\) and 5 are five units from zero and so are included in the solution. Numbers whose distance from zero is greater than five units would be less than \\(-5\\) and greater than 5 on the number line. See <a href=\"#CNX_IntAlg_Figure_02_07_010\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_010\"><span data-type=\"media\" id=\"fs-id1167834537398\" data-alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its left and a left bracket at 5 with shading to its right. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is greater than or equal to 5, then x is less than or equal to negative 5 or x is greater than or equal to 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its left and a left bracket at 5 with shading to its right. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is greater than or equal to 5, then x is less than or equal to negative 5 or x is greater than or equal to 5.\"><\/span><\/div><p id=\"fs-id1167835350484\">In a more general way, we can see that if \\(|u|\\ge a,\\) then \\(u\\le \\text{\u2212}a\\) or \\(u\\le a.\\) See <a href=\"#CNX_IntAlg_Figure_02_07_011\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_011\"><span data-type=\"media\" id=\"fs-id1167834340020\" data-alt=\"The figure is a number line with negative a, 0, and a displayed. There is a right bracket at negative a that has shading to its left and a left bracket at a with shading to its right. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is greater than or equal to a, then u is less than or equal to negative a or u is greater than or equal to a.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative a, 0, and a displayed. There is a right bracket at negative a that has shading to its left and a left bracket at a with shading to its right. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is greater than or equal to a, then u is less than or equal to negative a or u is greater than or equal to a.\"><\/span><\/div><p id=\"fs-id1167835511103\">This result is summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167834423855\"><div data-type=\"title\">Absolute Value Inequalities with &gt; or \u2265<\/div><p id=\"fs-id1167831880144\">For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<\/p><div data-type=\"equation\" id=\"fs-id1167832055469\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccccccccccc}\\text{if}\\hfill &amp; &amp; &amp; &amp; &amp; |u|&gt;a,\\hfill &amp; &amp; &amp; &amp; &amp; \\text{then}\\phantom{\\rule{0.2em}{0ex}}u&lt;-a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u&gt;a\\hfill \\\\ \\text{if}\\hfill &amp; &amp; &amp; &amp; &amp; |u|\\ge a,\\hfill &amp; &amp; &amp; &amp; &amp; \\text{then}\\phantom{\\rule{0.2em}{0ex}}u\\le \\text{\u2212}a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u\\ge a\\hfill \\end{array}\\)<\/div><\/div><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835326481\"><div data-type=\"problem\" id=\"fs-id1167835326484\"><p id=\"fs-id1167835216061\">Solve \\(|x|&gt;4.\\) Graph the solution and write the solution in interval notation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835339041\"><table id=\"fs-id1167835339044\" class=\"unnumbered unstyled\" summary=\"The absolute value of x is greater than 4. Write the equivalent inequality. They are x is less than negative 4 or x is greater than 4. Graph the solution. It is a right parenthesis at negative 4 with shading to its left and a parenthesis at 4 with shading to its right. Write the solution using interval notation. It is the union of negative infinity to negative 4 within parentheses and 4 to infinity with parentheses. Check. To verify, check a value in each section of the number line showing the solution. Choose numbers such as negative 6, 0, and 7.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(|x|&gt;4\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the equivalent inequality.<\/td><td data-valign=\"top\" data-align=\"center\">\\(x&lt;-4\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}x&gt;4\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Graph the solution.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835332104\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the solution using interval notation.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(\\text{\u2212}\\infty ,-4\\right)\\cup \\left(4,\\infty \\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167835378703\">To verify, check a value in each section of the number line showing the solution. Choose numbers such as \\(-6,\\) 0, and 7.<\/p><span data-type=\"media\" id=\"fs-id1167832053739\" data-alt=\"The figure is a number line with a right parenthesis at negative 4 with shading to its left and a left parenthesis at 4 shading to its right. The values negative 6, 0, and 7 are marked with points. The absolute value of negative 6 is greater than negative 4 is true. It does not satisfy the absolute value of x is greater than 4. The absolute value of 0 is greater than 4 is false. It does not satisfy the absolute value of x is greater than 4. The absolute value of 7 is less than 4 is true. It does satisfy the absolute value of x is greater than 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with a right parenthesis at negative 4 with shading to its left and a left parenthesis at 4 shading to its right. The values negative 6, 0, and 7 are marked with points. The absolute value of negative 6 is greater than negative 4 is true. It does not satisfy the absolute value of x is greater than 4. The absolute value of 0 is greater than 4 is false. It does not satisfy the absolute value of x is greater than 4. The absolute value of 7 is less than 4 is true. It does satisfy the absolute value of x is greater than 4.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835229280\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835334792\"><div data-type=\"problem\" id=\"fs-id1167835334794\"><p id=\"fs-id1167831239002\">Solve \\(|x|&gt;2.\\) Graph the solution and write the solution in interval notation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835311019\"><span data-type=\"media\" id=\"fs-id1167835410922\" data-alt=\"The solution is x is less than negative 2 or x is greater than 2. The number line shows an open circle at negative 2 with shading to its left and an open circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 2 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 2 or x is greater than 2. The number line shows an open circle at negative 2 with shading to its left and an open circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 2 to infinity within parentheses.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832006300\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167835343327\">Solve \\(|x|&gt;1.\\) Graph the solution and write the solution in interval notation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834535359\"><span data-type=\"media\" id=\"fs-id1167835334127\" data-alt=\"The solution is x is less than negative 1 or x is greater than 1. The number line shows an open circle at negative 1 with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to negative 1 within parentheses and 1 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 1 or x is greater than 1. The number line shows an open circle at negative 1 with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to negative 1 within parentheses and 1 to infinity within parentheses.\"><\/span><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167835530188\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832058615\"><div data-type=\"problem\" id=\"fs-id1167832058617\"><p id=\"fs-id1167835343965\">Solve \\(|2x-3|\\ge 5.\\) Graph the solution and write the solution in interval notation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832043290\"><table id=\"fs-id1167834228216\" class=\"unnumbered unstyled\" summary=\"The absolute value of the quantity 2 x minus 3 is greater than or equal to 5. Step 1 is to isolate the absolute value expression. It is isolated. Step 2 is to write the equivalent compound inequality. It is 2 x minus 3 is less than or equal to negative 5 or 2 x minus 3 is greater than or equal to 5. Step 3 is to solve the compound inequality. 2 x is less than or equal to negative 2 or 2 x is greater than or equal to 8. x is less than or equal to negative 1 or x is greater than or equal to 4. Step 4 is to graph the solution. On the number line, there is a closed circle at negative 1 with shading to its left and a closed circle at 4 with shading to its right. Step 5 is to write the solution using interval notation. It is the union of negative infinity to negative 1 with a parenthesis and a bracket and 4 to infinity within a bracket and a parenthesis.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(|2x-3|\\ge 5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Isolate the absolute value expression. It is isolated.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Write the equivalent compound inequality.<\/td><td data-valign=\"top\" data-align=\"center\">\\(2x-3\\le -5\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}2x-3\\ge 5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Solve the compound inequality.<\/td><td data-valign=\"top\" data-align=\"center\">\\(2x\\le \\text{\u2212}2\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}2x\\ge 8\\)<div data-type=\"newline\"><br><\/div>\\(x\\le \\text{\u2212}1\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}x\\ge 4\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4.<\/strong> Graph the solution.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835419666\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5.<\/strong> Write the solution using interval notation.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(\\text{\u2212}\\infty ,-1\\right]\\cup \\left[4,\\infty \\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div>The check is left to you.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830704068\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830704071\"><div data-type=\"problem\" id=\"fs-id1167826996782\"><p id=\"fs-id1167826996784\">Solve \\(|4x-3|\\ge 5.\\) Graph the solution and write the solution in interval notation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834185415\"><span data-type=\"media\" data-alt=\"The solution is x is less than or equal to negative one-half or x is greater than or equal 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to negative one-half or x is greater than or equal 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis\"><\/span><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834294495\"><div data-type=\"problem\" id=\"fs-id1167834294497\"><p id=\"fs-id1167834294499\">Solve \\(|3x-4|\\ge 2.\\) Graph the solution and write the solution in interval notation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834408311\"><span data-type=\"media\" id=\"fs-id1167834408314\" data-alt=\"The solution is x is less than or equal to two-thirds or x is greater than or equal 2. The number line shows a closed circle at two-thirds with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to two-thirds within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to two-thirds or x is greater than or equal 2. The number line shows a closed circle at two-thirds with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to two-thirds within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830963972\" class=\"howto\"><div data-type=\"title\">Solve absolute value inequalities with &gt; or \u2265.<\/div><ol id=\"fs-id1167835355451\" type=\"1\" class=\"stepwise\"><li>Isolate the absolute value expression.<\/li><li>Write the equivalent compound inequality.<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167835395542\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}|u|&gt;a\\phantom{\\rule{2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{2em}{0ex}}u&lt;-a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u&gt;a\\hfill \\\\ |u|\\ge a\\phantom{\\rule{2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{2em}{0ex}}u\\le \\text{\u2212}a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u\\ge a\\hfill \\end{array}\\)<\/div><\/li><li>Solve the compound inequality.<\/li><li>Graph the solution<\/li><li>Write the solution using interval notation.<\/li><\/ol><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167826783859\"><h3 data-type=\"title\">Solve Applications with Absolute Value<\/h3><p id=\"fs-id1167831086715\">Absolute value inequalities are often used in the manufacturing process. An item must be made with near perfect specifications. Usually there is a certain <em data-effect=\"italics\">tolerance<\/em> of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the item is rejected.<\/p><div data-type=\"equation\" id=\"fs-id1167834300306\" class=\"unnumbered\" data-label=\"\">\\(|\\text{actual-ideal}|\\le \\text{tolerance}\\)<\/div><div data-type=\"example\" id=\"fs-id1167835419283\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835419285\"><div data-type=\"problem\" id=\"fs-id1167831933994\"><p id=\"fs-id1167831933996\">The ideal diameter of a rod needed for a machine is 60 mm. The actual diameter can vary from the ideal diameter by \\(0.075\\) mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832055375\"><div data-type=\"equation\" id=\"fs-id1167835341915\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\text{Let}\\phantom{\\rule{0.2em}{0ex}}x=\\text{the actual measurement.}\\hfill \\\\ \\text{Use an absolute value inequality to express this situation.}\\hfill &amp; &amp; &amp; &amp; &amp; |\\text{actual-ideal}|\\le \\text{tolerance}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill |x-60|\\le 0.075\\hfill \\\\ \\text{Rewrite as a compound inequality.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\text{\u2212}0.075\\le x-60\\le 0.075\\hfill \\\\ \\text{Solve the inequality.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill 59.925\\le x\\le 60.075\\hfill \\\\ \\text{Answer the question.}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{The diameter of the rod can be between}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\text{59.925 mm and 60.075 mm.}\\hfill \\end{array}\\)<\/div><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835349645\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835306096\"><div data-type=\"problem\" id=\"fs-id1167835306098\"><p id=\"fs-id1167835306100\">The ideal diameter of a rod needed for a machine is 80 mm. The actual diameter can vary from the ideal diameter by 0.009 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835173711\"><p id=\"fs-id1167834395851\">The diameter of the rod can be between 79.991 and 80.009 mm.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835370683\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835370686\"><div data-type=\"problem\" id=\"fs-id1167835370688\"><p id=\"fs-id1167831871802\">The ideal diameter of a rod needed for a machine is 75 mm. The actual diameter can vary from the ideal diameter by 0.05 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835421096\"><p id=\"fs-id1167835421098\">The diameter of the rod can be between 74.95 and 75.05 mm.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835380767\" class=\"media-2\"><p id=\"fs-id1167835512794\">Access this online resource for additional instruction and practice with solving linear absolute value equations and inequalities.<\/p><ul id=\"fs-id1167835512799\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37solvlinabsol\">Solving Linear Absolute Value Equations and Inequalities<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167834111788\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167835244451\" data-bullet-style=\"bullet\"><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\">Absolute Value Equations<\/strong><div data-type=\"newline\"><br><\/div> For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{ccccc}\\text{if}\\hfill &amp; &amp; &amp; &amp; |u|=a\\hfill \\\\ \\text{then}\\hfill &amp; &amp; &amp; &amp; \\phantom{\\rule{0.3em}{0ex}}u=\\text{\u2212}a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u=a\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div> Remember that an absolute value cannot be a negative number.<\/li><li><strong data-effect=\"bold\">How to Solve Absolute Value Equations<\/strong><ol id=\"fs-id1167832056980\" type=\"1\" class=\"stepwise\"><li>Isolate the absolute value expression.<\/li><li>Write the equivalent equations.<\/li><li>Solve each equation.<\/li><li>Check each solution.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Equations with Two Absolute Values<\/strong><div data-type=\"newline\"><br><\/div> For any algebraic expressions, <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em>,<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{ccccc}\\text{if}\\hfill &amp; &amp; &amp; &amp; |u|=|v|\\hfill \\\\ \\text{then}\\hfill &amp; &amp; &amp; &amp; \\phantom{\\rule{0.3em}{0ex}}u=\\text{\u2212}v\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u=v\\hfill \\end{array}\\)<\/li><li><strong data-effect=\"bold\">Absolute Value Inequalities with<\/strong>\\(&lt;\\) or \\(\\le \\)<div data-type=\"newline\"><br><\/div> For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{ccccccccccc}\\text{if}\\hfill &amp; &amp; &amp; &amp; &amp; |u|&lt;a,\\hfill &amp; &amp; &amp; &amp; &amp; \\text{then}\\phantom{\\rule{0.5em}{0ex}}\\text{\u2212}a&lt;u&lt;a\\hfill \\\\ \\text{if}\\hfill &amp; &amp; &amp; &amp; &amp; |u|\\le a,\\hfill &amp; &amp; &amp; &amp; &amp; \\text{then}\\phantom{\\rule{0.5em}{0ex}}\\text{\u2212}a\\le u\\le a\\hfill \\end{array}\\)<\/li><li><strong data-effect=\"bold\">How To Solve Absolute Value Inequalities with<\/strong>\\(&lt;\\) or \\(\\le \\) <ol id=\"fs-id1167835358422\" type=\"1\" class=\"stepwise\"><li>Isolate the absolute value expression.<\/li><li>Write the equivalent compound inequality.<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{ccccccccccc}\\hfill |u|&lt;a\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\text{is equivalent to}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\text{\u2212}a&lt;u&lt;a\\hfill \\\\ \\hfill |u|\\le a\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\text{is equivalent to}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\text{\u2212}a\\le u\\le a\\hfill \\end{array}\\)<\/li><li>Solve the compound inequality.<\/li><li>Graph the solution<\/li><li>Write the solution using interval notation<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Absolute Value Inequalities with<\/strong>\\(&gt;\\) or \\(\\ge \\)<div data-type=\"newline\"><br><\/div> For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{ccccccccccc}\\text{if}\\hfill &amp; &amp; &amp; &amp; &amp; |u|&gt;a,\\hfill &amp; &amp; &amp; &amp; &amp; \\text{then}\\phantom{\\rule{0.2em}{0ex}}u&lt;\\text{\u2212}a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u&gt;a\\hfill \\\\ \\text{if}\\hfill &amp; &amp; &amp; &amp; &amp; |u|\\ge a,\\hfill &amp; &amp; &amp; &amp; &amp; \\text{then}\\phantom{\\rule{0.2em}{0ex}}u\\le \\text{\u2212}a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u\\ge a\\hfill \\end{array}\\)<\/li><li><strong data-effect=\"bold\">How To Solve Absolute Value Inequalities with<\/strong>\\(&gt;\\) or \\(\\ge \\) <ol id=\"fs-id1167835262429\" type=\"1\" class=\"stepwise\"><li>Isolate the absolute value expression.<\/li><li>Write the equivalent compound inequality.<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{ccccccccc}\\hfill |u|&gt;a\\hfill &amp; &amp; &amp; &amp; \\hfill \\text{is equivalent to}\\hfill &amp; &amp; &amp; &amp; \\hfill u&lt;\\text{\u2212}a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u&gt;a\\hfill \\\\ \\hfill |u|\\ge a\\hfill &amp; &amp; &amp; &amp; \\hfill \\text{is equivalent to}\\hfill &amp; &amp; &amp; &amp; \\hfill u\\le \\text{\u2212}a\\phantom{\\rule{0.5em}{0ex}}\\text{or}\\phantom{\\rule{0.5em}{0ex}}u\\ge a\\hfill \\end{array}\\)<\/li><li>Solve the compound inequality.<\/li><li>Graph the solution<\/li><li>Write the solution using interval notation<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831892771\"><h3 data-type=\"title\">Section Exercises<\/h3><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834426146\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167831823780\"><strong data-effect=\"bold\">Solve Absolute Value Equations<\/strong><\/p><p id=\"fs-id1167835241276\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167835241280\"><div data-type=\"problem\" id=\"fs-id1167835241282\"><p id=\"fs-id1167835417039\"><span class=\"token\">\u24d0<\/span>\\(|x|=6\\)<span class=\"token\">\u24d1<\/span>\\(|y|=-3\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(|z|=0\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835244537\"><div data-type=\"problem\" id=\"fs-id1167830964473\"><p id=\"fs-id1167830964475\"><span class=\"token\">\u24d0<\/span>\\(|x|=4\\)<span class=\"token\">\u24d1<\/span>\\(|y|=-5\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(|z|=0\\)<\/div><div data-type=\"solution\" id=\"fs-id1167826994174\"><p id=\"fs-id1167835339834\"><span class=\"token\">\u24d0<\/span>\\(x=4,x=-4\\)<span class=\"token\">\u24d1<\/span> no solution <span class=\"token\">\u24d2<\/span> \\(z=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832051939\"><div data-type=\"problem\" id=\"fs-id1167835287345\"><p id=\"fs-id1167835287347\"><span class=\"token\">\u24d0<\/span>\\(|x|=7\\)<span class=\"token\">\u24d1<\/span>\\(|y|=-11\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(|z|=0\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834308101\"><div data-type=\"problem\" id=\"fs-id1167834308103\"><p><span class=\"token\">\u24d0<\/span>\\(|x|=3\\)<span class=\"token\">\u24d1<\/span>\\(|y|=-1\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(|z|=0\\)<\/div><div data-type=\"solution\" id=\"fs-id1167831920204\"><p id=\"fs-id1167831920206\"><span class=\"token\">\u24d0<\/span>\\(x=3,x=-3\\)<span class=\"token\">\u24d1<\/span> no solution <span class=\"token\">\u24d2<\/span> \\(z=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835379168\"><div data-type=\"problem\" id=\"fs-id1167835379171\"><p id=\"fs-id1167835420231\">\\(|2x-3|-4=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826938162\"><div data-type=\"problem\" id=\"fs-id1167826938164\"><p id=\"fs-id1167826938166\">\\(|4x-1|-3=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834535598\"><p id=\"fs-id1167834535600\">\\(x=1,x=-\\frac{1}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834134944\"><div data-type=\"problem\" id=\"fs-id1167835422586\"><p id=\"fs-id1167835422589\">\\(|3x-4|+5=7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834227506\"><div data-type=\"problem\" id=\"fs-id1167832099513\"><p id=\"fs-id1167832099515\">\\(|4x+7|+2=5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831066058\"><p id=\"fs-id1167835357426\">\\(x=-1,x=-\\frac{5}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835355673\"><div data-type=\"problem\" id=\"fs-id1167835355675\"><p id=\"fs-id1167835355677\">\\(4|x-1|+2=10\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831896629\"><div data-type=\"problem\" id=\"fs-id1167835304820\"><p id=\"fs-id1167835304822\">\\(3|x-4|+2=11\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832031176\"><p id=\"fs-id1167832074584\">\\(x=7,x=1\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834179728\"><p id=\"fs-id1167826978921\">\\(3|4x-5|-4=11\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826874383\"><div data-type=\"problem\" id=\"fs-id1167832075587\"><p id=\"fs-id1167832075589\">\\(3|x+2|-5=4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835479692\"><p id=\"fs-id1167835479694\">\\(x=1,x=-5\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167831892748\">\\(-2|x-3|+8=-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835351901\"><div data-type=\"problem\"><p id=\"fs-id1167835351906\">\\(-3|x-4|+4=-5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832190069\"><p id=\"fs-id1167830694084\">\\(x=7,x=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834064105\"><div data-type=\"problem\" id=\"fs-id1167834064107\"><p id=\"fs-id1167832041919\">\\(|\\frac{3}{4}x-3|+7=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832075528\"><div data-type=\"problem\" id=\"fs-id1167832075530\"><p id=\"fs-id1167832075532\">\\(|\\frac{3}{5}x-2|+5=2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834279894\"><p id=\"fs-id1167834279896\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834279901\"><div data-type=\"problem\" id=\"fs-id1167834431365\"><p id=\"fs-id1167834431367\">\\(|\\frac{1}{2}x+5|+4=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835531744\"><div data-type=\"problem\" id=\"fs-id1167835531746\"><p id=\"fs-id1167835531748\">\\(|\\frac{1}{4}x+3|+3=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835420978\"><p id=\"fs-id1167835420980\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831883320\"><div data-type=\"problem\" id=\"fs-id1167831883322\"><p id=\"fs-id1167831883325\">\\(|3x-2|=|2x-3|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826987283\"><div data-type=\"problem\" id=\"fs-id1167834156944\"><p id=\"fs-id1167834156946\">\\(|4x+3|=|2x+1|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826788431\"><p id=\"fs-id1167826788434\">\\(x=-1,x=-\\frac{2}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830770225\"><div data-type=\"problem\" id=\"fs-id1167830770227\"><p>\\(|6x-5|=|2x+3|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834098374\"><div data-type=\"problem\" id=\"fs-id1167834098376\"><p id=\"fs-id1167834098378\">\\(|6-x|=|3-2x|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834534548\"><p id=\"fs-id1167834534550\">\\(x=-3,x=3\\)<\/p><\/div><\/div><p id=\"fs-id1167835312257\"><strong data-effect=\"bold\">Solve Absolute Value Inequalities with \u201cless than\u201d<\/strong><\/p><p id=\"fs-id1167835312263\">In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167834432808\"><div data-type=\"problem\" id=\"fs-id1167834432810\"><p id=\"fs-id1167834432812\">\\(|x|&lt;5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835417913\"><div data-type=\"problem\" id=\"fs-id1167835417915\"><p id=\"fs-id1167835417917\">\\(|x|&lt;1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826995902\"><span data-type=\"media\" id=\"fs-id1167828420273\" data-alt=\"The solution is negative 1 is less than x which is less than 1. The number line shows an open circle at negative 1, an open circle at 1, and shading between the circles. The interval notation is negative 1 to 1 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than 1. The number line shows an open circle at negative 1, an open circle at 1, and shading between the circles. The interval notation is negative 1 to 1 within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830697917\"><div data-type=\"problem\" id=\"fs-id1167830697919\"><p id=\"fs-id1167830697921\">\\(|x|\\le 8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831881549\"><div data-type=\"problem\" id=\"fs-id1167831881551\"><p id=\"fs-id1167831881553\">\\(|x|\\le 3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831970260\"><span data-type=\"media\" id=\"fs-id1167831970263\" data-alt=\"The solution is negative 3 is less than or equal to x which is less than or equal to 3. The number line shows a closed circle at negative 3, a closed circle at 3, and shading between the circles. The interval notation is negative 3 to 3 within brackets.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 3 is less than or equal to x which is less than or equal to 3. The number line shows a closed circle at negative 3, a closed circle at 3, and shading between the circles. The interval notation is negative 3 to 3 within brackets.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834430869\"><div data-type=\"problem\" id=\"fs-id1167834430871\"><p id=\"fs-id1167834430873\">\\(|3x-3|\\le 6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830961942\"><div data-type=\"problem\" id=\"fs-id1167830961944\"><p id=\"fs-id1167830961946\">\\(|2x-5|\\le 3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835304725\"><span data-type=\"media\" id=\"fs-id1167835304728\" data-alt=\"The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading between the circles. The interval notation is 1 to 4 within brackets.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading between the circles. The interval notation is 1 to 4 within brackets.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835489076\"><div data-type=\"problem\" id=\"fs-id1167835489078\"><p id=\"fs-id1167835489080\">\\(|2x+3|+5&lt;4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831920356\"><div data-type=\"problem\" id=\"fs-id1167831920358\"><p id=\"fs-id1167831920361\">\\(|3x-7|+3&lt;1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830865707\"><span data-type=\"media\" id=\"fs-id1167830865710\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832053707\"><div data-type=\"problem\" id=\"fs-id1167832053709\"><p id=\"fs-id1167832053711\">\\(|4x-3|&lt;1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831824995\"><div data-type=\"problem\" id=\"fs-id1167831824998\"><p id=\"fs-id1167831825000\">\\(|6x-5|&lt;7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835319170\"><span data-type=\"media\" id=\"fs-id1167835319173\" data-alt=\"The solution is negative one-third is less than x which is less than 2. The number line shows an open circle at negative one-half, an open circle at 2, and shading between the circles. The interval notation is negative one-third to 2 within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative one-third is less than x which is less than 2. The number line shows an open circle at negative one-half, an open circle at 2, and shading between the circles. The interval notation is negative one-third to 2 within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835327455\"><div data-type=\"problem\" id=\"fs-id1167835352659\"><p id=\"fs-id1167835352661\">\\(|x-4|\\le -1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831894282\"><div data-type=\"problem\" id=\"fs-id1167831894284\"><p id=\"fs-id1167831894286\">\\(|5x+1|\\le -2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835330461\"><span data-type=\"media\" id=\"fs-id1167835330464\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_320_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><\/span><\/div><\/div><p id=\"fs-id1167835498615\"><strong data-effect=\"bold\">Solve Absolute Value Inequalities with \u201cgreater than\u201d<\/strong><\/p><p id=\"fs-id1167835498622\">In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167834536113\"><div data-type=\"problem\" id=\"fs-id1167834536115\"><p id=\"fs-id1167834536117\">\\(|x|&gt;3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835589715\"><div data-type=\"problem\" id=\"fs-id1167835589717\"><p id=\"fs-id1167832015761\">\\(|x|&gt;6\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830703711\"><span data-type=\"media\" id=\"fs-id1167830703714\" data-alt=\"The solution is x is less than negative 6 or x is greater than 6. The number line shows an open circle at negative 6 with shading to its left and an open circle at 6 with shading to its right. The interval notation is the union of negative infinity to negative 6 within parentheses and 6 to infinity within parentheses\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_322_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 6 or x is greater than 6. The number line shows an open circle at negative 6 with shading to its left and an open circle at 6 with shading to its right. The interval notation is the union of negative infinity to negative 6 within parentheses and 6 to infinity within parentheses\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835235781\"><div data-type=\"problem\" id=\"fs-id1167835235783\"><p id=\"fs-id1167835235785\">\\(|x|\\ge 2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835479279\"><div data-type=\"problem\" id=\"fs-id1167835479282\"><p id=\"fs-id1167835479284\">\\(|x|\\ge 5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832058040\"><span data-type=\"media\" id=\"fs-id1167832058043\" data-alt=\"The solution is x is less than negative 5 or x is greater than 5. The number line shows an open circle at negative 5 with shading to its left and an open circle at 5 with shading to its right. The interval notation is the union of negative infinity to negative 5 within parentheses and 5 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_324_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 5 or x is greater than 5. The number line shows an open circle at negative 5 with shading to its left and an open circle at 5 with shading to its right. The interval notation is the union of negative infinity to negative 5 within parentheses and 5 to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826782509\"><div data-type=\"problem\" id=\"fs-id1167834300884\"><p id=\"fs-id1167834300886\">\\(|3x-8|&gt;\\text{\u2212}1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830693625\"><div data-type=\"problem\" id=\"fs-id1167830693627\"><p id=\"fs-id1167830693629\">\\(|x-5|&gt;\\text{\u2212}2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831148862\"><span data-type=\"media\" id=\"fs-id1167831148866\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_326_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835214240\"><div data-type=\"problem\" id=\"fs-id1167835214242\"><p id=\"fs-id1167835214244\">\\(|3x-2|&gt;4\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167832065991\">\\(|2x-1|&gt;5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828434603\"><span data-type=\"media\" id=\"fs-id1167828434606\" data-alt=\"The solution is x is less than negative 2 or x is greater than 3. The number line shows an open circle at negative 2 with shading to its left and an open circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 3 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_328_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 2 or x is greater than 3. The number line shows an open circle at negative 2 with shading to its left and an open circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 3 to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167828434610\"><p id=\"fs-id1167835375320\">\\(|x+3|\\ge 5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835301470\"><div data-type=\"problem\" id=\"fs-id1167834431326\"><p id=\"fs-id1167834431328\">\\(|x-7|\\ge 1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832087111\"><span data-type=\"media\" id=\"fs-id1167832087114\" data-alt=\"The solution is x is less than or equal to 6 or x is greater than or equal to 8. The number line shows a closed circle at 6 with shading to its left and a closed circle at 8 with shading to its right. The interval notation is the union of negative infinity to 6 within parenthesis and a bracket and 8 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_330_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to 6 or x is greater than or equal to 8. The number line shows a closed circle at 6 with shading to its left and a closed circle at 8 with shading to its right. The interval notation is the union of negative infinity to 6 within parenthesis and a bracket and 8 to infinity within a bracket and a parenthesis.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834377054\"><div data-type=\"problem\" id=\"fs-id1167834377056\"><p id=\"fs-id1167834377058\">\\(3|x|+4\\ge 1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832015454\"><div data-type=\"problem\" id=\"fs-id1167832015456\"><p id=\"fs-id1167832015458\">\\(5|x|+6\\ge 1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834531065\"><span data-type=\"media\" id=\"fs-id1167832074711\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_332_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><\/span><\/div><\/div><p id=\"fs-id1167834432597\">In the following exercises, solve. For each inequality, also graph the solution and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167834432602\"><div data-type=\"problem\" id=\"fs-id1167834432604\"><p id=\"fs-id1167834432606\">\\(2|x+6|+4=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832043932\"><div data-type=\"problem\" id=\"fs-id1167832043934\"><p id=\"fs-id1167835423440\">\\(|3x-4|\\ge 2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834534823\"><p>\\(x=4,x=\\frac{2}{7}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834192354\"><div data-type=\"problem\" id=\"fs-id1167834192356\"><p id=\"fs-id1167832043936\">\\(|6x-5|=|2x+3|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835375364\"><div data-type=\"problem\"><p id=\"fs-id1167830693540\">\\(|4x-3|&lt;5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831883636\"><p id=\"fs-id1167831883638\">\\(x=3,x=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835352698\"><div data-type=\"problem\" id=\"fs-id1167830693538\"><p id=\"fs-id1167835375368\">\\(|2x-5|+2=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834506132\"><div data-type=\"problem\" id=\"fs-id1167834506134\"><p id=\"fs-id1167834506136\">\\(|3x+1|-3=7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832153529\"><p id=\"fs-id1167832153531\">\\(x=3,x=-\\frac{11}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834120820\"><div data-type=\"problem\" id=\"fs-id1167834120822\"><p id=\"fs-id1167834120824\">\\(|7x+2|+8&lt;4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832067076\"><div data-type=\"problem\" id=\"fs-id1167832067078\"><p id=\"fs-id1167832067080\">\\(5|2x-1|-3=7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835303010\"><p id=\"fs-id1167835303012\">\\(x=\\frac{3}{2},x=-\\frac{1}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831911212\"><div data-type=\"problem\" id=\"fs-id1167831911214\"><p id=\"fs-id1167831884002\">\\(|x-7|&gt;\\text{\u2212}3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831883998\"><div data-type=\"problem\" id=\"fs-id1167831884000\"><p id=\"fs-id1167831911216\">\\(|8-x|=|4-3x|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835360752\"><span data-type=\"media\" id=\"fs-id1167835360755\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_336_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><\/span><\/div><\/div><p id=\"fs-id1167831919898\"><strong data-effect=\"bold\">Solve Applications with Absolute Value<\/strong><\/p><p id=\"fs-id1167831919904\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167830961506\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830961508\"><p id=\"fs-id1167830961510\">A chicken farm ideally produces 200,000 eggs per day. But this total can vary by as much as 25,000 eggs. What is the maximum and minimum expected production at the farm?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835534335\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835534337\"><p id=\"fs-id1167835534339\">An organic juice bottler ideally produces 215,000 bottle per day. But this total can vary by as much as 7,500 bottles. What is the maximum and minimum expected production at the bottling company?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835534345\"><p id=\"fs-id1167835287293\">The minimum to maximum expected production is 207,500 to 2,225,000 bottles<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835287299\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835287301\"><p id=\"fs-id1167835287303\">In order to insure compliance with the law, Miguel routinely overshoots the weight of his tortillas by 0.5 gram. He just received a report that told him that he could be losing as much as ?100,000 per year using this practice. He now plans to buy new equipment that guarantees the thickness of the tortilla within 0.005 inches. If the ideal thickness of the tortilla is 0.04 inches, what thickness of tortillas will be guaranteed?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835368288\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835368290\"><p id=\"fs-id1167835368292\">At Lilly\u2019s Bakery, the ideal weight of a loaf of bread is 24 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835217804\"><p id=\"fs-id1167835217806\">The acceptable weight is 22.5 to 25.5 ounces.<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835217813\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167835334267\"><div data-type=\"problem\" id=\"fs-id1167835334269\"><p id=\"fs-id1167835334271\">Write a graphical description of the absolute value of a number.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832153475\"><div data-type=\"problem\" id=\"fs-id1167832153477\"><p id=\"fs-id1167832153479\">In your own words, explain how to solve the absolute value inequality, \\(|3x-2|\\ge 4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835337602\"><p id=\"fs-id1167835337604\">Answers will vary.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835337611\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167834185038\"><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-id1167834185046\" data-alt=\"This table has four columns and five rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve absolute value equations. In row 3, the I can was solve absolute value inequalities with \u201cless than.\u201d In row 4, the I can was solve absolute value inequalities with \u201cgreater than.\u201d In row 5, the I can was solve applications with absolute value.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and five rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve absolute value equations. In row 3, the I can was solve absolute value inequalities with \u201cless than.\u201d In row 4, the I can was solve absolute value inequalities with \u201cgreater than.\u201d In row 5, the I can was solve applications with absolute value.\"><\/span><p id=\"fs-id1167826849544\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p><\/div><\/div><div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167834523724\"><h3 data-type=\"title\">Chapter Review Exercises<\/h3><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835350223\"><h4 data-type=\"title\"><a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c\" class=\"target-chapter\">Use a General Strategy to Solve Linear Equations<\/a><\/h4><p id=\"fs-id1167835350234\"><strong data-effect=\"bold\">Solve Equations Using the General Strategy for Solving Linear Equations<\/strong><\/p><p id=\"fs-id1167831837572\">In the following exercises, determine whether each number is a solution to the equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167831837576\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831837578\"><p id=\"fs-id1167831837580\">\\(10x-1=5x,x=\\frac{1}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835529828\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835529830\"><p id=\"fs-id1167835529832\">\\(-12n+5=8n,n=-\\frac{5}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831117374\"><p id=\"fs-id1167835339861\">no<\/p><\/div><\/div><p id=\"fs-id1167835339866\">In the following exercises, solve each linear equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167835339870\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835339872\"><p id=\"fs-id1167835339874\">\\(6\\left(x+6\\right)=24\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835515236\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835515239\"><p id=\"fs-id1167835515241\">\\(\\text{\u2212}\\left(s+4\\right)=18\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826874285\"><p id=\"fs-id1167826874287\">\\(s=-22\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835379630\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835379632\"><p id=\"fs-id1167835379634\">\\(23-3\\left(y-7\\right)=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826997377\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826997379\"><p id=\"fs-id1167826997381\">\\(\\frac{1}{3}\\left(6m+21\\right)=m-7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831025141\"><p id=\"fs-id1167831025143\">\\(m=-14\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834063690\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834063692\"><p id=\"fs-id1167834063694\">\\(4\\left(3.5y+0.25\\right)=365\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835354466\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835354468\"><p id=\"fs-id1167835354470\">\\(0.25\\left(q-8\\right)=0.1\\left(q+7\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830697964\"><p id=\"fs-id1167830697966\">\\(q=18\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834531012\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834531015\"><p id=\"fs-id1167834531017\">\\(8\\left(r-2\\right)=6\\left(r+10\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826851797\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835596119\"><p id=\"fs-id1167835596122\">\\(5+7\\left(2-5x\\right)=2\\left(9x+1\\right)-\\left(13x-57\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834183392\"><p id=\"fs-id1167834183394\">\\(x=-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834239022\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834239024\"><p id=\"fs-id1167834239026\">\\(\\left(9n+5\\right)-\\left(3n-7\\right)=20-\\left(4n-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835417519\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835417521\"><p id=\"fs-id1167835417523\">\\(2\\left[-16+5\\left(8k-6\\right)\\right]=8\\left(3-4k\\right)-32\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830925332\"><p id=\"fs-id1167830925334\">\\(k=\\frac{3}{4}\\)<\/p><\/div><\/div><p id=\"fs-id1167831832129\"><strong data-effect=\"bold\">Classify Equations<\/strong><\/p><p id=\"fs-id1167831832135\">In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.<\/p><div data-type=\"exercise\" id=\"fs-id1167831832139\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831832141\"><p id=\"fs-id1167831832144\">\\(17y-3\\left(4-2y\\right)=11\\left(y-1\\right)+12y-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835347858\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835347860\"><p id=\"fs-id1167835347862\">\\(9u+32=15\\left(u-4\\right)-3\\left(2u+21\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830838231\"><p id=\"fs-id1167834064014\">contradiction; no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834064019\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834064021\"><p id=\"fs-id1167834064023\">\\(-8\\left(7m+4\\right)=-6\\left(8m+9\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167834120288\"><strong data-effect=\"bold\">Solve Equations with Fraction or Decimal Coefficients<\/strong><\/p><p id=\"fs-id1167831839294\">In the following exercises, solve each equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167831839297\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831839299\"><p id=\"fs-id1167831839301\">\\(\\frac{2}{5}n-\\frac{1}{10}=\\frac{7}{10}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835510383\"><p id=\"fs-id1167835510385\">\\(n=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835513119\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835513121\"><p id=\"fs-id1167835513124\">\\(\\frac{3}{4}a-\\frac{1}{3}=\\frac{1}{2}a+\\frac{5}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835346782\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835346784\"><p id=\"fs-id1167835346786\">\\(\\frac{1}{2}\\left(k+3\\right)=\\frac{1}{3}\\left(k+16\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835187809\"><p id=\"fs-id1167835187811\">\\(k=23\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835355344\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835355346\"><p id=\"fs-id1167835355348\">\\(\\frac{5y-1}{3}+4=\\frac{-8y+4}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834065304\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834065306\"><p id=\"fs-id1167834065308\">\\(0.8x-0.3=0.7x+0.2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834209097\"><p id=\"fs-id1167834209100\">\\(x=5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834433422\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834433424\"><p id=\"fs-id1167834433426\">\\(0.10d+0.05\\left(d-4\\right)=2.05\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835267992\"><h4 data-type=\"title\"><a href=\"\/contents\/37489cba-b108-41fd-88b1-ab568fcea766\" class=\"target-chapter\">Use a Problem-Solving Strategy<\/a><\/h4><p id=\"fs-id1167835340717\"><strong data-effect=\"bold\">Use a Problem Solving Strategy for Word Problems<\/strong><\/p><p id=\"fs-id1167835340723\">In the following exercises, solve using the problem solving strategy for word problems.<\/p><div data-type=\"exercise\" id=\"fs-id1167835340726\"><div data-type=\"problem\" id=\"fs-id1167835340728\"><p id=\"fs-id1167835340730\">Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826849734\"><p id=\"fs-id1167826849736\">There are 116 people.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826849741\"><div data-type=\"problem\" id=\"fs-id1167826849744\"><p id=\"fs-id1167826849746\">There are nine saxophone players in the band. The number of saxophone players is one less than twice the number of tuba players. Find the number of tuba players.<\/p><\/div><\/div><p id=\"fs-id1167832138843\"><strong data-effect=\"bold\">Solve Number Word Problems<\/strong><\/p><p id=\"fs-id1167835319306\">In the following exercises, solve each number word problem.<\/p><div data-type=\"exercise\" id=\"fs-id1167835319310\"><div data-type=\"problem\" id=\"fs-id1167835319312\"><p id=\"fs-id1167835319314\">The sum of a number and three is forty-one. Find the number.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835319318\"><p id=\"fs-id1167835320424\">38<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835320429\"><div data-type=\"problem\" id=\"fs-id1167835320431\"><p id=\"fs-id1167835320433\">One number is nine less than another. Their sum is negative twenty-seven. Find the numbers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835180579\"><div data-type=\"problem\" id=\"fs-id1167835180581\"><p id=\"fs-id1167831186040\">One number is two more than four times another. Their sum is negative thirteen. Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831186044\"><p id=\"fs-id1167831186047\">\\(-3,-10\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834131070\"><div data-type=\"problem\" id=\"fs-id1167834131073\"><p id=\"fs-id1167834131075\">The sum of two consecutive integers is \\(-135.\\) Find the numbers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831892866\"><div data-type=\"problem\" id=\"fs-id1167831892868\"><p id=\"fs-id1167831892870\">Find three consecutive even integers whose sum is 234.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831892874\"><p id=\"fs-id1167835410247\">76, 78, 80<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835410252\"><div data-type=\"problem\" id=\"fs-id1167835410254\"><p id=\"fs-id1167835410257\">Find three consecutive odd integers whose sum is 51.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835623245\"><div data-type=\"problem\" id=\"fs-id1167835623247\"><p id=\"fs-id1167835623249\">Koji has ?5,502 in his savings account. This is ?30 less than six times the amount in his checking account. How much money does Koji have in his checking account?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832053195\"><p id=\"fs-id1167832053197\">?922<\/p><\/div><\/div><p id=\"fs-id1167832053202\"><strong data-effect=\"bold\">Solve Percent Applications<\/strong><\/p><p id=\"fs-id1167832053207\">In the following exercises, translate and solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167834324692\"><div data-type=\"problem\" id=\"fs-id1167834324694\"><p id=\"fs-id1167834324696\">What number is 67% of 250?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835216540\"><div data-type=\"problem\" id=\"fs-id1167835216543\"><p id=\"fs-id1167835216545\">12.5% of what number is 20?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835216555\"><p id=\"fs-id1167834536130\">160<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834536135\"><div data-type=\"problem\" id=\"fs-id1167834536137\"><p id=\"fs-id1167834536139\">What percent of 125 is 150?<\/p><\/div><\/div><p id=\"fs-id1167834060137\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167834060140\"><div data-type=\"problem\" id=\"fs-id1167834060142\"><p id=\"fs-id1167834060145\">The bill for Dino\u2019s lunch was ?19.45. He wanted to leave 20% of the total bill as a tip. How much should the tip be?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830962261\"><p id=\"fs-id1167830962263\">\\(\\text{?}3.89\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835339243\"><div data-type=\"problem\" id=\"fs-id1167835339245\"><p id=\"fs-id1167835339247\">Dolores bought a crib on sale for ?350. The sale price was 40% of the original price. What was the original price of the crib?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830898712\"><div data-type=\"problem\" id=\"fs-id1167830898714\"><p id=\"fs-id1167830898716\">Jaden earns ?2,680 per month. He pays ?938 a month for rent. What percent of his monthly pay goes to rent?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830898722\"><p id=\"fs-id1167830898724\">35%<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827966896\"><div data-type=\"problem\" id=\"fs-id1167827966898\"><p id=\"fs-id1167827966900\">Angel received a raise in his annual salary from ?55,400 to ?56,785. Find the percent change.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835511627\"><div data-type=\"problem\" id=\"fs-id1167835511630\"><p id=\"fs-id1167835511632\">Rowena\u2019s monthly gasoline bill dropped from ?83.75 last month to ?56.95 this month. Find the percent change.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834403054\"><p id=\"fs-id1167834403057\">32%<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834403062\"><div data-type=\"problem\" id=\"fs-id1167834403064\"><p id=\"fs-id1167835379835\">Emmett bought a pair of shoes on sale at 40% off from an original price of ?138. Find <span class=\"token\">\u24d0<\/span> the amount of discount and <span class=\"token\">\u24d1<\/span> the sale price.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835357750\"><div data-type=\"problem\" id=\"fs-id1167835357752\"><p id=\"fs-id1167835357754\">Lacey bought a pair of boots on sale for ?95. The original price of the boots was ?200. Find <span class=\"token\">\u24d0<\/span> the amount of discount and <span class=\"token\">\u24d1<\/span> the discount rate. (Round to the nearest tenth of a percent, if needed.)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834182448\"><p id=\"fs-id1167834182450\"><span class=\"token\">\u24d0<\/span> ?105 <span class=\"token\">\u24d1<\/span> \\(52.5\\text{%}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832115942\"><div data-type=\"problem\" id=\"fs-id1167832115944\"><p id=\"fs-id1167832115946\">Nga and Lauren bought a chest at a flea market for ?50. They re-finished it and then added a 350% mark-up. Find <span class=\"token\">\u24d0<\/span> the amount of the mark-up and <span class=\"token\">\u24d1<\/span> the list price.<\/p><\/div><\/div><p id=\"fs-id1167826802339\"><strong data-effect=\"bold\">Solve Simple Interest Applications<\/strong><em data-effect=\"italics\"><\/em><\/p><p id=\"fs-id1167826802346\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167826802349\"><div data-type=\"problem\" id=\"fs-id1167826802352\"><p id=\"fs-id1167834464242\">Winston deposited ?3,294 in a bank account with interest rate 2.6% How much interest was earned in five years?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834464247\"><p id=\"fs-id1167834464249\">?428.22<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834464255\"><div data-type=\"problem\" id=\"fs-id1167834464257\"><p id=\"fs-id1167835358219\">Moira borrowed ?4,500 from her grandfather to pay for her first year of college. Three years later, she repaid the ?4,500 plus ?243 interest. What was the rate of interest?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834061700\"><div data-type=\"problem\" id=\"fs-id1167834061702\"><p id=\"fs-id1167834061704\">Jaime\u2019s refrigerator loan statement said he would pay ?1,026 in interest for a four-year loan at 13.5%. How much did Jaime borrow to buy the refrigerator?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835321537\"><p id=\"fs-id1167835321539\">?1,900<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835321545\"><h4 data-type=\"title\"><a href=\"\/contents\/b03538a1-8a7b-4158-a68b-e0e8a24c9fd4\" class=\"target-chapter\">Solve a formula for a Specific Variable<\/a><\/h4><p id=\"fs-id1167835333094\"><strong data-effect=\"bold\">Solve a Formula for a Specific Variable<\/strong><\/p><p id=\"fs-id1167835333100\">In the following exercises, solve the formula for the specified variable.<\/p><div data-type=\"exercise\" id=\"fs-id1167835333103\"><div data-type=\"problem\" id=\"fs-id1167835333105\"><p id=\"fs-id1167834111808\">Solve the formula<\/p><div data-type=\"newline\"><br><\/div>\\(V=LWH\\) for <em data-effect=\"italics\">L<\/em>.<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835509163\"><div data-type=\"problem\" id=\"fs-id1167828396222\"><p id=\"fs-id1167828396224\">Solve the formula<\/p><div data-type=\"newline\"><br><\/div>\\(A=\\frac{1}{2}{d}_{1}{d}_{2}\\) for \\({d}_{2}.\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835341255\"><p id=\"fs-id1167835341258\">\\({d}_{2}=\\frac{2A}{{d}_{1}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826993960\"><div data-type=\"problem\" id=\"fs-id1167826993962\"><p id=\"fs-id1167826993964\">Solve the formula<\/p><div data-type=\"newline\"><br><\/div>\\(h=48t+\\frac{1}{2}a{t}^{2}\\) for <em data-effect=\"italics\">t<\/em>.<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834225907\"><div data-type=\"problem\" id=\"fs-id1167834225909\"><p id=\"fs-id1167835595718\">Solve the formula<\/p><div data-type=\"newline\"><br><\/div>\\(4x-3y=12\\) for <em data-effect=\"italics\">y<\/em>.<\/div><div data-type=\"solution\" id=\"fs-id1167826967385\"><p id=\"fs-id1167826967387\">\\(y=\\frac{4x}{3}-4\\)<\/p><\/div><\/div><p id=\"fs-id1167835489285\"><strong data-effect=\"bold\">Use Formulas to Solve Geometry Applications<\/strong><\/p><p id=\"fs-id1167835489291\">In the following exercises, solve using a geometry formula.<\/p><div data-type=\"exercise\" id=\"fs-id1167834516380\"><div data-type=\"problem\" id=\"fs-id1167834516382\"><p id=\"fs-id1167834516385\">What is the height of a triangle with area \\(67.5\\) square meters and base 9 meters?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834376462\"><div data-type=\"problem\" id=\"fs-id1167834376464\"><p id=\"fs-id1167834376466\">The measure of the smallest angle in a right triangle is \\(45\\text{\u00b0}\\) less than the measure of the next larger angle. Find the measures of all three angles.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831911722\"><p id=\"fs-id1167831911724\">\\(22.5\\text{\u00b0},\\phantom{\\rule{0.2em}{0ex}}67.5\\text{\u00b0},90\\text{\u00b0}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p>The perimeter of a triangle is 97 feet. One side of the triangle is eleven feet more than the smallest side. The third side is six feet more than twice the smallest side. Find the lengths of all sides.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831086741\"><div data-type=\"problem\" id=\"fs-id1167831086743\"><p id=\"fs-id1167831086746\">Find the length of the hypotenuse.<\/p><span data-type=\"media\" id=\"fs-id1167831086749\" data-alt=\"The figure is a right triangle with a base of 10 units and a height of 24 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a right triangle with a base of 10 units and a height of 24 units.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167834438989\"><p id=\"fs-id1167834438991\">26<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831888145\"><div data-type=\"problem\" id=\"fs-id1167831888147\"><p id=\"fs-id1167831888149\">Find the length of the missing side. Round to the nearest tenth, if necessary.<\/p><span data-type=\"media\" id=\"fs-id1167831888152\" data-alt=\"The figure is a right triangle with a height of 15 units and a hypotenuse of 17 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a right triangle with a height of 15 units and a hypotenuse of 17 units.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835346679\"><div data-type=\"problem\" id=\"fs-id1167835346681\"><p id=\"fs-id1167834222110\">Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is eight feet tall and Sergio has 10 feet of wire. How far from the base of the antenna can he attach the wire? Approximate to the nearest tenth, if necessary.<\/p><span data-type=\"media\" id=\"fs-id1167834222116\" data-alt=\"The figure is a right triangle with a height of 8 feet and a hypotenuse of 10 feet.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a right triangle with a height of 8 feet and a hypotenuse of 10 feet.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835370795\"><p id=\"fs-id1167835370797\">6 feet<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835370803\"><div data-type=\"problem\" id=\"fs-id1167835370805\"><p id=\"fs-id1167835370807\">Seong is building shelving in his garage. The shelves are 36 inches wide and 15 inches tall. He wants to put a diagonal brace across the back to stabilize the shelves, as shown. How long should the brace be?<\/p><span data-type=\"media\" id=\"fs-id1167834512816\" data-alt=\"The figure illustrates rectangular shelving whose width of 36 inch and height of 15 inches forms a right triangle with a diagonal brace.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure illustrates rectangular shelving whose width of 36 inch and height of 15 inches forms a right triangle with a diagonal brace.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831921595\"><div data-type=\"problem\" id=\"fs-id1167831921597\"><p id=\"fs-id1167831921599\">The length of a rectangle is 12 cm more than the width. The perimeter is 74 cm. Find the length and the width.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835509936\"><p id=\"fs-id1167835509938\">\\(24.5\\) cm, \\(12.5\\) cm<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835596592\"><div data-type=\"problem\" id=\"fs-id1167835596594\"><p id=\"fs-id1167835596596\">The width of a rectangle is three more than twice the length. The perimeter is 96 inches. Find the length and the width.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834502576\"><div data-type=\"problem\" id=\"fs-id1167834502578\"><p id=\"fs-id1167834502580\">The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834502586\"><p id=\"fs-id1167834152127\">9 ft, 14 ft, 12 ft<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834152134\"><h4 data-type=\"title\"><a href=\"\/contents\/36adea73-2201-46d3-b9b6-d13ef7df78b2\" class=\"target-chapter\">Solve Mixture and Uniform Motion Applications<\/a><\/h4><p id=\"fs-id1167826995576\"><strong data-effect=\"bold\">Solve Coin Word Problems<\/strong><\/p><p id=\"fs-id1167826995582\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167826995585\"><div data-type=\"problem\" id=\"fs-id1167826995587\"><p id=\"fs-id1167830703667\">Paulette has ?140 in ?5 and ?10 bills. The number of ?10 bills is one less than twice the number of ?5 bills. How many of each does she have?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830703680\"><div data-type=\"problem\" id=\"fs-id1167830703682\"><p id=\"fs-id1167835375493\">Lenny has ?3.69 in pennies, dimes, and quarters. The number of pennies is three more than the number of dimes. The number of quarters is twice the number of dimes. How many of each coin does he have?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835375507\"><p id=\"fs-id1167835333154\">nine pennies, six dimes, 12 quarters<\/p><\/div><\/div><p id=\"fs-id1167835333159\"><strong data-effect=\"bold\">Solve Ticket and Stamp Word Problems<\/strong><\/p><p id=\"fs-id1167835333166\">In the following exercises, solve each ticket or stamp word problem.<\/p><div data-type=\"exercise\" id=\"fs-id1167832150071\"><div data-type=\"problem\" id=\"fs-id1167832150073\"><p id=\"fs-id1167832150076\">Tickets for a basketball game cost ?2 for students and ?5 for adults. The number of students was three less than 10 times the number of adults. The total amount of money from ticket sales was ?619. How many of each ticket were sold?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834372263\"><div data-type=\"problem\" id=\"fs-id1167834372265\"><p id=\"fs-id1167834372267\">125 tickets were sold for the jazz band concert for a total of ?1,022. Student tickets cost ?6 each and general admission tickets cost ?10 each. How many of each kind of ticket were sold?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834372273\"><p id=\"fs-id1167834372275\">57 students, 68 adults<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830699855\"><div data-type=\"problem\" id=\"fs-id1167830699857\"><p id=\"fs-id1167830699859\">Yumi spent ?34.15 buying stamps. The number of ?0.56 stamps she bought was 10 less than four times the number of ?0.41 stamps. How many of each did she buy?<\/p><\/div><\/div><p id=\"fs-id1167831031227\"><strong data-effect=\"bold\">Solve Mixture Word Problems<\/strong><\/p><p id=\"fs-id1167826986123\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167826986126\"><div data-type=\"problem\" id=\"fs-id1167826986128\"><p id=\"fs-id1167826986130\">Marquese is making 10 pounds of trail mix from raisins and nuts. Raisins cost ?3.45 per pound and nuts cost ?7.95 per pound. How many pounds of raisins and how many pounds of nuts should Marquese use for the trail mix to cost him ?6.96 per pound?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832015658\"><p id=\"fs-id1167832015660\">\\(2.2\\) lbs of raisins, \\(7.8\\) lbs of nuts<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835417630\"><div data-type=\"problem\" id=\"fs-id1167835417632\"><p id=\"fs-id1167835417635\">Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tile. She will use basic tiles that cost ?8 per square foot and decorator tiles that cost ?20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be ?10 per square foot?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834301117\"><div data-type=\"problem\" id=\"fs-id1167834301120\"><p id=\"fs-id1167834301122\">Enrique borrowed ?23,500 to buy a car. He pays his uncle 2% interest on the ?4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total ?23,500? (Round your answer to the nearest tenth of a percent.)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835338550\"><p id=\"fs-id1167835338552\">\\(9.7\\text{%}\\)<\/p><\/div><\/div><p id=\"fs-id1167834058806\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p><p id=\"fs-id1167834058812\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167835357470\"><div data-type=\"problem\" id=\"fs-id1167835357472\"><p id=\"fs-id1167835357474\">When Gabe drives from Sacramento to Redding it takes him 2.2 hours. It takes Elsa two hours to drive the same distance. Elsa\u2019s speed is seven miles per hour faster than Gabe\u2019s speed. Find Gabe\u2019s speed and Elsa\u2019s speed.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835331664\"><div data-type=\"problem\" id=\"fs-id1167835331666\"><p id=\"fs-id1167835331668\">Louellen and Tracy met at a restaurant on the road between Chicago and Nashville. Louellen had left Chicago and drove 3.2 hours towards Nashville. Tracy had left Nashville and drove 4 hours towards Chicago, at a speed one mile per hour faster than Louellen\u2019s speed. The distance between Chicago and Nashville is 472 miles. Find Louellen\u2019s speed and Tracy\u2019s speed.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828421648\"><p id=\"fs-id1167828421650\">Louellen 65 mph, Tracy 66 mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832134182\"><div data-type=\"problem\" id=\"fs-id1167832134185\"><p id=\"fs-id1167832134187\">Two busses leave Amarillo at the same time. The Albuquerque bus heads west on the I-40 at a speed of 72 miles per hour, and the Oklahoma City bus heads east on the I-40 at a speed of 78 miles per hour. How many hours will it take them to be 375 miles apart?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835301341\"><div data-type=\"problem\" id=\"fs-id1167835301344\"><p id=\"fs-id1167835301346\">Kyle rowed his boat upstream for 50 minutes. It took him 30 minutes to row back downstream. His speed going upstream is two miles per hour slower than his speed going downstream. Find Kyle\u2019s upstream and downstream speeds.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835301353\"><p id=\"fs-id1167831970367\">upstream 3 mph, downstream 5 mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831970372\"><div data-type=\"problem\" id=\"fs-id1167831970374\"><p id=\"fs-id1167831970377\">At 6:30, Devon left her house and rode her bike on the flat road until 7:30. Then she started riding uphill and rode until 8:00. She rode a total of 15 miles. Her speed on the flat road was three miles per hour faster than her speed going uphill. Find Devon\u2019s speed on the flat road and riding uphill.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831892888\"><div data-type=\"problem\" id=\"fs-id1167831892890\"><p id=\"fs-id1167831892892\">Anthony drove from New York City to Baltimore, which is a distance of 192 miles. He left at 3:45 and had heavy traffic until 5:30. Traffic was light for the rest of the drive, and he arrived at 7:30. His speed in light traffic was four miles per hour more than twice his speed in heavy traffic. Find Anthony\u2019s driving speed in heavy traffic and light traffic.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834132162\"><p id=\"fs-id1167834132164\">heavy traffic 32 mph, light traffic 66 mph<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834526168\"><h4 data-type=\"title\"><a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7\" class=\"target-chapter\">Solve Linear Inequalities<\/a><\/h4><p id=\"fs-id1167834526178\"><strong data-effect=\"bold\">Graph Inequalities on the Number Line<\/strong><\/p><p id=\"fs-id1167831893359\">In the following exercises, graph the inequality on the number line and write in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167831893362\"><div data-type=\"problem\" id=\"fs-id1167831893365\"><p id=\"fs-id1167831893367\">\\(x&lt;\\text{\u2212}1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834227427\"><div data-type=\"problem\" id=\"fs-id1167834227429\"><p id=\"fs-id1167834227431\">\\(x\\ge -2.5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834225921\"><span data-type=\"media\" id=\"fs-id1167834225924\" data-alt=\"The solution is x is greater than or equal to negative 2.5. The number line shows a left bracket at negative 2.5 with shading to its right. The interval notation is negative 2.5 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_338_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than or equal to negative 2.5. The number line shows a left bracket at negative 2.5 with shading to its right. The interval notation is negative 2.5 to infinity within a bracket and a parenthesis.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834084430\"><div data-type=\"problem\" id=\"fs-id1167834084432\"><p id=\"fs-id1167834395944\">\\(x\\le \\frac{5}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835595469\"><div data-type=\"problem\" id=\"fs-id1167835595471\"><p id=\"fs-id1167835595473\">\\(x&gt;2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835381796\"><span data-type=\"media\" id=\"fs-id1167835381800\" data-alt=\"The solution is x is greater than 2. The number line shows a left parenthesis at 2 with shading to its right. The interval notation is 2 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than 2. The number line shows a left parenthesis at 2 with shading to its right. The interval notation is 2 to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834340090\"><div data-type=\"problem\" id=\"fs-id1167834340092\"><p id=\"fs-id1167834340094\">\\(-2&lt;x&lt;0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831239513\"><div data-type=\"problem\" id=\"fs-id1167831239515\"><p id=\"fs-id1167831239517\">\\(-5\\le x&lt;\\text{\u2212}3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830704579\"><span data-type=\"media\" id=\"fs-id1167830704582\" data-alt=\"The solution is negative 5 is less than or equal to x which is less than negative 3. The number line shows a closed circle at negative 5, an open circle at negative 3, and shading between the circles. The interval notation is negative 5 to negative 3 within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_342_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 5 is less than or equal to x which is less than negative 3. The number line shows a closed circle at negative 5, an open circle at negative 3, and shading between the circles. The interval notation is negative 5 to negative 3 within a bracket and a parenthesis.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835358365\"><div data-type=\"problem\" id=\"fs-id1167835358367\"><p id=\"fs-id1167835358369\">\\(0\\le x\\le 3.5\\)<\/p><\/div><\/div><p id=\"fs-id1167831880112\"><strong data-effect=\"bold\">Solve Linear Inequalities<\/strong><\/p><p id=\"fs-id1167831880118\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167830699525\"><div data-type=\"problem\" id=\"fs-id1167830699528\"><p id=\"fs-id1167830699530\">\\(n-12\\le 23\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830868612\"><span data-type=\"media\" id=\"fs-id1167830868615\" data-alt=\"The solution is n is less than or equal to 35. The number line shows a a right bracket at 35 with shading to its left. The interval notation is negative infinity to 35 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_344_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is n is less than or equal to 35. The number line shows a a right bracket at 35 with shading to its left. The interval notation is negative infinity to 35 within a parenthesis and a bracket.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827943826\"><div data-type=\"problem\" id=\"fs-id1167827943828\"><p id=\"fs-id1167827943830\">\\(a+\\frac{2}{3}\\ge \\frac{7}{12}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834186134\"><div data-type=\"problem\" id=\"fs-id1167834186136\"><p id=\"fs-id1167834186138\">\\(9x&gt;54\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834119890\"><span data-type=\"media\" id=\"fs-id1167834119893\" data-alt=\"The solution is x is greater than 6. The number line shows a left parenthesis at 6 with shading to its right. The interval notation is 6 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_346_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than 6. The number line shows a left parenthesis at 6 with shading to its right. The interval notation is 6 to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831103887\"><div data-type=\"problem\" id=\"fs-id1167831103889\"><p id=\"fs-id1167831103892\">\\(\\frac{q}{-2}\\ge -24\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834432727\"><div data-type=\"problem\" id=\"fs-id1167830698724\"><p id=\"fs-id1167830698727\">\\(6p&gt;15p-30\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832081955\"><span data-type=\"media\" id=\"fs-id1167832081958\" data-alt=\"The solution is p is less than ten-thirds. The number line shows a right parenthesis at ten-thirds with shading to its left. The interval notation is negative infinity to ten-thirds within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_348_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is p is less than ten-thirds. The number line shows a right parenthesis at ten-thirds with shading to its left. The interval notation is negative infinity to ten-thirds within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835317539\"><div data-type=\"problem\" id=\"fs-id1167835317541\"><p id=\"fs-id1167835317544\">\\(9h-7\\left(h-1\\right)\\le 4h-23\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834376443\"><div data-type=\"problem\" id=\"fs-id1167834376445\"><p id=\"fs-id1167834376447\">\\(5n-15\\left(4-n\\right)&lt;10\\left(n-6\\right)+10n\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826940750\"><span data-type=\"media\" id=\"fs-id1167826940753\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_350_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832096934\"><div data-type=\"problem\" id=\"fs-id1167832096937\"><p id=\"fs-id1167835213526\">\\(\\frac{3}{8}a-\\frac{1}{12}a&gt;\\frac{5}{12}a+\\frac{3}{4}\\)<\/p><\/div><\/div><p id=\"fs-id1167835216800\"><strong data-effect=\"bold\">Translate Words to an Inequality and Solve<\/strong><\/p><p id=\"fs-id1167835216807\">In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.<\/p><div data-type=\"exercise\" id=\"fs-id1167832076547\"><div data-type=\"problem\" id=\"fs-id1167832076549\"><p id=\"fs-id1167832076551\">Five more than <em data-effect=\"italics\">z<\/em> is at most 19.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834502686\"><span data-type=\"media\" id=\"fs-id1167834502689\" data-alt=\"The inequality is z plus 5 is less than or equal to 19. Its solution is z is less than or equal to 14. The number line shows a right bracket at 14 with shading to its left. The interval notation is negative infinity to 14 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_352_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is z plus 5 is less than or equal to 19. Its solution is z is less than or equal to 14. The number line shows a right bracket at 14 with shading to its left. The interval notation is negative infinity to 14 within a parenthesis and a bracket.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832117857\"><div data-type=\"problem\" id=\"fs-id1167832117859\"><p id=\"fs-id1167832117861\">Three less than <em data-effect=\"italics\">c<\/em> is at least 360.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834189544\"><div data-type=\"problem\" id=\"fs-id1167834189546\"><p id=\"fs-id1167834189548\">Nine times <em data-effect=\"italics\">n<\/em> exceeds 42.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834189557\"><span data-type=\"media\" id=\"fs-id1167834130099\" data-alt=\"The inequality is 9 n is greater than 42. Its solution is n is greater than fourteen-thirds. The number line shows a left parentheses at fourteen-thirds with shading to its right. The interval notation is fourteen-thirds to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_354_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is 9 n is greater than 42. Its solution is n is greater than fourteen-thirds. The number line shows a left parentheses at fourteen-thirds with shading to its right. The interval notation is fourteen-thirds to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834060372\"><div data-type=\"problem\" id=\"fs-id1167834060374\"><p id=\"fs-id1167834060376\">Negative two times <em data-effect=\"italics\">a<\/em> is no more than eight.<\/p><\/div><\/div><p id=\"fs-id1167835381679\"><strong data-effect=\"bold\">Solve Applications with Linear Inequalities<\/strong><\/p><p id=\"fs-id1167835347503\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167835347506\"><div data-type=\"problem\" id=\"fs-id1167835347509\"><p id=\"fs-id1167835347511\">Julianne has a weekly food budget of ?231 for her family. If she plans to budget the same amount for each of the seven days of the week, what is the maximum amount she can spend on food each day?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826997072\"><p id=\"fs-id1167826997074\">?33 per day<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826997079\"><div data-type=\"problem\" id=\"fs-id1167826997082\"><p id=\"fs-id1167826997084\">Rogelio paints watercolors. He got a ?100 gift card to the art supply store and wants to use it to buy 12\u2033 \u00d7 16\u2033 canvases. Each canvas costs ?10.99. What is the maximum number of canvases he can buy with his gift card?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835283185\"><div data-type=\"problem\" id=\"fs-id1167835283187\"><p id=\"fs-id1167826799367\">Briana has been offered a sales job in another city. The offer was for ?42,500 plus 8% of her total sales. In order to make it worth the move, Briana needs to have an annual salary of at least ?66,500. What would her total sales need to be for her to move?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826799373\"><p id=\"fs-id1167826799375\">at least ?300,000<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826799381\"><div data-type=\"problem\" id=\"fs-id1167831823842\"><p id=\"fs-id1167831823844\">Renee\u2019s car costs her ?195 per month plus ?0.09 per mile. How many miles can Renee drive so that her monthly car expenses are no more than ?250?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835217782\"><div data-type=\"problem\" id=\"fs-id1167835217784\"><p id=\"fs-id1167835217786\">Costa is an accountant. During tax season, he charges ?125 to do a simple tax return. His expenses for buying software, renting an office, and advertising are ?6,000. How many tax returns must he do if he wants to make a profit of at least ?8,000?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835534279\"><p id=\"fs-id1167835534281\">at least 112 jobs<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835534286\"><div data-type=\"problem\" id=\"fs-id1167835534288\"><p id=\"fs-id1167835534290\">Jenna is planning a five-day resort vacation with three of her friends. It will cost her ?279 for airfare, ?300 for food and entertainment, and ?65 per day for her share of the hotel. She has ?550 saved towards her vacation and can earn ?25 per hour as an assistant in her uncle\u2019s photography studio. How many hours must she work in order to have enough money for her vacation?<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834517700\"><h4 data-type=\"title\"><a href=\"\/contents\/730fb57f-9832-4993-a96f-39ecb47c371d\" class=\"target-chapter\">Solve Compound Inequalities<\/a><\/h4><p id=\"fs-id1167834252729\"><strong data-effect=\"bold\">Solve Compound Inequalities with \u201cand\u201d<\/strong><\/p><p id=\"fs-id1167834252736\">In each of the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167834324650\"><div data-type=\"problem\" id=\"fs-id1167834324652\"><p id=\"fs-id1167834324654\">\\(x\\le 5\\) and \\(x&gt;\\text{\u2212}3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835254053\"><span data-type=\"media\" id=\"fs-id1167835254056\" data-alt=\"The solution is negative 3 is less than x which is less than or equal to 5. The number line shows an open circle at negative 3 and a closed circle at 5. The interval notation is negative 3 to 5 within a parenthesis and a bracket.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_356_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 3 is less than x which is less than or equal to 5. The number line shows an open circle at negative 3 and a closed circle at 5. The interval notation is negative 3 to 5 within a parenthesis and a bracket.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834433653\"><div data-type=\"problem\" id=\"fs-id1167834433655\"><p id=\"fs-id1167835615128\">\\(4x-2\\le 4\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(7x-1&gt;\\text{\u2212}8\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834194611\"><div data-type=\"problem\" id=\"fs-id1167834194614\"><p id=\"fs-id1167834194616\">\\(5\\left(3x-2\\right)\\le 5\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(4\\left(x+2\\right)&lt;3\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834536900\"><span data-type=\"media\" id=\"fs-id1167834536904\" data-alt=\"The solution is negative x is less than negative five-fourths. The number line shows an open circle at negative five-fourths with shading to its left. The interval notation is negative infinity to negative five-fourths within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_358_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative x is less than negative five-fourths. The number line shows an open circle at negative five-fourths with shading to its left. The interval notation is negative infinity to negative five-fourths within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835302985\"><div data-type=\"problem\" id=\"fs-id1167835302988\"><p id=\"fs-id1167835302990\">\\(\\frac{3}{4}\\left(x-8\\right)\\le 3\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(\\frac{1}{5}\\left(x-5\\right)\\le 3\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835320407\"><div data-type=\"problem\" id=\"fs-id1167835361804\"><p id=\"fs-id1167835361806\">\\(\\frac{3}{4}x-5\\ge -2\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(-3\\left(x+1\\right)\\ge 6\\)<\/div><div data-type=\"solution\" id=\"fs-id1167831920623\"><span data-type=\"media\" id=\"fs-id1167831920626\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_360_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835346616\"><div data-type=\"problem\" id=\"fs-id1167835346618\"><p id=\"fs-id1167835433642\">\\(-5\\le 4x-1&lt;7\\)<\/p><\/div><\/div><p id=\"fs-id1167834403456\"><strong data-effect=\"bold\">Solve Compound Inequalities with \u201cor\u201d<\/strong><\/p><p id=\"fs-id1167834403462\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167831116499\"><div data-type=\"problem\" id=\"fs-id1167831116501\"><p id=\"fs-id1167831116503\">\\(5-2x\\le -1\\) or<\/p><div data-type=\"newline\"><br><\/div>\\(6+3x\\le 4\\)<\/div><div data-type=\"solution\" id=\"fs-id1167831822966\"><span data-type=\"media\" id=\"fs-id1167831822969\" data-alt=\"The solution is x is less than negative two-thirds or x is greater than or equal to 3. The number line shows a closed circle at negative two-thirds with shading to its left and a closed circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative two-thirds within a parenthesis and a bracket and 3 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_362_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative two-thirds or x is greater than or equal to 3. The number line shows a closed circle at negative two-thirds with shading to its left and a closed circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative two-thirds within a parenthesis and a bracket and 3 to infinity within a bracket and a parenthesis.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835514661\"><div data-type=\"problem\" id=\"fs-id1167835514663\"><p id=\"fs-id1167831822973\">\\(3\\left(2x-3\\right)&lt;\\text{\u2212}5\\) or<\/p><div data-type=\"newline\"><br><\/div>\\(4x-1&gt;3\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835355098\"><div data-type=\"problem\" id=\"fs-id1167835355100\"><p id=\"fs-id1167835363594\">\\(\\frac{3}{4}x-2&gt;4\\) or \\(4\\left(2-x\\right)&gt;0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835415809\"><span data-type=\"media\" id=\"fs-id1167835415812\" data-alt=\"The solution is x is less than 2 or x is greater than 8. The number line shows an open circle at 2 with shading to its left and an open circle at 8 with shading to its right. The interval notation is the union of negative infinity to 8 within parentheses and 8 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_364_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than 2 or x is greater than 8. The number line shows an open circle at 2 with shading to its left and an open circle at 8 with shading to its right. The interval notation is the union of negative infinity to 8 within parentheses and 8 to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834226120\"><div data-type=\"problem\" id=\"fs-id1167834226122\"><p id=\"fs-id1167834226124\">\\(2\\left(x+3\\right)\\ge 0\\) or<\/p><div data-type=\"newline\"><br><\/div>\\(3\\left(x+4\\right)\\le 6\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830963955\"><div data-type=\"problem\" id=\"fs-id1167830963957\"><p id=\"fs-id1167830963959\">\\(\\frac{1}{2}x-3\\le 4\\) or<\/p><div data-type=\"newline\"><br><\/div>\\(\\frac{1}{3}\\left(x-6\\right)\\ge -2\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834221993\"><span data-type=\"media\" id=\"fs-id1167834221996\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_366_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><\/span><\/div><\/div><p id=\"fs-id1167834526656\"><strong data-effect=\"bold\">Solve Applications with Compound Inequalities<\/strong><\/p><p id=\"fs-id1167831894427\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167831894430\"><div data-type=\"problem\" id=\"fs-id1167831894432\"><p id=\"fs-id1167831894434\">Liam is playing a number game with his sister Audry. Liam is thinking of a number and wants Audry to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Liam might be thinking of.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831922049\"><div data-type=\"problem\" id=\"fs-id1167831922051\"><p id=\"fs-id1167831922054\">Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831922060\"><p id=\"fs-id1167831922062\">\\(6\\le w\\le 12\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834306707\"><h4 data-type=\"title\"><a href=\"\/contents\/6cfd2376-f945-4ebb-bdb6-92ac7398c9ad\" class=\"target-chapter\">Solve Absolute Value Inequalities<\/a><\/h4><p id=\"fs-id1167834394984\"><strong data-effect=\"bold\">Solve Absolute Value Equations<\/strong><\/p><p id=\"fs-id1167834394990\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167826978048\"><div data-type=\"problem\" id=\"fs-id1167826978050\"><p id=\"fs-id1167826978053\">\\(|x|=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831910773\"><div data-type=\"problem\" id=\"fs-id1167831910775\"><p id=\"fs-id1167831910777\">\\(|y|=-14\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835531584\"><p id=\"fs-id1167835356798\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835356803\"><div data-type=\"problem\" id=\"fs-id1167835356805\"><p id=\"fs-id1167835356807\">\\(|z|=0\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834346568\"><p id=\"fs-id1167834346570\">\\(|3x-4|+5=7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831148746\"><p id=\"fs-id1167831148748\">\\(x=2,x=\\frac{2}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834184739\"><div data-type=\"problem\" id=\"fs-id1167834184741\"><p id=\"fs-id1167834184743\">\\(4|x-1|+2=10\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831912139\"><div data-type=\"problem\" id=\"fs-id1167831912142\"><p id=\"fs-id1167831912144\">\\(-2|x-3|+8=-4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830769668\"><p id=\"fs-id1167830769670\">\\(x=9,x=-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826875112\"><div data-type=\"problem\" id=\"fs-id1167826875114\"><p id=\"fs-id1167826875116\">\\(|\\frac{1}{2}x+5|+4=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831115234\"><div data-type=\"problem\" id=\"fs-id1167831115237\"><p id=\"fs-id1167831115239\">\\(|6x-5|=|2x+3|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835302391\"><p id=\"fs-id1167835302394\">\\(x=2,x=\\frac{1}{4}\\)<\/p><\/div><\/div><p id=\"fs-id1167835509790\"><strong data-effect=\"bold\">Solve Absolute Value Inequalities with \u201cless than\u201d<\/strong><\/p><p id=\"fs-id1167835509795\">In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167835509799\"><div data-type=\"problem\" id=\"fs-id1167835509801\"><p id=\"fs-id1167835360885\">\\(|x|\\le 8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835623478\"><div data-type=\"problem\" id=\"fs-id1167835623481\"><p id=\"fs-id1167835623483\">\\(|2x-5|\\le 3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835230321\"><span data-type=\"media\" id=\"fs-id1167835340944\" data-alt=\"The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading in between the circles. The interval notation is 1 to 4 within brackets.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_368_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading in between the circles. The interval notation is 1 to 4 within brackets.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835340958\"><div data-type=\"problem\" id=\"fs-id1167835340960\"><p id=\"fs-id1167835390176\">\\(|6x-5|&lt;7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835512107\"><div data-type=\"problem\" id=\"fs-id1167835512109\"><p id=\"fs-id1167835512111\">\\(|5x+1|\\le -2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835489105\"><span data-type=\"media\" id=\"fs-id1167835489108\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_370_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><\/span><\/div><\/div><p id=\"fs-id1167835367695\"><strong data-effect=\"bold\">Solve Absolute Value Inequalities with \u201cgreater than\u201d<\/strong><\/p><p id=\"fs-id1167835367701\">In the following exercises, solve. Graph the solution and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167835367705\"><div data-type=\"problem\" id=\"fs-id1167834229123\"><p id=\"fs-id1167834229125\">\\(|x|&gt;6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834523740\"><div data-type=\"problem\" id=\"fs-id1167834523742\"><p id=\"fs-id1167834523744\">\\(|x|\\ge 2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832067007\"><span data-type=\"media\" id=\"fs-id1167832067010\" data-alt=\"The solution is x is less than negative 2 or x is greater than 6. The number line shows a closed circle at negative 2 with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_372_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 2 or x is greater than 6. The number line shows a closed circle at negative 2 with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826996917\"><div data-type=\"problem\" id=\"fs-id1167826996919\"><p id=\"fs-id1167832067013\">\\(|x-5|&gt;\\text{\u2212}2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835230288\"><div data-type=\"problem\" id=\"fs-id1167835230290\"><p id=\"fs-id1167835230292\">\\(|x-7|\\ge 1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835281147\"><span data-type=\"media\" id=\"fs-id1167834329824\" data-alt=\"The solution is x is less than or equal to 6 or x is greater than or equal to 8. The number line shows a closed circle at 6 with shading to its left and a closed circle at 8 with shading to its right. The interval notation is the union of negative infinity to negative 6 within a parenthesis and a bracket and 8 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_374_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to 6 or x is greater than or equal to 8. The number line shows a closed circle at 6 with shading to its left and a closed circle at 8 with shading to its right. The interval notation is the union of negative infinity to negative 6 within a parenthesis and a bracket and 8 to infinity within a bracket and a parenthesis.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835182598\"><div data-type=\"problem\" id=\"fs-id1167835182600\"><p id=\"fs-id1167835182602\">\\(3|x|+4\\ge 1\\)<\/p><\/div><\/div><p id=\"fs-id1167834515323\"><strong data-effect=\"bold\">Solve Applications with Absolute Value<\/strong><\/p><p id=\"fs-id1167826801815\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167826801818\"><div data-type=\"problem\" id=\"fs-id1167826801820\"><p id=\"fs-id1167826801823\">A craft beer brewer needs 215,000 bottle per day. But this total can vary by as much as 5,000 bottles. What is the maximum and minimum expected usage at the bottling company?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826801828\"><p id=\"fs-id1167826801831\">The minimum to maximum expected usage is 210,000 to 220,000 bottles<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832056602\"><div data-type=\"problem\" id=\"fs-id1167832056604\"><p id=\"fs-id1167832056606\">At Fancy Grocery, the ideal weight of a loaf of bread is 16 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?<\/p><\/div><\/div><\/div><\/div><div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167826977992\"><h3 data-type=\"title\">Practice Test<\/h3><p id=\"fs-id1167826977999\">In the following exercises, solve each equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167826978003\"><div data-type=\"problem\" id=\"fs-id1167835534241\"><p id=\"fs-id1167835534244\">\\(-5\\left(2x+1\\right)=45\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826808724\"><p id=\"fs-id1167826808726\">\\(x=-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826986742\"><div data-type=\"problem\" id=\"fs-id1167826986744\"><p id=\"fs-id1167826986746\">\\(\\frac{1}{4}\\left(12m+28\\right)=6+2\\left(3m+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835360007\"><div data-type=\"problem\" id=\"fs-id1167835356019\"><p id=\"fs-id1167835356021\">\\(8\\left(3a+5\\right)-7\\left(4a-3\\right)=20-3a\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832042649\"><p id=\"fs-id1167832042651\">\\(a=41\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834422892\"><div data-type=\"problem\"><p>\\(0.1d+0.25\\left(d+8\\right)=4.1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834394761\"><div data-type=\"problem\" id=\"fs-id1167834394763\"><p id=\"fs-id1167834394765\">\\(14n-3\\left(4n+5\\right)=-9+2\\left(n-8\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835297404\"><p id=\"fs-id1167835297406\">contradiction; no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835297412\"><div data-type=\"problem\" id=\"fs-id1167835297414\"><p id=\"fs-id1167835297416\">\\(3\\left(3u+2\\right)+4\\left[6-8\\left(u-1\\right)\\right]=3\\left(u-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834059385\"><div data-type=\"problem\" id=\"fs-id1167835609691\"><p id=\"fs-id1167835609693\">\\(\\frac{3}{4}x-\\frac{2}{3}=\\frac{1}{2}x+\\frac{5}{6}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834489813\"><p id=\"fs-id1167834489815\">\\(x=6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832153843\"><div data-type=\"problem\" id=\"fs-id1167832153845\"><p id=\"fs-id1167832153847\">\\(|3x-4|=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831076534\"><div data-type=\"problem\" id=\"fs-id1167831076536\"><p id=\"fs-id1167831076538\">\\(|2x-1|=|4x+3|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835479596\"><p id=\"fs-id1167835479599\">\\(x=-2,x=-\\frac{1}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835514170\"><div data-type=\"problem\" id=\"fs-id1167835514172\"><p id=\"fs-id1167835514174\">Solve the formula<\/p><div data-type=\"newline\"><br><\/div>\\(x+2y=5\\) for <em data-effect=\"italics\">y<\/em>.<\/div><\/div><p id=\"fs-id1167835387100\">In the following exercises, graph the inequality on the number line and write in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167835387104\"><div data-type=\"problem\" id=\"fs-id1167831919540\"><p id=\"fs-id1167831919543\">\\(x\\ge -3.5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831919554\"><span data-type=\"media\" id=\"fs-id1167834464190\" data-alt=\"The inequality is x is greater than or equal to negative 3.5. The number line shows a left bracket at negative 3.5 and shading to the right. The interval notation is negative 3.5 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_376_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is x is greater than or equal to negative 3.5. The number line shows a left bracket at negative 3.5 and shading to the right. The interval notation is negative 3.5 to infinity within a bracket and a parenthesis.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834464204\"><div data-type=\"problem\" id=\"fs-id1167834189855\"><p id=\"fs-id1167834189857\">\\(x&lt;\\frac{11}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835327400\"><div data-type=\"problem\" id=\"fs-id1167835327402\"><p id=\"fs-id1167835327404\">\\(-2\\le x&lt;5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832052659\"><span data-type=\"media\" id=\"fs-id1167832052662\" data-alt=\"The inequality is negative two is less than or equal to x which is less than 5. The number line shows a closed circle at negative 2 and an open circle at 5 with shading between the circles. The interval notation is negative 2 to 5 within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_378_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is negative two is less than or equal to x which is less than 5. The number line shows a closed circle at negative 2 and an open circle at 5 with shading between the circles. The interval notation is negative 2 to 5 within a bracket and a parenthesis.\"><\/span><\/div><\/div><p id=\"fs-id1167831883587\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167831883591\"><div data-type=\"problem\" id=\"fs-id1167831883593\"><p id=\"fs-id1167831883595\">\\(8k\\ge 5k-120\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835376497\"><div data-type=\"problem\" id=\"fs-id1167835376499\"><p id=\"fs-id1167835376501\">\\(3c-10\\left(c-2\\right)&lt;5c+16\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835373843\"><span data-type=\"media\" id=\"fs-id1167835373846\" data-alt=\"The solution is c is greater than one-third. The number line shows a left parenthesis at one-third with shading to its right. The interval notation is one-third to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_380_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is c is greater than one-third. The number line shows a left parenthesis at one-third with shading to its right. The interval notation is one-third to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834132955\"><div data-type=\"problem\" id=\"fs-id1167834132957\"><p id=\"fs-id1167834132959\">\\(\\frac{3}{4}x-5\\ge -2\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(-3\\left(x+1\\right)\\ge 6\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831023918\"><div data-type=\"problem\" id=\"fs-id1167831023920\"><p id=\"fs-id1167831023922\">\\(3\\left(2x-3\\right)&lt;\\text{\u2212}5\\) or<\/p><div data-type=\"newline\"><br><\/div>\\(4x-1&gt;3\\)<\/div><div data-type=\"solution\" id=\"fs-id1167826978335\"><span data-type=\"media\" id=\"fs-id1167826978338\" data-alt=\"The solution is x is less than two-thirds or x is greater than 1. The number line shows an open circle at two-thirds with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 to infinity within parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_382_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than two-thirds or x is greater than 1. The number line shows an open circle at two-thirds with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 to infinity within parentheses.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834062394\"><div data-type=\"problem\" id=\"fs-id1167834062396\"><p id=\"fs-id1167834062398\">\\(\\frac{1}{2}x-3\\le 4\\) or<\/p><div data-type=\"newline\"><br><\/div>\\(\\frac{1}{3}\\left(x-6\\right)\\ge -2\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835352568\"><div data-type=\"problem\" id=\"fs-id1167835352570\"><p id=\"fs-id1167835352572\">\\(|4x-3|\\ge 5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832134036\"><span data-type=\"media\" id=\"fs-id1167832134040\" data-alt=\"The solution is x is less than or equal to negative one-half or x is greater than or equal to 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and bracket and 2 to infinity within a bracket and a parenthesis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_384_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to negative one-half or x is greater than or equal to 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and bracket and 2 to infinity within a bracket and a parenthesis.\"><\/span><\/div><\/div><p id=\"fs-id1167832152980\">In the following exercises, translate to an equation or inequality and solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167832152983\"><div data-type=\"problem\" id=\"fs-id1167832152985\"><p id=\"fs-id1167832152987\">Four less than twice <em data-effect=\"italics\">x<\/em> is 16.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826967326\"><div data-type=\"problem\" id=\"fs-id1167826967328\"><p id=\"fs-id1167826967330\">Find the length of the missing side.<\/p><span data-type=\"media\" id=\"fs-id1167826967334\" data-alt=\"The figure is a right triangle with a base of 6 units and a height of 9 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a right triangle with a base of 6 units and a height of 9 units.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167835329113\"><p id=\"fs-id1167835329115\">\\(10.8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835329124\"><div data-type=\"problem\" id=\"fs-id1167835329126\"><p id=\"fs-id1167835329128\">One number is four more than twice another. Their sum is \\(-47.\\) Find the numbers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831116947\"><div data-type=\"problem\" id=\"fs-id1167831116949\"><p id=\"fs-id1167831116951\">The sum of two consecutive odd integers is \\(-112.\\) Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826803036\"><p id=\"fs-id1167826803038\">\\(-57,-55\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834219646\"><div data-type=\"problem\" id=\"fs-id1167834219648\"><p id=\"fs-id1167834219650\">Marcus bought a television on sale for ?626.50 The original price of the television was ?895. Find <span class=\"token\">\u24d0<\/span> the amount of discount and <span class=\"token\">\u24d1<\/span> the discount rate.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834532431\"><div data-type=\"problem\" id=\"fs-id1167834532433\"><p id=\"fs-id1167835281905\">Bonita has ?2.95 in dimes and quarters in her pocket. If she has five more dimes than quarters, how many of each coin does she have?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835281911\"><p id=\"fs-id1167835281913\">12 dimes, seven quarters<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835281918\"><div data-type=\"problem\" id=\"fs-id1167835281920\"><p id=\"fs-id1167835329472\">Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs ?6.04 per gallon and the soda costs ?4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs ?5.71 per gallon?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832075936\"><div data-type=\"problem\" id=\"fs-id1167832075938\"><p id=\"fs-id1167832075940\">The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832150928\"><p id=\"fs-id1167832150930\">\\(30\\text{\u00b0},60\\text{\u00b0},90\\text{\u00b0}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835529713\"><div data-type=\"problem\" id=\"fs-id1167835529715\"><p id=\"fs-id1167835529717\">The length of a rectangle is five feet more than four times the width. The perimeter is 60 feet. Find the dimensions of the rectangle.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830705577\"><div data-type=\"problem\" id=\"fs-id1167830705579\"><p id=\"fs-id1167830705581\">Two planes leave Dallas at the same time. One heads east at a speed of 428 miles per hour. The other plane heads west at a speed of 382 miles per hour. How many hours will it take them to be 2,025 miles apart?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831076614\"><p id=\"fs-id1167831076616\">\\(2.5\\) hours<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831076625\"><div data-type=\"problem\" id=\"fs-id1167831076628\"><p id=\"fs-id1167830925372\">Leon drove from his house in Cincinnati to his sister\u2019s house in Cleveland, a distance of 252 miles. It took him \\(4\\frac{1}{2}\\) hours. For the first half hour, he had heavy traffic, and the rest of the time his speed was five miles per hour less than twice his speed in heavy traffic. What was his speed in heavy traffic?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831887239\"><div data-type=\"problem\" id=\"fs-id1167831887241\"><p id=\"fs-id1167831887243\">Sara has a budget of ?1,000 for costumes for the 18 members of her musical theater group. What is the maximum she can spend for each costume?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834183595\"><p id=\"fs-id1167834183597\">At most ?55.56 per costume.<\/p><\/div><\/div><\/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>Solve absolute value equations<\/li>\n<li>Solve absolute value inequalities with \u201cless than\u201d<\/li>\n<li>Solve absolute value inequalities with \u201cgreater than\u201d<\/li>\n<li>Solve applications with absolute value<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831894154\" class=\"be-prepared\">\n<p id=\"fs-id1167834111770\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167831890429\" type=\"1\">\n<li>Evaluate: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3930243d5c5d3c69c55b7e1471f8c38b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#124;&#55;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167835595046\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Fill in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89033c4d1cdb8cf4801c1c66ea47b2ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#60;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#62;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"24\" 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.\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-f773696ccf783d0f0eaf473c509ac4ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#56;&#124;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#125;&#45;&#124;&#45;&#56;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" 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-4ff0fc8509d4af25ea2e5daa1d1e7219_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#125;&#45;&#124;&#45;&#49;&#50;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" 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-c4f178d946e75a2cece6849d42bddb8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#54;&#124;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#125;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" 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-9f7973206fbbd83b2ed219246ef613c7_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;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#125;&#45;&#124;&#45;&#49;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167835595046\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ef979b7673e77bf6fa5826dc4a24c92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#45;&#50;&#124;&#56;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"160\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167835319324\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834065781\">\n<h3 data-type=\"title\">Solve Absolute Value Equations<\/h3>\n<p id=\"fs-id1167835333127\">As we prepare to solve absolute value equations, we review our definition of <span data-type=\"term\">absolute value<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167830962404\">\n<div data-type=\"title\">Absolute Value<\/div>\n<p>The absolute value of a number is its distance from zero on the number line.<\/p>\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<p id=\"fs-id1167834120880\">Absolute values are always greater than or equal to zero.<\/p>\n<\/div>\n<p id=\"fs-id1167831923735\">We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:<\/p>\n<p id=\"fs-id1167835338742\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e056724bd23f796924453f7cff901484_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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: 0px;\" \/> is 5 units away from 0, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87a3814c4da576fb50b0dce4763b902a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#53;&#124;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6d0c36e6a54759464c4a21eb1153052_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;&#51;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> is 5 units away from 0, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9a60168ffeccebf5ac020c60244df92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#53;&#124;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835339214\"><a href=\"#CNX_IntAlg_Figure_02_07_001\" class=\"autogenerated-content\">(Figure)<\/a> illustrates this idea.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_001\">\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 both five units away from zero.<\/div>\n<p><span data-type=\"media\" data-alt=\"The figure is a number line with tick marks at negative 5, 0, and 5. The distance between negative 5 and 0 is given as 5 units, so the absolute value of negative 5 is 5. The distance between 5 and 0 is 5 units, so the absolute value of 5 is 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with tick marks at negative 5, 0, and 5. The distance between negative 5 and 0 is given as 5 units, so the absolute value of negative 5 is 5. The distance between 5 and 0 is 5 units, so the absolute value of 5 is 5.\" \/><\/span><\/div>\n<p id=\"fs-id1167835357520\">For the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9df8b5612796719005d2c6431830daa5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 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 five units from zero on the number line. They are the solutions to the equation.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835215234\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c7e44d4efbe4d26a78750d5a9fc07c7_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;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#120;&#124;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#120;&#61;&#45;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#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=\"37\" width=\"253\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"fs-id1167834300003\">The solution can be simplified to a single statement by writing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0a4bd08aeae4b1262cd6c85c03d90a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: 0px;\" \/> This is read, \u201c<em data-effect=\"italics\">x<\/em> is equal to positive or negative 5\u201d.<\/p>\n<p id=\"fs-id1167830837184\">We can generalize this to the following property for absolute value equations.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834299926\">\n<div data-type=\"title\">Absolute Value Equations<\/div>\n<p id=\"fs-id1167832041537\">For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835284995\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6055ae5901096402e44fd5b4199d3c2_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;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#61;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#117;&#61;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#61;&#97;&#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=\"240\" style=\"vertical-align: -12px;\" \/><\/div>\n<p>Remember that an absolute value cannot be a negative number.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167831112021\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835363618\">\n<p id=\"fs-id1167834495232\">Solve: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47fe8cdef5ea7773cc75b95fb9ceae0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" 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-48d44d130e6922fac896da575c40f521_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#121;&#124;&#61;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" 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-bff1356215b10f5e9799486f4268133d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#122;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834124409\">\n<p id=\"fs-id1167835400249\"><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-d055105a92d9c748f133e002d1ea979b_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;&#38;&#32;&#38;&#32;&#124;&#120;&#124;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#52;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#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;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#52;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"457\" style=\"vertical-align: -22px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><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-7fdfbe29a24ac33089f3b01bd6b5e1cf_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;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#50;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#121;&#124;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#50;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#111;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#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=\"36\" width=\"90\" style=\"vertical-align: -11px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Since an absolute value is always positive, there are no solutions to this equation.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <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-04ec9ce2fe17c321ed17138db5f00838_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;&#38;&#32;&#38;&#32;&#124;&#122;&#124;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#52;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#122;&#61;&#45;&#48;&#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;&#111;&#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;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#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;&#45;&#48;&#61;&#48;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#52;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"451\" style=\"vertical-align: -26px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Both equations tell us that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e381c273c9f7ca5b93029ee2e8bab16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> and so there is only one solution.<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831106822\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831922992\">Solve: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-479dba4ade36e26b05f99aef03411be2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" 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-5c3e162c2e1b08d486ea20f67402b29d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#121;&#124;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" 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-bff1356215b10f5e9799486f4268133d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#122;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835336858\">\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-809fa2f20190e7801ec07e8a753ab352_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span> no solution <span class=\"token\">\u24d2<\/span> 0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835595443\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835348491\">\n<div data-type=\"problem\" id=\"fs-id1167834430172\">\n<p id=\"fs-id1167834133052\">Solve: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c196a8965d2b70156adf94221d67d69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" 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-19e5e71fcc0853201cc6fe31126232a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#121;&#124;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" 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-bff1356215b10f5e9799486f4268133d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#122;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830894511\">\n<p id=\"fs-id1167834528036\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e47d37202d9577ff54785acdd059b8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span> no solution <span class=\"token\">\u24d2<\/span> 0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To solve an <span data-type=\"term\" class=\"no-emphasis\">absolute value equation<\/span>, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834339985\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve Absolute Value Equations<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826801729\">\n<div data-type=\"problem\" id=\"fs-id1167835339566\">\n<p id=\"fs-id1167831910493\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17d8ddd88ea8bad96acc5166cde6e9ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#53;&#120;&#45;&#52;&#124;&#45;&#51;&#61;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835258843\"><span data-type=\"media\" id=\"fs-id1167835288367\" data-alt=\"Step 1 is to isolate the absolute value expression. The difference between the absolute value of the quantity 5 x minus 4 and 3 is equal to 8. Add 3 to both sides. The result is the absolute value of the quantity 5 x minus 4 is equal to 11.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to isolate the absolute value expression. The difference between the absolute value of the quantity 5 x minus 4 and 3 is equal to 8. Add 3 to both sides. The result is the absolute value of the quantity 5 x minus 4 is equal to 11.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167826995876\" data-alt=\"Step 2 is to write the equivalent equations, 5 x minus 4 is equal to negative 11 and 5 x minus 4 is equal to 11.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to write the equivalent equations, 5 x minus 4 is equal to negative 11 and 5 x minus 4 is equal to 11.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835362546\" data-alt=\"Step 3 is to solve each equation. Add 4 to each side. 5 x is equal to negative 7 or 5 x is equal to 15. Divide each side by 5. The result is x is equal to negative seven-fifths or x is equal to 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve each equation. Add 4 to each side. 5 x is equal to negative 7 or 5 x is equal to 15. Divide each side by 5. The result is x is equal to negative seven-fifths or x is equal to 3.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835264946\" data-alt=\"Step 4 is to check each solution. Substitute 3 and negative seven-fifths into the original equation, the difference between the absolute value of the quantity 5 x minus 4 and 3 is equal to 8. Substitute 3 for x. Is the difference between the absolute value of the quantity 5 times 3 minus 4 and 3 equal to 8? Is the difference between the absolute value of the quantity 15 minus 4 and 3 equal to 8? Is the difference between the absolute value of the 11 and 3 equal to 8? Is 11 minus 3 equal to 8? 8 is equal to 8, so the solution x is equal to 3 checks. Substitute negative seven-fifths for x. Is the difference between the absolute value of the quantity 5 times negative seven-fifths minus 4 and 3 equal to 8? Is the difference between the absolute value of the quantity negative 7 minus 4 and 3 equal to 8? Is the difference between the absolute value of the negative 11 and 3 equal to 8? Is 11 minus 3 equal to 8? 8 is equal to 8, so the solution x is equal to negative seven-fifths checks.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check each solution. Substitute 3 and negative seven-fifths into the original equation, the difference between the absolute value of the quantity 5 x minus 4 and 3 is equal to 8. Substitute 3 for x. Is the difference between the absolute value of the quantity 5 times 3 minus 4 and 3 equal to 8? Is the difference between the absolute value of the quantity 15 minus 4 and 3 equal to 8? Is the difference between the absolute value of the 11 and 3 equal to 8? Is 11 minus 3 equal to 8? 8 is equal to 8, so the solution x is equal to 3 checks. Substitute negative seven-fifths for x. Is the difference between the absolute value of the quantity 5 times negative seven-fifths minus 4 and 3 equal to 8? Is the difference between the absolute value of the quantity negative 7 minus 4 and 3 equal to 8? Is the difference between the absolute value of the negative 11 and 3 equal to 8? Is 11 minus 3 equal to 8? 8 is equal to 8, so the solution x is equal to negative seven-fifths checks.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835304704\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835511680\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6135e10e3bf73871bf86081c61d2154c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#53;&#124;&#45;&#49;&#61;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835596157\">\n<p id=\"fs-id1167835327087\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c9a98153477faa0177c3ffe1722d2f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835320274\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835353084\">\n<div data-type=\"problem\" id=\"fs-id1167831024455\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c0ca9b1b65fa6bbd8af8737040da596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#120;&#45;&#51;&#124;&#45;&#53;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835304970\">\n<p id=\"fs-id1167831923395\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c931d8c1008dde5da348b4c1e6bcc19b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835338375\">The steps for solving an absolute value equation are summarized here.<\/p>\n<div data-type=\"note\" class=\"howto\">\n<div data-type=\"title\">Solve absolute value equations.<\/div>\n<ol id=\"fs-id1167835262974\" type=\"1\" class=\"stepwise\">\n<li>Isolate the absolute value expression.<\/li>\n<li>Write the equivalent equations.<\/li>\n<li>Solve each equation.<\/li>\n<li>Check each solution.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835364757\">\n<div data-type=\"problem\" id=\"fs-id1167832043509\">\n<p id=\"fs-id1167832153416\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7473e1456da192fe0f04aaec24e9f50e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#124;&#120;&#45;&#55;&#124;&#43;&#53;&#61;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171790373440\">\n<table id=\"fs-id1167835335371\" class=\"unnumbered unstyled can-break\" summary=\"The sum of 2 times the absolute value of the quantity x minus 7 and 5 is equal to 9. Isolate the absolute value expression. 2 times the absolute value of the quantity x minus 7 is equal to 4 simplifies to the absolute value of the quantity x minus 7 is equal to 2. Write the equivalent equations. They are x minus 7 is equal to negative 2 or x minus 7 is equal to 2. Solve each equation. The solutions x is equal to 5 or x is equal to 9. Check using the original equation, the sum of 2 times the absolute value of the quantity x minus 7 and 5 is equal to 9. Is the sum of 2 times the absolute value of the quantity 5 minus 7 and 5 equal to 9? Is the sum of 2 times the absolute value of negative 2 and 5 equal to 9? Is 2 times 2 plus 5 equal to 9? Is 4 plus 5 equal to 9? 9 is equal to 9, so the solution x is equal to 5 checks. Is the sum of 2 times the absolute value of the quantity 9 minus 7 and 5 equal to 9? Is the sum of 2 times the absolute value of 2 and 5 equal to 9? Is 2 times 2 plus 5 equal to 9? Is 4 plus 5 equal to 9? 9 is equal to 9, so the solution x is equal to 9 checks.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e71aa0da47f984fd957a77c7d535f639_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#124;&#120;&#45;&#55;&#124;&#43;&#53;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Isolate the absolute value expression.\u2003\u2003\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e372c0014b3aaaaf7c638927b7ec670_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#124;&#120;&#45;&#55;&#124;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0ed89f9331d9722f143d85966be94e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#45;&#55;&#124;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the equivalent equations.<\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40d111b5af8b229b9a75da7f0f1d39d_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;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#45;&#55;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"72\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18b12289e13398812d9d78fa80ec8bb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#55;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"72\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve each equation.<\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b89a18be96b2b9ff4bbf2b2c1b66ec15_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;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-079728bea87b9616bbee4c5b7961c095_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;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835333456\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835198774\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835371191\">\n<div data-type=\"problem\" id=\"fs-id1167835419823\">\n<p id=\"fs-id1167835368941\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4cc6ef1ce477bd86e1db36dac55a948d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#124;&#120;&#45;&#52;&#124;&#45;&#52;&#61;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834308115\">\n<p id=\"fs-id1167834228781\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d6362b3a7160a9d07817c26cb1b90c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;&#44;&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"93\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835181971\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835349582\">\n<div data-type=\"problem\" id=\"fs-id1167830962365\">\n<p id=\"fs-id1167826880268\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6890b612e5ac076b04915e403412f43e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#124;&#120;&#45;&#53;&#124;&#43;&#51;&#61;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834228481\">\n<p id=\"fs-id1167835370272\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9041ad3617ea369933ece4a8d12962d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;&#44;&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1166400945810\">Remember, an absolute value is always positive!<\/p>\n<div data-type=\"example\" id=\"fs-id1167835281948\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834539505\">\n<div data-type=\"problem\" id=\"fs-id1167834274143\">\n<p id=\"fs-id1167834279299\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6a9ec1e072cd4d734713aeb5a0db64b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#124;&#43;&#49;&#49;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"135\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835529802\">\n<p id=\"fs-id1167835183544\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dda3c70168efb6f905ee79070cfe806c_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;&#38;&#32;&#38;&#32;&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#124;&#43;&#49;&#49;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#111;&#108;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#97;&#98;&#115;&#111;&#108;&#117;&#116;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#32;&#116;&#101;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#52;&#124;&#61;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#110;&#32;&#97;&#98;&#115;&#111;&#108;&#117;&#116;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#32;&#99;&#97;&#110;&#110;&#111;&#116;&#32;&#98;&#101;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#111;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#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=\"63\" width=\"524\" style=\"vertical-align: -25px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831025415\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834192376\">\n<div data-type=\"problem\" id=\"fs-id1167835274654\">\n<p id=\"fs-id1167831823492\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c27f2fbd66e0968ebe869a2796bd7b0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#53;&#124;&#43;&#57;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835236935\">\n<p id=\"fs-id1167826994048\">No solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831923617\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834489662\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832060267\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b60c366068754234d7283de5e9c2fc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#120;&#43;&#51;&#124;&#43;&#56;&#61;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p>No solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167830960678\">Some of our absolute value equations could be of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a8e669629b2a4410de8c7ccbd87edc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#124;&#61;&#124;&#118;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> where <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em> are algebraic expressions. For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5be747962ad829d9bcc47b7c5d21b1ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#45;&#51;&#124;&#61;&#124;&#50;&#120;&#43;&#49;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835361268\">How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, <em data-effect=\"italics\">u<\/em>, and a positive real number, <em data-effect=\"italics\">a<\/em>, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ddb8f519f4ca8dda394073ec2507d130_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#124;&#61;&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9aeabf4cc9f37420ec63237122918e4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"43\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9da17670ddd6ffdc611fb6f7e9a0667_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167835339561\">This tell us that<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835213198\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4379602d46c42862c1b164856c95038_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;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#124;&#117;&#124;&#61;&#124;&#118;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#117;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#118;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#61;&#118;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#117;&#61;&#118;&#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=\"416\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"fs-id1167835288119\">This leads us to the following property for equations with two absolute values.<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\">Equations with Two Absolute Values<\/div>\n<p id=\"fs-id1167834557281\">For any algebraic expressions, <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em>,<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834523020\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffa9a06d62e046b107298cd430dfb996_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;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#61;&#124;&#118;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#117;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#118;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#61;&#118;&#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=\"240\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1166400208360\">When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835269461\">\n<div data-type=\"problem\" id=\"fs-id1167835186322\">\n<p id=\"fs-id1167835326417\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87f93d9ff4a8606de1a02493b4a0b4ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#53;&#120;&#45;&#49;&#124;&#61;&#124;&#50;&#120;&#43;&#51;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1171792882089\">\n<div data-type=\"equation\" id=\"fs-id1167836729601\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-febf635f16891d7c797f82f3b729cf44_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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#53;&#120;&#45;&#49;&#124;&#61;&#124;&#50;&#120;&#43;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167834189093\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82df84c391d8d7bec594ca27c97f64ec_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;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#120;&#45;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#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;&#111;&#114;&#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;&#53;&#120;&#45;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#120;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#101;&#97;&#99;&#104;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#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;&#53;&#120;&#45;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#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;&#111;&#114;&#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;&#51;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#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;&#55;&#120;&#45;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#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;&#55;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#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;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#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;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#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;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#108;&#101;&#97;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#99;&#104;&#101;&#99;&#107;&#32;&#116;&#111;&#32;&#121;&#111;&#117;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"171\" width=\"735\" style=\"vertical-align: -80px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835361950\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835310674\">\n<div data-type=\"problem\" id=\"fs-id1167835194778\">\n<p id=\"fs-id1167834228660\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2f6695338855afe4ab202bb3b626b40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#55;&#120;&#45;&#51;&#124;&#61;&#124;&#51;&#120;&#43;&#55;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831040602\">\n<p id=\"fs-id1167835360737\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-012ed67f22c9ec07ffda503a54922261_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fa72629f044a280114c8466fb1eee02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831891635\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167826857424\">\n<p id=\"fs-id1167835370427\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-210d7914c1970545ad26fca4148357d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#54;&#120;&#45;&#53;&#124;&#61;&#124;&#51;&#120;&#43;&#52;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835610057\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7d62a78a97a33b2bfba1ddcfe9dbb8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b2f8392e1ee82c3c91af916527b8b70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Solve Absolute Value Inequalities with \u201cLess Than\u201d<\/h3>\n<p>Let\u2019s look now at what happens when we have an <span data-type=\"term\" class=\"no-emphasis\">absolute value inequality<\/span>. Everything we\u2019ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.<\/p>\n<p id=\"fs-id1167835378734\">Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line. For the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9df8b5612796719005d2c6431830daa5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> we saw that both 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 five units from zero on the number line. They are the solutions to the equation.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835283055\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99afa8478d8a415b561ab67f91176e55_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;&#104;&#102;&#105;&#108;&#108;&#32;&#124;&#120;&#124;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#61;&#45;&#53;&#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;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#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;&#120;&#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=\"37\" width=\"256\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"fs-id1167835365479\">What about the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5f4dd65838dedf8744a1f60fe5e124e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#92;&#108;&#101;&#32;&#53;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> Where are the numbers whose distance is less than or equal to 5? We know <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 5 are both five units from zero. All the numbers between <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 5 are less than five units from zero. See <a href=\"#CNX_IntAlg_Figure_02_07_004\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_004\"><span data-type=\"media\" id=\"fs-id1167835333448\" data-alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a left bracket at negative 5 and a right bracket at 5. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x which is less than or equal to 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a left bracket at negative 5 and a right bracket at 5. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x which is less than or equal to 5.\" \/><\/span><\/div>\n<p id=\"fs-id1167835421120\">In a more general way, we can see that if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc30be7ab27e487fe9d97baf4a99b46a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#124;&#92;&#108;&#101;&#32;&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e2b0cd4f8c4a4ef2f33b1139b2354d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#108;&#101;&#32;&#117;&#92;&#108;&#101;&#32;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"80\" style=\"vertical-align: -3px;\" \/> See <a href=\"#CNX_IntAlg_Figure_02_07_005\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_005\"><span data-type=\"media\" id=\"fs-id1167834432227\" data-alt=\"The figure is a number line with negative a 0, and a displayed. There is a left bracket at negative a and a right bracket at a. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is less than or equal to a, then negative a is less than or equal to u which is less than or equal to a.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative a 0, and a displayed. There is a left bracket at negative a and a right bracket at a. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is less than or equal to a, then negative a is less than or equal to u which is less than or equal to a.\" \/><\/span><\/div>\n<p id=\"fs-id1167835301566\">This result is summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835331002\">\n<div data-type=\"title\">Absolute Value Inequalities with <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;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54e2b165150917469474a6d203f27e67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"fs-id1167835381129\">For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834526452\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fab943061f617b8c993adb97c63a4674_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;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#60;&#97;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#60;&#117;&#60;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#92;&#108;&#101;&#32;&#97;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#108;&#101;&#32;&#117;&#92;&#108;&#101;&#32;&#97;&#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=\"333\" style=\"vertical-align: -15px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835349608\">After solving an inequality, it is often helpful to check some points to see if the solution makes sense. The graph of the solution divides the number line into three sections. Choose a value in each section and substitute it in the original inequality to see if it makes the inequality true or not. While this is not a complete check, it often helps verify the solution.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835352166\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167826993390\">\n<p id=\"fs-id1167826779359\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c3a590ee6a314a49d4571eef828cee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#60;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835354119\">\n<table id=\"fs-id1167835344288\" class=\"unnumbered unstyled\" summary=\"The absolute value of x is less than 7. Write the equivalent inequality. It is negative 7 is less than x which is less than 7. Graph the solution. It is a number line with a left parenthesis at negative 7, a right parenthesis at 7 and shading between the parentheses. Write the solution using interval notation. It is negative 7 to 7 within parentheses.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834228084\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the equivalent inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835343024\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the solution.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the solution using interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831238937\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835483823\">Check:<\/p>\n<p id=\"fs-id1167835304135\">To verify, check a value in each section of the number line showing the solution. Choose numbers such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95b507f07085049276cbc062895c8807_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> 1, and 9.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832066987\" data-alt=\"The figure is a number line with a left parenthesis at negative 7, a right parenthesis at 7 and shading between the parentheses. The values negative 8, 1, and 9 are marked with points. The absolute value of negative 8 is less than 7 is false. It does not satisfy the absolute value of x is less than 7. The absolute value of 1 is less than 7 is true. It does satisfy the absolute value of x is less than 7. The absolute value of 9 is less than 7 is false. It does not satisfy the absolute value of x is less than 7.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with a left parenthesis at negative 7, a right parenthesis at 7 and shading between the parentheses. The values negative 8, 1, and 9 are marked with points. The absolute value of negative 8 is less than 7 is false. It does not satisfy the absolute value of x is less than 7. The absolute value of 1 is less than 7 is true. It does satisfy the absolute value of x is less than 7. The absolute value of 9 is less than 7 is false. It does not satisfy the absolute value of x is less than 7.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835375988\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832066840\">\n<div data-type=\"problem\" id=\"fs-id1167834141948\">\n<p id=\"fs-id1167831107059\">Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-adbb1b7360eb38babeb0d85c6a6e7ecf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#60;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835225764\"><span data-type=\"media\" id=\"fs-id1167826967468\" data-alt=\"The solution is negative 9 is less than x which is less than 9. The number line shows open circles at negative 9 and 9 with shading in between the circles. The interval notation is negative 9 to 9 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_301_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 9 is less than x which is less than 9. The number line shows open circles at negative 9 and 9 with shading in between the circles. The interval notation is negative 9 to 9 within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835358565\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835350323\">\n<div data-type=\"problem\" id=\"fs-id1167835341270\">\n<p id=\"fs-id1167834535568\">Graph the solution and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0db4655f84ae1191ed445e2ee3b5a605_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#60;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835307947\"><span data-type=\"media\" id=\"fs-id1167831923510\" data-alt=\"The solution is negative 1 is less than x which is less than 1. The number line shows open circles at negative 1 and 1 with shading in between the circles. The interval notation is negative 1 to 1 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than 1. The number line shows open circles at negative 1 and 1 with shading in between the circles. The interval notation is negative 1 to 1 within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835363985\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167826967314\">\n<div data-type=\"problem\" id=\"fs-id1167834132141\">\n<p id=\"fs-id1167834097907\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16957909cdcdb07807133296ceedaf49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#53;&#120;&#45;&#54;&#124;&#92;&#108;&#101;&#32;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835364122\">\n<table id=\"fs-id1167835366818\" class=\"unnumbered unstyled can-break\" summary=\"Step 1 is to isolate the absolute value expression, the absolute value of the quantity 5 x minus 6 is less than or equal to 4. It is isolated. Step 2 is to write the equivalent compound inequality. Negative 4 is less than or equal to 5 x minus 6 which is less than 4. Step 3 is to solve the compound inequality. 2 is less than or equal to 5 x which is less than or equal to 10. Step 4 is to graph the solution. The graph showed closed points at two-fifths and 2 with shading between the circles. Step 5 is to write the solution using interval notation. It is two-fifths to 2 within brackets. The check is left to you.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Isolate the absolute value expression.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>It is isolated.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ea53b2687149dc8b611ff06aea8b673_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#53;&#120;&#45;&#54;&#124;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Write the equivalent compound inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf1851502bbca8b15f3d50360fdc37ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#92;&#108;&#101;&#32;&#53;&#120;&#45;&#54;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Solve the compound inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20371d24cbdd6c2ccaf7a41006f20e32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#32;&#53;&#120;&#92;&#108;&#101;&#32;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"93\" style=\"vertical-align: -3px;\" \/><\/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-6974cc34e4bae286a7685de6ac987503_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"74\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4.<\/strong> Graph the solution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835416933\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5.<\/strong> Write the solution using interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd8b18d69b58d35004b0ad3d74e67ea1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"36\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The check is left to you.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834189360\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835358273\">\n<div data-type=\"problem\" id=\"fs-id1167834238700\">\n<p id=\"fs-id1167835596723\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c71f8fdcefbaec427ae93adb6f378b97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#45;&#49;&#124;&#92;&#108;&#101;&#32;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation:<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834238687\"><span data-type=\"media\" id=\"fs-id1167835327750\" data-alt=\"The solution is negative 2 is less than or equal to x which is less than or equal to 3. The number line shows closed circles at negative 2 and 3 with shading between the circles. The interval notation is negative 2 to 3 within brackets.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 2 is less than or equal to x which is less than or equal to 3. The number line shows closed circles at negative 2 and 3 with shading between the circles. The interval notation is negative 2 to 3 within brackets.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832051824\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826782420\">\n<div data-type=\"problem\" id=\"fs-id1167835513688\">\n<p id=\"fs-id1167834094549\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81bfd1d594aa287d9ee417880e28a1ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#120;&#45;&#53;&#124;&#92;&#108;&#101;&#32;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation:<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834432147\"><span data-type=\"media\" id=\"fs-id1167834593016\" data-alt=\"The solution is one-half is less than or equal to x which is less than or equal to 2. The number line shows closed circles at one-half and 2 with shading between the circles. The interval notation is one-half to 2 within brackets.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is one-half is less than or equal to x which is less than or equal to 2. The number line shows closed circles at one-half and 2 with shading between the circles. The interval notation is one-half to 2 within brackets.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834246739\" class=\"howto\">\n<div data-type=\"title\">Solve absolute value inequalities with &lt; or \u2264.<\/div>\n<ol id=\"fs-id1167832067486\" type=\"1\" class=\"stepwise\">\n<li>Isolate the absolute value expression.<\/li>\n<li>Write the equivalent compound inequality.\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167835419866\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19b770fee117add045aeb15da55a0a88_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;&#99;&#125;&#124;&#117;&#124;&#60;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#60;&#117;&#60;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#124;&#117;&#124;&#92;&#108;&#101;&#32;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#108;&#101;&#32;&#117;&#92;&#108;&#101;&#32;&#97;&#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=\"407\" style=\"vertical-align: -15px;\" \/><\/div>\n<\/li>\n<li>Solve the compound inequality.<\/li>\n<li>Graph the solution<\/li>\n<li>Write the solution using interval notation.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834308172\">\n<h3 data-type=\"title\">Solve Absolute Value Inequalities with \u201cGreater Than\u201d<\/h3>\n<p id=\"fs-id1167832060088\">What happens for absolute value inequalities that have \u201cgreater than\u201d? Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.<\/p>\n<p id=\"fs-id1167835337199\">We started with the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18d7bfd113bcb5aaa2c6d7d7431a4f6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#92;&#108;&#101;&#32;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> We saw that the numbers whose distance is less than or equal to five from zero on the number line were <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 5 and all the numbers between <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 5. See <a href=\"#CNX_IntAlg_Figure_02_07_009\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_009\"><span data-type=\"media\" id=\"fs-id1167835320124\" data-alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its right and a right bracket at 5 with shading to its left. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x is less than or equal to 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its right and a right bracket at 5 with shading to its left. It illustrates that if the absolute value of x is less than or equal to 5, then negative 5 is less than or equal to x is less than or equal to 5.\" \/><\/span><\/div>\n<p id=\"fs-id1167835304662\">Now we want to look at the inequality <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fac335a8dc8987c9fabe1f34a03d384f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#92;&#103;&#101;&#32;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> Where are the numbers whose distance from zero is greater than or equal to five?<\/p>\n<p id=\"fs-id1167826994597\">Again both <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 5 are five units from zero and so are included in the solution. Numbers whose distance from zero is greater than five units would be less than <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 greater than 5 on the number line. See <a href=\"#CNX_IntAlg_Figure_02_07_010\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_010\"><span data-type=\"media\" id=\"fs-id1167834537398\" data-alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its left and a left bracket at 5 with shading to its right. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is greater than or equal to 5, then x is less than or equal to negative 5 or x is greater than or equal to 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative 5, 0, and 5 displayed. There is a right bracket at negative 5 that has shading to its left and a left bracket at 5 with shading to its right. The distance between negative 5 and 0 is given as 5 units and the distance between 5 and 0 is given as 5 units. It illustrates that if the absolute value of x is greater than or equal to 5, then x is less than or equal to negative 5 or x is greater than or equal to 5.\" \/><\/span><\/div>\n<p id=\"fs-id1167835350484\">In a more general way, we can see that if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0838202e33e3fe205fb159c4804af4c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#117;&#124;&#92;&#103;&#101;&#32;&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d8792765cc8e68bfaec9202983473b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ce62d59d9e4e513f19a592fed496d80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#92;&#108;&#101;&#32;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: -3px;\" \/> See <a href=\"#CNX_IntAlg_Figure_02_07_011\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_02_07_011\"><span data-type=\"media\" id=\"fs-id1167834340020\" data-alt=\"The figure is a number line with negative a, 0, and a displayed. There is a right bracket at negative a that has shading to its left and a left bracket at a with shading to its right. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is greater than or equal to a, then u is less than or equal to negative a or u is greater than or equal to a.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with negative a, 0, and a displayed. There is a right bracket at negative a that has shading to its left and a left bracket at a with shading to its right. The distance between negative a and 0 is given as a units and the distance between a and 0 is given as a units. It illustrates that if the absolute value of u is greater than or equal to a, then u is less than or equal to negative a or u is greater than or equal to a.\" \/><\/span><\/div>\n<p id=\"fs-id1167835511103\">This result is summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834423855\">\n<div data-type=\"title\">Absolute Value Inequalities with &gt; or \u2265<\/div>\n<p id=\"fs-id1167831880144\">For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<\/p>\n<div data-type=\"equation\" id=\"fs-id1167832055469\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d7a1c3c67942dd45acba5dff13f332c_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;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#62;&#97;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#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;&#117;&#60;&#45;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#62;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#92;&#103;&#101;&#32;&#97;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#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;&#117;&#92;&#108;&#101;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#92;&#103;&#101;&#32;&#97;&#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=\"402\" style=\"vertical-align: -15px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835326481\">\n<div data-type=\"problem\" id=\"fs-id1167835326484\">\n<p id=\"fs-id1167835216061\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c55d1f86737605658278c63987ab10a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#62;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835339041\">\n<table id=\"fs-id1167835339044\" class=\"unnumbered unstyled\" summary=\"The absolute value of x is greater than 4. Write the equivalent inequality. They are x is less than negative 4 or x is greater than 4. Graph the solution. It is a right parenthesis at negative 4 with shading to its left and a parenthesis at 4 with shading to its right. Write the solution using interval notation. It is the union of negative infinity to negative 4 within parentheses and 4 to infinity with parentheses. Check. To verify, check a value in each section of the number line showing the solution. Choose numbers such as negative 6, 0, and 7.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd05c6275d3f1ed0b5001e7652f6a52b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#62;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the equivalent inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31e672b1bafaa0423d9203fcb6bd1d88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#45;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#62;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"133\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Graph the solution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835332104\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the solution using interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d903f218f1c79a940c0a73033d721df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167835378703\">To verify, check a value in each section of the number line showing the solution. Choose numbers such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-350a72d4937e92680f68559bba291ce8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"26\" style=\"vertical-align: -4px;\" \/> 0, and 7.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832053739\" data-alt=\"The figure is a number line with a right parenthesis at negative 4 with shading to its left and a left parenthesis at 4 shading to its right. The values negative 6, 0, and 7 are marked with points. The absolute value of negative 6 is greater than negative 4 is true. It does not satisfy the absolute value of x is greater than 4. The absolute value of 0 is greater than 4 is false. It does not satisfy the absolute value of x is greater than 4. The absolute value of 7 is less than 4 is true. It does satisfy the absolute value of x is greater than 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a number line with a right parenthesis at negative 4 with shading to its left and a left parenthesis at 4 shading to its right. The values negative 6, 0, and 7 are marked with points. The absolute value of negative 6 is greater than negative 4 is true. It does not satisfy the absolute value of x is greater than 4. The absolute value of 0 is greater than 4 is false. It does not satisfy the absolute value of x is greater than 4. The absolute value of 7 is less than 4 is true. It does satisfy the absolute value of x is greater than 4.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835229280\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835334792\">\n<div data-type=\"problem\" id=\"fs-id1167835334794\">\n<p id=\"fs-id1167831239002\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6843ad579f3822d9307a693c389f02c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#62;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835311019\"><span data-type=\"media\" id=\"fs-id1167835410922\" data-alt=\"The solution is x is less than negative 2 or x is greater than 2. The number line shows an open circle at negative 2 with shading to its left and an open circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 2 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 2 or x is greater than 2. The number line shows an open circle at negative 2 with shading to its left and an open circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 2 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832006300\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835343327\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d2d20335889f7e86713685dd6ea9960_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#62;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834535359\"><span data-type=\"media\" id=\"fs-id1167835334127\" data-alt=\"The solution is x is less than negative 1 or x is greater than 1. The number line shows an open circle at negative 1 with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to negative 1 within parentheses and 1 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_306_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 1 or x is greater than 1. The number line shows an open circle at negative 1 with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to negative 1 within parentheses and 1 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835530188\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832058615\">\n<div data-type=\"problem\" id=\"fs-id1167832058617\">\n<p id=\"fs-id1167835343965\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6c4253707710046728989138a24ccb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#45;&#51;&#124;&#92;&#103;&#101;&#32;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832043290\">\n<table id=\"fs-id1167834228216\" class=\"unnumbered unstyled\" summary=\"The absolute value of the quantity 2 x minus 3 is greater than or equal to 5. Step 1 is to isolate the absolute value expression. It is isolated. Step 2 is to write the equivalent compound inequality. It is 2 x minus 3 is less than or equal to negative 5 or 2 x minus 3 is greater than or equal to 5. Step 3 is to solve the compound inequality. 2 x is less than or equal to negative 2 or 2 x is greater than or equal to 8. x is less than or equal to negative 1 or x is greater than or equal to 4. Step 4 is to graph the solution. On the number line, there is a closed circle at negative 1 with shading to its left and a closed circle at 4 with shading to its right. Step 5 is to write the solution using interval notation. It is the union of negative infinity to negative 1 with a parenthesis and a bracket and 4 to infinity within a bracket and a parenthesis.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e2df65b541fca2e580e4f17556e436e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#45;&#51;&#124;&#92;&#103;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1.<\/strong> Isolate the absolute value expression. It is isolated.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2.<\/strong> Write the equivalent compound inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3ebdefb6b99b489620f1d47f9b395c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#51;&#92;&#108;&#101;&#32;&#45;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#120;&#45;&#51;&#92;&#103;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"211\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3.<\/strong> Solve the compound inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506da6a64b29426e56a0640d3361b546_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#120;&#92;&#103;&#101;&#32;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"137\" style=\"vertical-align: -3px;\" \/><\/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-a468bba91e001e2a45c66544d3ebeb4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#103;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4.<\/strong> Graph the solution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835419666\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_021a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5.<\/strong> Write the solution using interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60c850f521060be027bbd72479ed7d3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#52;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The check is left to you.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830704068\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830704071\">\n<div data-type=\"problem\" id=\"fs-id1167826996782\">\n<p id=\"fs-id1167826996784\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3d4e6be3000d705ebb7ffd16e0eb752_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#120;&#45;&#51;&#124;&#92;&#103;&#101;&#32;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834185415\"><span data-type=\"media\" data-alt=\"The solution is x is less than or equal to negative one-half or x is greater than or equal 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to negative one-half or x is greater than or equal 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834294495\">\n<div data-type=\"problem\" id=\"fs-id1167834294497\">\n<p id=\"fs-id1167834294499\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62d1a8c3574accef4be9efe8f7fe17af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#52;&#124;&#92;&#103;&#101;&#32;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/> Graph the solution and write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834408311\"><span data-type=\"media\" id=\"fs-id1167834408314\" data-alt=\"The solution is x is less than or equal to two-thirds or x is greater than or equal 2. The number line shows a closed circle at two-thirds with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to two-thirds within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_308_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to two-thirds or x is greater than or equal 2. The number line shows a closed circle at two-thirds with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to two-thirds within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830963972\" class=\"howto\">\n<div data-type=\"title\">Solve absolute value inequalities with &gt; or \u2265.<\/div>\n<ol id=\"fs-id1167835355451\" type=\"1\" class=\"stepwise\">\n<li>Isolate the absolute value expression.<\/li>\n<li>Write the equivalent compound inequality.\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167835395542\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb8984be5bf52ac100e10f0ad10e99fc_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;&#124;&#117;&#124;&#62;&#97;&#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;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#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;&#117;&#60;&#45;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#62;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#124;&#117;&#124;&#92;&#103;&#101;&#32;&#97;&#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;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#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;&#117;&#92;&#108;&#101;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#92;&#103;&#101;&#32;&#97;&#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=\"373\" style=\"vertical-align: -15px;\" \/><\/div>\n<\/li>\n<li>Solve the compound inequality.<\/li>\n<li>Graph the solution<\/li>\n<li>Write the solution using interval notation.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167826783859\">\n<h3 data-type=\"title\">Solve Applications with Absolute Value<\/h3>\n<p id=\"fs-id1167831086715\">Absolute value inequalities are often used in the manufacturing process. An item must be made with near perfect specifications. Usually there is a certain <em data-effect=\"italics\">tolerance<\/em> of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the item is rejected.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834300306\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3a9ead597edcf037dd498e955d2739e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#99;&#116;&#117;&#97;&#108;&#45;&#105;&#100;&#101;&#97;&#108;&#125;&#124;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#108;&#101;&#114;&#97;&#110;&#99;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"192\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"example\" id=\"fs-id1167835419283\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835419285\">\n<div data-type=\"problem\" id=\"fs-id1167831933994\">\n<p id=\"fs-id1167831933996\">The ideal diameter of a rod needed for a machine is 60 mm. The actual diameter can vary from the ideal diameter by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f078c1e67a0cb7af166cbe913cbde1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#48;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/> mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832055375\">\n<div data-type=\"equation\" id=\"fs-id1167835341915\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0177be8e8b980005cf57ccad4e83521d_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;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#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;&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#97;&#99;&#116;&#117;&#97;&#108;&#32;&#109;&#101;&#97;&#115;&#117;&#114;&#101;&#109;&#101;&#110;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#97;&#110;&#32;&#97;&#98;&#115;&#111;&#108;&#117;&#116;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#114;&#101;&#115;&#115;&#32;&#116;&#104;&#105;&#115;&#32;&#115;&#105;&#116;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#99;&#116;&#117;&#97;&#108;&#45;&#105;&#100;&#101;&#97;&#108;&#125;&#124;&#92;&#108;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#108;&#101;&#114;&#97;&#110;&#99;&#101;&#125;&#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;&#124;&#120;&#45;&#54;&#48;&#124;&#92;&#108;&#101;&#32;&#48;&#46;&#48;&#55;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#97;&#115;&#32;&#97;&#32;&#99;&#111;&#109;&#112;&#111;&#117;&#110;&#100;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#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;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#48;&#46;&#48;&#55;&#53;&#92;&#108;&#101;&#32;&#120;&#45;&#54;&#48;&#92;&#108;&#101;&#32;&#48;&#46;&#48;&#55;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#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;&#53;&#57;&#46;&#57;&#50;&#53;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#54;&#48;&#46;&#48;&#55;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#110;&#115;&#119;&#101;&#114;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#101;&#115;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#100;&#105;&#97;&#109;&#101;&#116;&#101;&#114;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#114;&#111;&#100;&#32;&#99;&#97;&#110;&#32;&#98;&#101;&#32;&#98;&#101;&#116;&#119;&#101;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#57;&#46;&#57;&#50;&#53;&#32;&#109;&#109;&#32;&#97;&#110;&#100;&#32;&#54;&#48;&#46;&#48;&#55;&#53;&#32;&#109;&#109;&#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=\"145\" width=\"842\" style=\"vertical-align: -66px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835349645\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835306096\">\n<div data-type=\"problem\" id=\"fs-id1167835306098\">\n<p id=\"fs-id1167835306100\">The ideal diameter of a rod needed for a machine is 80 mm. The actual diameter can vary from the ideal diameter by 0.009 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835173711\">\n<p id=\"fs-id1167834395851\">The diameter of the rod can be between 79.991 and 80.009 mm.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835370683\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835370686\">\n<div data-type=\"problem\" id=\"fs-id1167835370688\">\n<p id=\"fs-id1167831871802\">The ideal diameter of a rod needed for a machine is 75 mm. The actual diameter can vary from the ideal diameter by 0.05 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835421096\">\n<p id=\"fs-id1167835421098\">The diameter of the rod can be between 74.95 and 75.05 mm.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835380767\" class=\"media-2\">\n<p id=\"fs-id1167835512794\">Access this online resource for additional instruction and practice with solving linear absolute value equations and inequalities.<\/p>\n<ul id=\"fs-id1167835512799\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37solvlinabsol\">Solving Linear Absolute Value Equations and Inequalities<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167834111788\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167835244451\" data-bullet-style=\"bullet\">\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\">Absolute Value Equations<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<\/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-0084010100bb74c9d4ac8e8c9f345a35_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;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#61;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#117;&#61;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#61;&#97;&#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=\"224\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Remember that an absolute value cannot be a negative number.<\/li>\n<li><strong data-effect=\"bold\">How to Solve Absolute Value Equations<\/strong>\n<ol id=\"fs-id1167832056980\" type=\"1\" class=\"stepwise\">\n<li>Isolate the absolute value expression.<\/li>\n<li>Write the equivalent equations.<\/li>\n<li>Solve each equation.<\/li>\n<li>Check each solution.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Equations with Two Absolute Values<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> For any algebraic expressions, <em data-effect=\"italics\">u<\/em> and <em data-effect=\"italics\">v<\/em>,<\/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-a825331bdc47aabca82578dabad84f4d_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;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#61;&#124;&#118;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#117;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#118;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#61;&#118;&#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=\"224\" style=\"vertical-align: -12px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Absolute Value Inequalities with<\/strong><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;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54e2b165150917469474a6d203f27e67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<\/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-ffe63633925582a6f4d3ebba938864a7_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;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#60;&#97;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#60;&#117;&#60;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#92;&#108;&#101;&#32;&#97;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#108;&#101;&#32;&#117;&#92;&#108;&#101;&#32;&#97;&#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=\"349\" style=\"vertical-align: -15px;\" \/><\/li>\n<li><strong data-effect=\"bold\">How To Solve Absolute Value Inequalities with<\/strong><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;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54e2b165150917469474a6d203f27e67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>\n<ol id=\"fs-id1167835358422\" type=\"1\" class=\"stepwise\">\n<li>Isolate the absolute value expression.<\/li>\n<li>Write the equivalent compound inequality.\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-e213ac560dc549ab2d51df63bfcce4dd_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;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#124;&#117;&#124;&#60;&#97;&#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;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#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;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#60;&#117;&#60;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#124;&#117;&#124;&#92;&#108;&#101;&#32;&#97;&#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;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#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;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#108;&#101;&#32;&#117;&#92;&#108;&#101;&#32;&#97;&#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=\"407\" style=\"vertical-align: -15px;\" \/><\/li>\n<li>Solve the compound inequality.<\/li>\n<li>Graph the solution<\/li>\n<li>Write the solution using interval notation<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Absolute Value Inequalities with<\/strong><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;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bed2a7b3a4cb8554a7ceb3c94e90a24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> For any algebraic expression, <em data-effect=\"italics\">u<\/em>, and any positive real number, <em data-effect=\"italics\">a<\/em>,<\/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-90b7007886db345926d3ccde3656c31b_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;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#62;&#97;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#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;&#117;&#60;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#62;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#124;&#117;&#124;&#92;&#103;&#101;&#32;&#97;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#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;&#117;&#92;&#108;&#101;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#92;&#103;&#101;&#32;&#97;&#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=\"388\" style=\"vertical-align: -15px;\" \/><\/li>\n<li><strong data-effect=\"bold\">How To Solve Absolute Value Inequalities with<\/strong><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;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bed2a7b3a4cb8554a7ceb3c94e90a24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"12\" style=\"vertical-align: -3px;\" \/>\n<ol id=\"fs-id1167835262429\" type=\"1\" class=\"stepwise\">\n<li>Isolate the absolute value expression.<\/li>\n<li>Write the equivalent compound inequality.\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-57851e10b5e79a0bdb2501cd7aa183b2_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;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#124;&#117;&#124;&#62;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#117;&#60;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#62;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#124;&#117;&#124;&#92;&#103;&#101;&#32;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#117;&#92;&#108;&#101;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#117;&#92;&#103;&#101;&#32;&#97;&#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=\"418\" style=\"vertical-align: -15px;\" \/><\/li>\n<li>Solve the compound inequality.<\/li>\n<li>Graph the solution<\/li>\n<li>Write the solution using interval notation<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831892771\">\n<h3 data-type=\"title\">Section Exercises<\/h3>\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834426146\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167831823780\"><strong data-effect=\"bold\">Solve Absolute Value Equations<\/strong><\/p>\n<p id=\"fs-id1167835241276\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835241280\">\n<div data-type=\"problem\" id=\"fs-id1167835241282\">\n<p id=\"fs-id1167835417039\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2eaaec082db62bf723aeb6afacc42038_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" 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-16ba2dda669ff441c3ce057ef68f9672_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#121;&#124;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" 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-bff1356215b10f5e9799486f4268133d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#122;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835244537\">\n<div data-type=\"problem\" id=\"fs-id1167830964473\">\n<p id=\"fs-id1167830964475\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21014db593ed1b908bc5ffadaef69f24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" 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-19e5e71fcc0853201cc6fe31126232a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#121;&#124;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" 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-bff1356215b10f5e9799486f4268133d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#122;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167826994174\">\n<p id=\"fs-id1167835339834\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-720df1ccc86adea77a6e99faf5ee8b90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#44;&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"107\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span> no solution <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e381c273c9f7ca5b93029ee2e8bab16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832051939\">\n<div data-type=\"problem\" id=\"fs-id1167835287345\">\n<p id=\"fs-id1167835287347\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-438c3a5ae79680945266ef1422fe8294_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" 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-4318ad7611194351850e0173bdb17d1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#121;&#124;&#61;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" 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-bff1356215b10f5e9799486f4268133d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#122;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834308101\">\n<div data-type=\"problem\" id=\"fs-id1167834308103\">\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb0e2a9a02d97775962ecc83d44b502e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" 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-0f2b42edf679f807e2b10283a244be5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#121;&#124;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" 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-bff1356215b10f5e9799486f4268133d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#122;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167831920204\">\n<p id=\"fs-id1167831920206\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57e451c55f060ffc4c47c8f35bf7d451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;&#120;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"107\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span> no solution <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e381c273c9f7ca5b93029ee2e8bab16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835379168\">\n<div data-type=\"problem\" id=\"fs-id1167835379171\">\n<p id=\"fs-id1167835420231\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b833f907ce5942be6e5fb3cc40dab3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#45;&#51;&#124;&#45;&#52;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826938162\">\n<div data-type=\"problem\" id=\"fs-id1167826938164\">\n<p id=\"fs-id1167826938166\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ffc0e7aaeea73342b42afd4562cd76e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#120;&#45;&#49;&#124;&#45;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834535598\">\n<p id=\"fs-id1167834535600\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4aa3be300036e817ade4a922ff4353bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834134944\">\n<div data-type=\"problem\" id=\"fs-id1167835422586\">\n<p id=\"fs-id1167835422589\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5f4eb79ecf4fc464cf210ffb56af574_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#52;&#124;&#43;&#53;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834227506\">\n<div data-type=\"problem\" id=\"fs-id1167832099513\">\n<p id=\"fs-id1167832099515\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f5ffa88bc2eb680e18b27fc106d4dc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#120;&#43;&#55;&#124;&#43;&#50;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831066058\">\n<p id=\"fs-id1167835357426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a908428a99891c8cf56ee6259dc7d967_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835355673\">\n<div data-type=\"problem\" id=\"fs-id1167835355675\">\n<p id=\"fs-id1167835355677\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d60d19d57a21fd1716683cb8fe39515e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#124;&#120;&#45;&#49;&#124;&#43;&#50;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831896629\">\n<div data-type=\"problem\" id=\"fs-id1167835304820\">\n<p id=\"fs-id1167835304822\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6cdecefb5270eec165543993203885ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#124;&#120;&#45;&#52;&#124;&#43;&#50;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832031176\">\n<p id=\"fs-id1167832074584\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccc17beef91f6292ccd968d5367aa683_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#55;&#44;&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834179728\">\n<p id=\"fs-id1167826978921\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee0ddf51079ccfd6b69bce85d6a01ef6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#124;&#52;&#120;&#45;&#53;&#124;&#45;&#52;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826874383\">\n<div data-type=\"problem\" id=\"fs-id1167832075587\">\n<p id=\"fs-id1167832075589\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e08c8efbecdbef3d0ff9ae3cc1f5133_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#124;&#120;&#43;&#50;&#124;&#45;&#53;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835479692\">\n<p id=\"fs-id1167835479694\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efb243dd61b79c0636f31cbdbb8a75ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#44;&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831892748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8294302b40904c68b66260c9ec89563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#124;&#120;&#45;&#51;&#124;&#43;&#56;&#61;&#45;&#52;\" 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-id1167835351901\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835351906\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed6850f6092af98231f8265edd6feb73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#124;&#120;&#45;&#52;&#124;&#43;&#52;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832190069\">\n<p id=\"fs-id1167830694084\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccc17beef91f6292ccd968d5367aa683_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#55;&#44;&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834064105\">\n<div data-type=\"problem\" id=\"fs-id1167834064107\">\n<p id=\"fs-id1167832041919\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d9163ad761399c3b6eda35ef4c4d3df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#51;&#124;&#43;&#55;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832075528\">\n<div data-type=\"problem\" id=\"fs-id1167832075530\">\n<p id=\"fs-id1167832075532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dac87709a6ba1310c73fe83761ab5177_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#120;&#45;&#50;&#124;&#43;&#53;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834279894\">\n<p id=\"fs-id1167834279896\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834279901\">\n<div data-type=\"problem\" id=\"fs-id1167834431365\">\n<p id=\"fs-id1167834431367\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85829cfd33923513f937f2f28531aad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#53;&#124;&#43;&#52;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835531744\">\n<div data-type=\"problem\" id=\"fs-id1167835531746\">\n<p id=\"fs-id1167835531748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98db6494bc1d11f588eab4ddbff4f709_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#43;&#51;&#124;&#43;&#51;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835420978\">\n<p id=\"fs-id1167835420980\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831883320\">\n<div data-type=\"problem\" id=\"fs-id1167831883322\">\n<p id=\"fs-id1167831883325\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0a4b119a54b32e8e14f54e7ca19d209_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#50;&#124;&#61;&#124;&#50;&#120;&#45;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826987283\">\n<div data-type=\"problem\" id=\"fs-id1167834156944\">\n<p id=\"fs-id1167834156946\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-439f907e9f14c4d6f85af99f9edd6298_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#120;&#43;&#51;&#124;&#61;&#124;&#50;&#120;&#43;&#49;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826788431\">\n<p id=\"fs-id1167826788434\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdb639f8872d7fa9cdba1281f4eb5c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830770225\">\n<div data-type=\"problem\" id=\"fs-id1167830770227\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d2b5b8ef4630065a6133ca4d8fb3404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#54;&#120;&#45;&#53;&#124;&#61;&#124;&#50;&#120;&#43;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834098374\">\n<div data-type=\"problem\" id=\"fs-id1167834098376\">\n<p id=\"fs-id1167834098378\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aeea8c5a31b72b455d1ded1828540300_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#54;&#45;&#120;&#124;&#61;&#124;&#51;&#45;&#50;&#120;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834534548\">\n<p id=\"fs-id1167834534550\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec0458fd8edba53ed9e3600670d105c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#44;&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835312257\"><strong data-effect=\"bold\">Solve Absolute Value Inequalities with \u201cless than\u201d<\/strong><\/p>\n<p id=\"fs-id1167835312263\">In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834432808\">\n<div data-type=\"problem\" id=\"fs-id1167834432810\">\n<p id=\"fs-id1167834432812\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a71836e3d6a97c8f838aa82e39736517_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835417913\">\n<div data-type=\"problem\" id=\"fs-id1167835417915\">\n<p id=\"fs-id1167835417917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eea925f4887f9b8e9efb33fd4207b679_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#60;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826995902\"><span data-type=\"media\" id=\"fs-id1167828420273\" data-alt=\"The solution is negative 1 is less than x which is less than 1. The number line shows an open circle at negative 1, an open circle at 1, and shading between the circles. The interval notation is negative 1 to 1 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_310_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 1 is less than x which is less than 1. The number line shows an open circle at negative 1, an open circle at 1, and shading between the circles. The interval notation is negative 1 to 1 within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830697917\">\n<div data-type=\"problem\" id=\"fs-id1167830697919\">\n<p id=\"fs-id1167830697921\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ae5d20c1fa5416e0afdda592d92557c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#92;&#108;&#101;&#32;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831881549\">\n<div data-type=\"problem\" id=\"fs-id1167831881551\">\n<p id=\"fs-id1167831881553\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ed9923d8e80ddded9bf65b407825aa0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831970260\"><span data-type=\"media\" id=\"fs-id1167831970263\" data-alt=\"The solution is negative 3 is less than or equal to x which is less than or equal to 3. The number line shows a closed circle at negative 3, a closed circle at 3, and shading between the circles. The interval notation is negative 3 to 3 within brackets.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_312_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 3 is less than or equal to x which is less than or equal to 3. The number line shows a closed circle at negative 3, a closed circle at 3, and shading between the circles. The interval notation is negative 3 to 3 within brackets.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834430869\">\n<div data-type=\"problem\" id=\"fs-id1167834430871\">\n<p id=\"fs-id1167834430873\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-881838a19cd7cd2544ed79eb2ccd8b41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#51;&#124;&#92;&#108;&#101;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830961942\">\n<div data-type=\"problem\" id=\"fs-id1167830961944\">\n<p id=\"fs-id1167830961946\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d95407653924d8de57efed6d87e34d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#45;&#53;&#124;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835304725\"><span data-type=\"media\" id=\"fs-id1167835304728\" data-alt=\"The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading between the circles. The interval notation is 1 to 4 within brackets.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading between the circles. The interval notation is 1 to 4 within brackets.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835489076\">\n<div data-type=\"problem\" id=\"fs-id1167835489078\">\n<p id=\"fs-id1167835489080\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5820f2e5bc0684cd478235bcb013243a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#43;&#51;&#124;&#43;&#53;&#60;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920356\">\n<div data-type=\"problem\" id=\"fs-id1167831920358\">\n<p id=\"fs-id1167831920361\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1add90e752235037c0c07adede929cd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#55;&#124;&#43;&#51;&#60;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830865707\"><span data-type=\"media\" id=\"fs-id1167830865710\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832053707\">\n<div data-type=\"problem\" id=\"fs-id1167832053709\">\n<p id=\"fs-id1167832053711\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c81f7d5211ce8ec47610015ad2c0772_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#120;&#45;&#51;&#124;&#60;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831824995\">\n<div data-type=\"problem\" id=\"fs-id1167831824998\">\n<p id=\"fs-id1167831825000\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-945956595bc7c7058e368c51ac1a3609_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#54;&#120;&#45;&#53;&#124;&#60;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835319170\"><span data-type=\"media\" id=\"fs-id1167835319173\" data-alt=\"The solution is negative one-third is less than x which is less than 2. The number line shows an open circle at negative one-half, an open circle at 2, and shading between the circles. The interval notation is negative one-third to 2 within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative one-third is less than x which is less than 2. The number line shows an open circle at negative one-half, an open circle at 2, and shading between the circles. The interval notation is negative one-third to 2 within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835327455\">\n<div data-type=\"problem\" id=\"fs-id1167835352659\">\n<p id=\"fs-id1167835352661\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-314a90907fe963681e3dd24351ead22c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#45;&#52;&#124;&#92;&#108;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831894282\">\n<div data-type=\"problem\" id=\"fs-id1167831894284\">\n<p id=\"fs-id1167831894286\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef50e46a395dac5c85243f776270266f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#53;&#120;&#43;&#49;&#124;&#92;&#108;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835330461\"><span data-type=\"media\" id=\"fs-id1167835330464\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_320_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167835498615\"><strong data-effect=\"bold\">Solve Absolute Value Inequalities with \u201cgreater than\u201d<\/strong><\/p>\n<p id=\"fs-id1167835498622\">In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834536113\">\n<div data-type=\"problem\" id=\"fs-id1167834536115\">\n<p id=\"fs-id1167834536117\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-560f064a18fbcea9b147ddcb2a87301c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835589715\">\n<div data-type=\"problem\" id=\"fs-id1167835589717\">\n<p id=\"fs-id1167832015761\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0961bd5dd5da1a564a8f65f0e3b3986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#62;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830703711\"><span data-type=\"media\" id=\"fs-id1167830703714\" data-alt=\"The solution is x is less than negative 6 or x is greater than 6. The number line shows an open circle at negative 6 with shading to its left and an open circle at 6 with shading to its right. The interval notation is the union of negative infinity to negative 6 within parentheses and 6 to infinity within parentheses\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_322_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 6 or x is greater than 6. The number line shows an open circle at negative 6 with shading to its left and an open circle at 6 with shading to its right. The interval notation is the union of negative infinity to negative 6 within parentheses and 6 to infinity within parentheses\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835235781\">\n<div data-type=\"problem\" id=\"fs-id1167835235783\">\n<p id=\"fs-id1167835235785\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb21df9ef3ed1d8c966f4c88056b3406_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#92;&#103;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835479279\">\n<div data-type=\"problem\" id=\"fs-id1167835479282\">\n<p id=\"fs-id1167835479284\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b2712c1fec85255d7533ba91cae6c5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#92;&#103;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832058040\"><span data-type=\"media\" id=\"fs-id1167832058043\" data-alt=\"The solution is x is less than negative 5 or x is greater than 5. The number line shows an open circle at negative 5 with shading to its left and an open circle at 5 with shading to its right. The interval notation is the union of negative infinity to negative 5 within parentheses and 5 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_324_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 5 or x is greater than 5. The number line shows an open circle at negative 5 with shading to its left and an open circle at 5 with shading to its right. The interval notation is the union of negative infinity to negative 5 within parentheses and 5 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826782509\">\n<div data-type=\"problem\" id=\"fs-id1167834300884\">\n<p id=\"fs-id1167834300886\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60071d11f4454bd88e10f7c5e0b2b352_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#56;&#124;&#62;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830693625\">\n<div data-type=\"problem\" id=\"fs-id1167830693627\">\n<p id=\"fs-id1167830693629\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9283f32b3d73e63d26f44983b964136f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#45;&#53;&#124;&#62;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831148862\"><span data-type=\"media\" id=\"fs-id1167831148866\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_326_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835214240\">\n<div data-type=\"problem\" id=\"fs-id1167835214242\">\n<p id=\"fs-id1167835214244\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a0159f0212c609c5dae600950fe7f84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#50;&#124;&#62;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832065991\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1820bc696f2e965be2a4d8edc4700c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#45;&#49;&#124;&#62;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828434603\"><span data-type=\"media\" id=\"fs-id1167828434606\" data-alt=\"The solution is x is less than negative 2 or x is greater than 3. The number line shows an open circle at negative 2 with shading to its left and an open circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 3 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_328_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 2 or x is greater than 3. The number line shows an open circle at negative 2 with shading to its left and an open circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 3 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167828434610\">\n<p id=\"fs-id1167835375320\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-159e9525337805b8fec8c8b356f56635_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#43;&#51;&#124;&#92;&#103;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835301470\">\n<div data-type=\"problem\" id=\"fs-id1167834431326\">\n<p id=\"fs-id1167834431328\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e0cd4109eb7172b37182f0cafa27e87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#45;&#55;&#124;&#92;&#103;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832087111\"><span data-type=\"media\" id=\"fs-id1167832087114\" data-alt=\"The solution is x is less than or equal to 6 or x is greater than or equal to 8. The number line shows a closed circle at 6 with shading to its left and a closed circle at 8 with shading to its right. The interval notation is the union of negative infinity to 6 within parenthesis and a bracket and 8 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_330_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to 6 or x is greater than or equal to 8. The number line shows a closed circle at 6 with shading to its left and a closed circle at 8 with shading to its right. The interval notation is the union of negative infinity to 6 within parenthesis and a bracket and 8 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834377054\">\n<div data-type=\"problem\" id=\"fs-id1167834377056\">\n<p id=\"fs-id1167834377058\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4afb8b8693a1bfe4792ca65527496bb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#124;&#120;&#124;&#43;&#52;&#92;&#103;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832015454\">\n<div data-type=\"problem\" id=\"fs-id1167832015456\">\n<p id=\"fs-id1167832015458\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ffb2b831d763ad0bad7e6b659cc2cef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#124;&#120;&#124;&#43;&#54;&#92;&#103;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834531065\"><span data-type=\"media\" id=\"fs-id1167832074711\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_332_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167834432597\">In the following exercises, solve. For each inequality, also graph the solution and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834432602\">\n<div data-type=\"problem\" id=\"fs-id1167834432604\">\n<p id=\"fs-id1167834432606\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d78b35e93a0177085b7e27fc8104919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#124;&#120;&#43;&#54;&#124;&#43;&#52;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832043932\">\n<div data-type=\"problem\" id=\"fs-id1167832043934\">\n<p id=\"fs-id1167835423440\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e561f015dea75a9871db1c654690033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#52;&#124;&#92;&#103;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834534823\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-663fb4133c6e3b475fe607d3225c4377_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834192354\">\n<div data-type=\"problem\" id=\"fs-id1167834192356\">\n<p id=\"fs-id1167832043936\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d2b5b8ef4630065a6133ca4d8fb3404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#54;&#120;&#45;&#53;&#124;&#61;&#124;&#50;&#120;&#43;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835375364\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167830693540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13bc558aa493ba796f2fc8f9c157a263_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#120;&#45;&#51;&#124;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831883636\">\n<p id=\"fs-id1167831883638\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51589fb037893f5de698bafa5ebf4a5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835352698\">\n<div data-type=\"problem\" id=\"fs-id1167830693538\">\n<p id=\"fs-id1167835375368\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-567d598a5bacde13390b60d98f9c3835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#45;&#53;&#124;&#43;&#50;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834506132\">\n<div data-type=\"problem\" id=\"fs-id1167834506134\">\n<p id=\"fs-id1167834506136\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e08501898baaad028629efb531fb5e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#43;&#49;&#124;&#45;&#51;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832153529\">\n<p id=\"fs-id1167832153531\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3fdd883a02cd2311f90c0523ad57c15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"114\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834120820\">\n<div data-type=\"problem\" id=\"fs-id1167834120822\">\n<p id=\"fs-id1167834120824\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6feed948ac3c0ccfa8575ff301738d7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#55;&#120;&#43;&#50;&#124;&#43;&#56;&#60;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832067076\">\n<div data-type=\"problem\" id=\"fs-id1167832067078\">\n<p id=\"fs-id1167832067080\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3313ba45a4677f7142d71c2709179c81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#124;&#50;&#120;&#45;&#49;&#124;&#45;&#51;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835303010\">\n<p id=\"fs-id1167835303012\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d887eb440cf253f97d9a6a8975bdcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"109\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831911212\">\n<div data-type=\"problem\" id=\"fs-id1167831911214\">\n<p id=\"fs-id1167831884002\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-477229bc28d54cc78f6dd330563c6b4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#45;&#55;&#124;&#62;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831883998\">\n<div data-type=\"problem\" id=\"fs-id1167831884000\">\n<p id=\"fs-id1167831911216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec28f502c910232d505ad77b2e4689f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#56;&#45;&#120;&#124;&#61;&#124;&#52;&#45;&#51;&#120;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835360752\"><span data-type=\"media\" id=\"fs-id1167835360755\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_336_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167831919898\"><strong data-effect=\"bold\">Solve Applications with Absolute Value<\/strong><\/p>\n<p id=\"fs-id1167831919904\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830961506\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830961508\">\n<p id=\"fs-id1167830961510\">A chicken farm ideally produces 200,000 eggs per day. But this total can vary by as much as 25,000 eggs. What is the maximum and minimum expected production at the farm?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835534335\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835534337\">\n<p id=\"fs-id1167835534339\">An organic juice bottler ideally produces 215,000 bottle per day. But this total can vary by as much as 7,500 bottles. What is the maximum and minimum expected production at the bottling company?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835534345\">\n<p id=\"fs-id1167835287293\">The minimum to maximum expected production is 207,500 to 2,225,000 bottles<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835287299\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835287301\">\n<p id=\"fs-id1167835287303\">In order to insure compliance with the law, Miguel routinely overshoots the weight of his tortillas by 0.5 gram. He just received a report that told him that he could be losing as much as ?100,000 per year using this practice. He now plans to buy new equipment that guarantees the thickness of the tortilla within 0.005 inches. If the ideal thickness of the tortilla is 0.04 inches, what thickness of tortillas will be guaranteed?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835368288\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835368290\">\n<p id=\"fs-id1167835368292\">At Lilly\u2019s Bakery, the ideal weight of a loaf of bread is 24 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835217804\">\n<p id=\"fs-id1167835217806\">The acceptable weight is 22.5 to 25.5 ounces.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835217813\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167835334267\">\n<div data-type=\"problem\" id=\"fs-id1167835334269\">\n<p id=\"fs-id1167835334271\">Write a graphical description of the absolute value of a number.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832153475\">\n<div data-type=\"problem\" id=\"fs-id1167832153477\">\n<p id=\"fs-id1167832153479\">In your own words, explain how to solve the absolute value inequality, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b7bbdca0d94057d9729825c72423790_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#50;&#124;&#92;&#103;&#101;&#32;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835337602\">\n<p id=\"fs-id1167835337604\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835337611\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167834185038\"><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-id1167834185046\" data-alt=\"This table has four columns and five rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve absolute value equations. In row 3, the I can was solve absolute value inequalities with \u201cless than.\u201d In row 4, the I can was solve absolute value inequalities with \u201cgreater than.\u201d In row 5, the I can was solve applications with absolute value.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and five rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve absolute value equations. In row 3, the I can was solve absolute value inequalities with \u201cless than.\u201d In row 4, the I can was solve absolute value inequalities with \u201cgreater than.\u201d In row 5, the I can was solve applications with absolute value.\" \/><\/span><\/p>\n<p id=\"fs-id1167826849544\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\n<\/div>\n<\/div>\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167834523724\">\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835350223\">\n<h4 data-type=\"title\"><a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c\" class=\"target-chapter\">Use a General Strategy to Solve Linear Equations<\/a><\/h4>\n<p id=\"fs-id1167835350234\"><strong data-effect=\"bold\">Solve Equations Using the General Strategy for Solving Linear Equations<\/strong><\/p>\n<p id=\"fs-id1167831837572\">In the following exercises, determine whether each number is a solution to the equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831837576\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831837578\">\n<p id=\"fs-id1167831837580\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72fce6d583ea9875a4ef9c90c9bf5300_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#120;&#45;&#49;&#61;&#53;&#120;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"151\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835529828\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835529830\">\n<p id=\"fs-id1167835529832\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e3f8452737d6e8d7e2885e9819763ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#110;&#43;&#53;&#61;&#56;&#110;&#44;&#110;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"180\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831117374\">\n<p id=\"fs-id1167835339861\">no<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835339866\">In the following exercises, solve each linear equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835339870\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835339872\">\n<p id=\"fs-id1167835339874\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd3f06e2e3ea36574854a9f62961c019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835515236\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835515239\">\n<p id=\"fs-id1167835515241\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cff8ead5eaaa93a410f368366dba235a_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;&#115;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826874285\">\n<p id=\"fs-id1167826874287\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7db7a04b9122e6bdac12ae26aa1698f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#45;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"63\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835379630\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835379632\">\n<p id=\"fs-id1167835379634\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72918154efe35812134d927c54733e87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#51;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997377\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826997379\">\n<p id=\"fs-id1167826997381\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5e6f10f23a2866c2ec5aeaf5f1fa647_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#109;&#43;&#50;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#109;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"159\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831025141\">\n<p id=\"fs-id1167831025143\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca2b2e2596b41957ac11de0d019a61b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"71\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834063690\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834063692\">\n<p id=\"fs-id1167834063694\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87af26d7ec906c328864a4455f39a140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#53;&#121;&#43;&#48;&#46;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#54;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"160\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835354466\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835354468\">\n<p id=\"fs-id1167835354470\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f92e0aaa198949fa5b5e66d899c176c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830697964\">\n<p id=\"fs-id1167830697966\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-184e8df7f1cb1b19297c51d48731aeb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834531012\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834531015\">\n<p id=\"fs-id1167834531017\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8669cfb4fa37ae2f64e65c733e331ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"161\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826851797\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835596119\">\n<p id=\"fs-id1167835596122\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32de4e3cc4d1beda525be118288b76e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#53;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#120;&#45;&#53;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"307\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834183392\">\n<p id=\"fs-id1167834183394\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad143a0d979362a51b48a48c9ca9f59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834239022\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834239024\">\n<p id=\"fs-id1167834239026\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e85c1bd630cff988e1cb15f90e800973_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#110;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#110;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#48;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#110;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"274\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835417519\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835417521\">\n<p id=\"fs-id1167835417523\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77112f5ab4d9be069d4cadcbcdfbfe8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#49;&#54;&#43;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#107;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#52;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"287\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830925332\">\n<p id=\"fs-id1167830925334\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-014f1dc607354b114d0953f0e5d218c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831832129\"><strong data-effect=\"bold\">Classify Equations<\/strong><\/p>\n<p id=\"fs-id1167831832135\">In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831832139\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831832141\">\n<p id=\"fs-id1167831832144\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-465886ed927a0366fc831e66d8505907_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;&#121;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#50;&#121;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"298\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835347858\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835347860\">\n<p id=\"fs-id1167835347862\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b39380a49c5412ba4e1cc1e976046890_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#117;&#43;&#51;&#50;&#61;&#49;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#117;&#43;&#50;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"262\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830838231\">\n<p id=\"fs-id1167834064014\">contradiction; no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834064019\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834064021\">\n<p id=\"fs-id1167834064023\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-780373b0bf008cd9af60e500d83a6dd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#109;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#109;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"211\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834120288\"><strong data-effect=\"bold\">Solve Equations with Fraction or Decimal Coefficients<\/strong><\/p>\n<p id=\"fs-id1167831839294\">In the following exercises, solve each equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831839297\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831839299\">\n<p id=\"fs-id1167831839301\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f33fae974205ff0ae2ab5d5c0199084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#110;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"98\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835510383\">\n<p id=\"fs-id1167835510385\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b4c6cd9d27ba344abe355a47d378bb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835513119\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835513121\">\n<p id=\"fs-id1167835513124\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-245dc187c0165086d5cbb4a2683cf6cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#97;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#97;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"125\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835346782\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835346784\">\n<p id=\"fs-id1167835346786\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59f21953f4716c97ea2621fd4cbfbda4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#43;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"166\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835187809\">\n<p id=\"fs-id1167835187811\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97a4b3b63464d180f2dadc8b9d008409_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835355344\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835355346\">\n<p id=\"fs-id1167835355348\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4156892d006b66e9982b0dfb659d7ea4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#121;&#45;&#49;&#125;&#123;&#51;&#125;&#43;&#52;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#56;&#121;&#43;&#52;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"133\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834065304\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834065306\">\n<p id=\"fs-id1167834065308\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43bc6722d333d87eb6663128d16d64af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#56;&#120;&#45;&#48;&#46;&#51;&#61;&#48;&#46;&#55;&#120;&#43;&#48;&#46;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"177\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834209097\">\n<p id=\"fs-id1167834209100\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ddab230605c435eb8b7408a736d3e77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834433422\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834433424\">\n<p id=\"fs-id1167834433426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6f4b8fbd72383584a0ffb661cd1b474_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#49;&#48;&#100;&#43;&#48;&#46;&#48;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#46;&#48;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"205\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835267992\">\n<h4 data-type=\"title\"><a href=\"\/contents\/37489cba-b108-41fd-88b1-ab568fcea766\" class=\"target-chapter\">Use a Problem-Solving Strategy<\/a><\/h4>\n<p id=\"fs-id1167835340717\"><strong data-effect=\"bold\">Use a Problem Solving Strategy for Word Problems<\/strong><\/p>\n<p id=\"fs-id1167835340723\">In the following exercises, solve using the problem solving strategy for word problems.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835340726\">\n<div data-type=\"problem\" id=\"fs-id1167835340728\">\n<p id=\"fs-id1167835340730\">Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826849734\">\n<p id=\"fs-id1167826849736\">There are 116 people.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826849741\">\n<div data-type=\"problem\" id=\"fs-id1167826849744\">\n<p id=\"fs-id1167826849746\">There are nine saxophone players in the band. The number of saxophone players is one less than twice the number of tuba players. Find the number of tuba players.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832138843\"><strong data-effect=\"bold\">Solve Number Word Problems<\/strong><\/p>\n<p id=\"fs-id1167835319306\">In the following exercises, solve each number word problem.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835319310\">\n<div data-type=\"problem\" id=\"fs-id1167835319312\">\n<p id=\"fs-id1167835319314\">The sum of a number and three is forty-one. Find the number.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835319318\">\n<p id=\"fs-id1167835320424\">38<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835320429\">\n<div data-type=\"problem\" id=\"fs-id1167835320431\">\n<p id=\"fs-id1167835320433\">One number is nine less than another. Their sum is negative twenty-seven. Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835180579\">\n<div data-type=\"problem\" id=\"fs-id1167835180581\">\n<p id=\"fs-id1167831186040\">One number is two more than four times another. Their sum is negative thirteen. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831186044\">\n<p id=\"fs-id1167831186047\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79455350666f88cc96c8d695ed7b7c02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"62\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834131070\">\n<div data-type=\"problem\" id=\"fs-id1167834131073\">\n<p id=\"fs-id1167834131075\">The sum of two consecutive integers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6997e1446088e9f17765551a0cbabda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"44\" style=\"vertical-align: -1px;\" \/> Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831892866\">\n<div data-type=\"problem\" id=\"fs-id1167831892868\">\n<p id=\"fs-id1167831892870\">Find three consecutive even integers whose sum is 234.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831892874\">\n<p id=\"fs-id1167835410247\">76, 78, 80<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835410252\">\n<div data-type=\"problem\" id=\"fs-id1167835410254\">\n<p id=\"fs-id1167835410257\">Find three consecutive odd integers whose sum is 51.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835623245\">\n<div data-type=\"problem\" id=\"fs-id1167835623247\">\n<p id=\"fs-id1167835623249\">Koji has ?5,502 in his savings account. This is ?30 less than six times the amount in his checking account. How much money does Koji have in his checking account?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832053195\">\n<p id=\"fs-id1167832053197\">?922<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832053202\"><strong data-effect=\"bold\">Solve Percent Applications<\/strong><\/p>\n<p id=\"fs-id1167832053207\">In the following exercises, translate and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834324692\">\n<div data-type=\"problem\" id=\"fs-id1167834324694\">\n<p id=\"fs-id1167834324696\">What number is 67% of 250?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835216540\">\n<div data-type=\"problem\" id=\"fs-id1167835216543\">\n<p id=\"fs-id1167835216545\">12.5% of what number is 20?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835216555\">\n<p id=\"fs-id1167834536130\">160<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834536135\">\n<div data-type=\"problem\" id=\"fs-id1167834536137\">\n<p id=\"fs-id1167834536139\">What percent of 125 is 150?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834060137\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834060140\">\n<div data-type=\"problem\" id=\"fs-id1167834060142\">\n<p id=\"fs-id1167834060145\">The bill for Dino\u2019s lunch was ?19.45. He wanted to leave 20% of the total bill as a tip. How much should the tip be?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830962261\">\n<p id=\"fs-id1167830962263\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5a3d39b4f5bdceb828441a656356fd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#51;&#46;&#56;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835339243\">\n<div data-type=\"problem\" id=\"fs-id1167835339245\">\n<p id=\"fs-id1167835339247\">Dolores bought a crib on sale for ?350. The sale price was 40% of the original price. What was the original price of the crib?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830898712\">\n<div data-type=\"problem\" id=\"fs-id1167830898714\">\n<p id=\"fs-id1167830898716\">Jaden earns ?2,680 per month. He pays ?938 a month for rent. What percent of his monthly pay goes to rent?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830898722\">\n<p id=\"fs-id1167830898724\">35%<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827966896\">\n<div data-type=\"problem\" id=\"fs-id1167827966898\">\n<p id=\"fs-id1167827966900\">Angel received a raise in his annual salary from ?55,400 to ?56,785. Find the percent change.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835511627\">\n<div data-type=\"problem\" id=\"fs-id1167835511630\">\n<p id=\"fs-id1167835511632\">Rowena\u2019s monthly gasoline bill dropped from ?83.75 last month to ?56.95 this month. Find the percent change.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834403054\">\n<p id=\"fs-id1167834403057\">32%<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834403062\">\n<div data-type=\"problem\" id=\"fs-id1167834403064\">\n<p id=\"fs-id1167835379835\">Emmett bought a pair of shoes on sale at 40% off from an original price of ?138. Find <span class=\"token\">\u24d0<\/span> the amount of discount and <span class=\"token\">\u24d1<\/span> the sale price.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835357750\">\n<div data-type=\"problem\" id=\"fs-id1167835357752\">\n<p id=\"fs-id1167835357754\">Lacey bought a pair of boots on sale for ?95. The original price of the boots was ?200. Find <span class=\"token\">\u24d0<\/span> the amount of discount and <span class=\"token\">\u24d1<\/span> the discount rate. (Round to the nearest tenth of a percent, if needed.)<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834182448\">\n<p id=\"fs-id1167834182450\"><span class=\"token\">\u24d0<\/span> ?105 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0c753e00ea2b35f8a9c237d6cedac33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#50;&#46;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#37;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832115942\">\n<div data-type=\"problem\" id=\"fs-id1167832115944\">\n<p id=\"fs-id1167832115946\">Nga and Lauren bought a chest at a flea market for ?50. They re-finished it and then added a 350% mark-up. Find <span class=\"token\">\u24d0<\/span> the amount of the mark-up and <span class=\"token\">\u24d1<\/span> the list price.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826802339\"><strong data-effect=\"bold\">Solve Simple Interest Applications<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"fs-id1167826802346\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826802349\">\n<div data-type=\"problem\" id=\"fs-id1167826802352\">\n<p id=\"fs-id1167834464242\">Winston deposited ?3,294 in a bank account with interest rate 2.6% How much interest was earned in five years?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834464247\">\n<p id=\"fs-id1167834464249\">?428.22<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834464255\">\n<div data-type=\"problem\" id=\"fs-id1167834464257\">\n<p id=\"fs-id1167835358219\">Moira borrowed ?4,500 from her grandfather to pay for her first year of college. Three years later, she repaid the ?4,500 plus ?243 interest. What was the rate of interest?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834061700\">\n<div data-type=\"problem\" id=\"fs-id1167834061702\">\n<p id=\"fs-id1167834061704\">Jaime\u2019s refrigerator loan statement said he would pay ?1,026 in interest for a four-year loan at 13.5%. How much did Jaime borrow to buy the refrigerator?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835321537\">\n<p id=\"fs-id1167835321539\">?1,900<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835321545\">\n<h4 data-type=\"title\"><a href=\"\/contents\/b03538a1-8a7b-4158-a68b-e0e8a24c9fd4\" class=\"target-chapter\">Solve a formula for a Specific Variable<\/a><\/h4>\n<p id=\"fs-id1167835333094\"><strong data-effect=\"bold\">Solve a Formula for a Specific Variable<\/strong><\/p>\n<p id=\"fs-id1167835333100\">In the following exercises, solve the formula for the specified variable.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835333103\">\n<div data-type=\"problem\" id=\"fs-id1167835333105\">\n<p id=\"fs-id1167834111808\">Solve the formula<\/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-70441d53f3f0f6969e5be735ccb8b3f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#76;&#87;&#72;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"85\" style=\"vertical-align: 0px;\" \/> for <em data-effect=\"italics\">L<\/em>.<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835509163\">\n<div data-type=\"problem\" id=\"fs-id1167828396222\">\n<p id=\"fs-id1167828396224\">Solve the formula<\/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-5fdad0cf7dcfb87f40734deb99647d1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#123;&#100;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"81\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93e860f227b56edaf8e3e95498cee86e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#100;&#125;&#95;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835341255\">\n<p id=\"fs-id1167835341258\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bc27dec206437045101ba1bb0030c12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#100;&#125;&#95;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#65;&#125;&#123;&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"60\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826993960\">\n<div data-type=\"problem\" id=\"fs-id1167826993962\">\n<p id=\"fs-id1167826993964\">Solve the formula<\/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-944460697cc806861b0c6e0213feedd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#52;&#56;&#116;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#97;&#123;&#116;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"112\" style=\"vertical-align: -6px;\" \/> for <em data-effect=\"italics\">t<\/em>.<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834225907\">\n<div data-type=\"problem\" id=\"fs-id1167834225909\">\n<p id=\"fs-id1167835595718\">Solve the formula<\/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-efedfb1adc9ce250cfc862e4fb07e730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#51;&#121;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/> for <em data-effect=\"italics\">y<\/em>.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826967385\">\n<p id=\"fs-id1167826967387\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-429092098228ff599523155069a1271e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#125;&#123;&#51;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835489285\"><strong data-effect=\"bold\">Use Formulas to Solve Geometry Applications<\/strong><\/p>\n<p id=\"fs-id1167835489291\">In the following exercises, solve using a geometry formula.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834516380\">\n<div data-type=\"problem\" id=\"fs-id1167834516382\">\n<p id=\"fs-id1167834516385\">What is the height of a triangle with area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e22ed61a7065f006b6a992ba8876c9dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#55;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/> square meters and base 9 meters?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834376462\">\n<div data-type=\"problem\" id=\"fs-id1167834376464\">\n<p id=\"fs-id1167834376466\">The measure of the smallest angle in a right triangle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e104257ca81ccb9329a0b9fa023fdba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/> less than the measure of the next larger angle. Find the measures of all three angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831911722\">\n<p id=\"fs-id1167831911724\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd84cf4e58aa88e0b6d670d93a0109a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#50;&#46;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;&#44;&#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;&#54;&#55;&#46;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;&#44;&#57;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p>The perimeter of a triangle is 97 feet. One side of the triangle is eleven feet more than the smallest side. The third side is six feet more than twice the smallest side. Find the lengths of all sides.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831086741\">\n<div data-type=\"problem\" id=\"fs-id1167831086743\">\n<p id=\"fs-id1167831086746\">Find the length of the hypotenuse.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831086749\" data-alt=\"The figure is a right triangle with a base of 10 units and a height of 24 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a right triangle with a base of 10 units and a height of 24 units.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834438989\">\n<p id=\"fs-id1167834438991\">26<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831888145\">\n<div data-type=\"problem\" id=\"fs-id1167831888147\">\n<p id=\"fs-id1167831888149\">Find the length of the missing side. Round to the nearest tenth, if necessary.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831888152\" data-alt=\"The figure is a right triangle with a height of 15 units and a hypotenuse of 17 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a right triangle with a height of 15 units and a hypotenuse of 17 units.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835346679\">\n<div data-type=\"problem\" id=\"fs-id1167835346681\">\n<p id=\"fs-id1167834222110\">Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is eight feet tall and Sergio has 10 feet of wire. How far from the base of the antenna can he attach the wire? Approximate to the nearest tenth, if necessary.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834222116\" data-alt=\"The figure is a right triangle with a height of 8 feet and a hypotenuse of 10 feet.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a right triangle with a height of 8 feet and a hypotenuse of 10 feet.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835370795\">\n<p id=\"fs-id1167835370797\">6 feet<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835370803\">\n<div data-type=\"problem\" id=\"fs-id1167835370805\">\n<p id=\"fs-id1167835370807\">Seong is building shelving in his garage. The shelves are 36 inches wide and 15 inches tall. He wants to put a diagonal brace across the back to stabilize the shelves, as shown. How long should the brace be?<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834512816\" data-alt=\"The figure illustrates rectangular shelving whose width of 36 inch and height of 15 inches forms a right triangle with a diagonal brace.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure illustrates rectangular shelving whose width of 36 inch and height of 15 inches forms a right triangle with a diagonal brace.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831921595\">\n<div data-type=\"problem\" id=\"fs-id1167831921597\">\n<p id=\"fs-id1167831921599\">The length of a rectangle is 12 cm more than the width. The perimeter is 74 cm. Find the length and the width.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835509936\">\n<p id=\"fs-id1167835509938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-489995a58c4385ac7b99b19401609729_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/> cm, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0154243e4669a75b5cdfd1ed3f8759ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/> cm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835596592\">\n<div data-type=\"problem\" id=\"fs-id1167835596594\">\n<p id=\"fs-id1167835596596\">The width of a rectangle is three more than twice the length. The perimeter is 96 inches. Find the length and the width.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834502576\">\n<div data-type=\"problem\" id=\"fs-id1167834502578\">\n<p id=\"fs-id1167834502580\">The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834502586\">\n<p id=\"fs-id1167834152127\">9 ft, 14 ft, 12 ft<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834152134\">\n<h4 data-type=\"title\"><a href=\"\/contents\/36adea73-2201-46d3-b9b6-d13ef7df78b2\" class=\"target-chapter\">Solve Mixture and Uniform Motion Applications<\/a><\/h4>\n<p id=\"fs-id1167826995576\"><strong data-effect=\"bold\">Solve Coin Word Problems<\/strong><\/p>\n<p id=\"fs-id1167826995582\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826995585\">\n<div data-type=\"problem\" id=\"fs-id1167826995587\">\n<p id=\"fs-id1167830703667\">Paulette has ?140 in ?5 and ?10 bills. The number of ?10 bills is one less than twice the number of ?5 bills. How many of each does she have?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830703680\">\n<div data-type=\"problem\" id=\"fs-id1167830703682\">\n<p id=\"fs-id1167835375493\">Lenny has ?3.69 in pennies, dimes, and quarters. The number of pennies is three more than the number of dimes. The number of quarters is twice the number of dimes. How many of each coin does he have?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835375507\">\n<p id=\"fs-id1167835333154\">nine pennies, six dimes, 12 quarters<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835333159\"><strong data-effect=\"bold\">Solve Ticket and Stamp Word Problems<\/strong><\/p>\n<p id=\"fs-id1167835333166\">In the following exercises, solve each ticket or stamp word problem.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832150071\">\n<div data-type=\"problem\" id=\"fs-id1167832150073\">\n<p id=\"fs-id1167832150076\">Tickets for a basketball game cost ?2 for students and ?5 for adults. The number of students was three less than 10 times the number of adults. The total amount of money from ticket sales was ?619. How many of each ticket were sold?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834372263\">\n<div data-type=\"problem\" id=\"fs-id1167834372265\">\n<p id=\"fs-id1167834372267\">125 tickets were sold for the jazz band concert for a total of ?1,022. Student tickets cost ?6 each and general admission tickets cost ?10 each. How many of each kind of ticket were sold?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834372273\">\n<p id=\"fs-id1167834372275\">57 students, 68 adults<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830699855\">\n<div data-type=\"problem\" id=\"fs-id1167830699857\">\n<p id=\"fs-id1167830699859\">Yumi spent ?34.15 buying stamps. The number of ?0.56 stamps she bought was 10 less than four times the number of ?0.41 stamps. How many of each did she buy?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831031227\"><strong data-effect=\"bold\">Solve Mixture Word Problems<\/strong><\/p>\n<p id=\"fs-id1167826986123\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826986126\">\n<div data-type=\"problem\" id=\"fs-id1167826986128\">\n<p id=\"fs-id1167826986130\">Marquese is making 10 pounds of trail mix from raisins and nuts. Raisins cost ?3.45 per pound and nuts cost ?7.95 per pound. How many pounds of raisins and how many pounds of nuts should Marquese use for the trail mix to cost him ?6.96 per pound?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832015658\">\n<p id=\"fs-id1167832015660\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f660b802d2f7d82d22e350fcb188c68d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> lbs of raisins, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6578adc4c7ff8a4a5147446911541c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#46;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: 0px;\" \/> lbs of nuts<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835417630\">\n<div data-type=\"problem\" id=\"fs-id1167835417632\">\n<p id=\"fs-id1167835417635\">Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tile. She will use basic tiles that cost ?8 per square foot and decorator tiles that cost ?20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be ?10 per square foot?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834301117\">\n<div data-type=\"problem\" id=\"fs-id1167834301120\">\n<p id=\"fs-id1167834301122\">Enrique borrowed ?23,500 to buy a car. He pays his uncle 2% interest on the ?4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total ?23,500? (Round your answer to the nearest tenth of a percent.)<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835338550\">\n<p id=\"fs-id1167835338552\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#57;&#46;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#37;&#125;\n\n*** Error message:\n&#70;&#105;&#108;&#101;&#32;&#101;&#110;&#100;&#101;&#100;&#32;&#119;&#104;&#105;&#108;&#101;&#32;&#115;&#99;&#97;&#110;&#110;&#105;&#110;&#103;&#32;&#117;&#115;&#101;&#32;&#111;&#102;&#32;&#92;&#116;&#101;&#120;&#116;&#64;&#46;\r\n&#69;&#109;&#101;&#114;&#103;&#101;&#110;&#99;&#121;&#32;&#115;&#116;&#111;&#112;&#46;\r\n\n<\/pre>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834058806\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p>\n<p id=\"fs-id1167834058812\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835357470\">\n<div data-type=\"problem\" id=\"fs-id1167835357472\">\n<p id=\"fs-id1167835357474\">When Gabe drives from Sacramento to Redding it takes him 2.2 hours. It takes Elsa two hours to drive the same distance. Elsa\u2019s speed is seven miles per hour faster than Gabe\u2019s speed. Find Gabe\u2019s speed and Elsa\u2019s speed.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835331664\">\n<div data-type=\"problem\" id=\"fs-id1167835331666\">\n<p id=\"fs-id1167835331668\">Louellen and Tracy met at a restaurant on the road between Chicago and Nashville. Louellen had left Chicago and drove 3.2 hours towards Nashville. Tracy had left Nashville and drove 4 hours towards Chicago, at a speed one mile per hour faster than Louellen\u2019s speed. The distance between Chicago and Nashville is 472 miles. Find Louellen\u2019s speed and Tracy\u2019s speed.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828421648\">\n<p id=\"fs-id1167828421650\">Louellen 65 mph, Tracy 66 mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832134182\">\n<div data-type=\"problem\" id=\"fs-id1167832134185\">\n<p id=\"fs-id1167832134187\">Two busses leave Amarillo at the same time. The Albuquerque bus heads west on the I-40 at a speed of 72 miles per hour, and the Oklahoma City bus heads east on the I-40 at a speed of 78 miles per hour. How many hours will it take them to be 375 miles apart?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835301341\">\n<div data-type=\"problem\" id=\"fs-id1167835301344\">\n<p id=\"fs-id1167835301346\">Kyle rowed his boat upstream for 50 minutes. It took him 30 minutes to row back downstream. His speed going upstream is two miles per hour slower than his speed going downstream. Find Kyle\u2019s upstream and downstream speeds.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835301353\">\n<p id=\"fs-id1167831970367\">upstream 3 mph, downstream 5 mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831970372\">\n<div data-type=\"problem\" id=\"fs-id1167831970374\">\n<p id=\"fs-id1167831970377\">At 6:30, Devon left her house and rode her bike on the flat road until 7:30. Then she started riding uphill and rode until 8:00. She rode a total of 15 miles. Her speed on the flat road was three miles per hour faster than her speed going uphill. Find Devon\u2019s speed on the flat road and riding uphill.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831892888\">\n<div data-type=\"problem\" id=\"fs-id1167831892890\">\n<p id=\"fs-id1167831892892\">Anthony drove from New York City to Baltimore, which is a distance of 192 miles. He left at 3:45 and had heavy traffic until 5:30. Traffic was light for the rest of the drive, and he arrived at 7:30. His speed in light traffic was four miles per hour more than twice his speed in heavy traffic. Find Anthony\u2019s driving speed in heavy traffic and light traffic.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834132162\">\n<p id=\"fs-id1167834132164\">heavy traffic 32 mph, light traffic 66 mph<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834526168\">\n<h4 data-type=\"title\"><a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7\" class=\"target-chapter\">Solve Linear Inequalities<\/a><\/h4>\n<p id=\"fs-id1167834526178\"><strong data-effect=\"bold\">Graph Inequalities on the Number Line<\/strong><\/p>\n<p id=\"fs-id1167831893359\">In the following exercises, graph the inequality on the number line and write in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831893362\">\n<div data-type=\"problem\" id=\"fs-id1167831893365\">\n<p id=\"fs-id1167831893367\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b843bada776d2328df7301957d0f16c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834227427\">\n<div data-type=\"problem\" id=\"fs-id1167834227429\">\n<p id=\"fs-id1167834227431\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fef1e6db02c3957bd00ec928e63fe3cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#45;&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834225921\"><span data-type=\"media\" id=\"fs-id1167834225924\" data-alt=\"The solution is x is greater than or equal to negative 2.5. The number line shows a left bracket at negative 2.5 with shading to its right. The interval notation is negative 2.5 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_338_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than or equal to negative 2.5. The number line shows a left bracket at negative 2.5 with shading to its right. The interval notation is negative 2.5 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834084430\">\n<div data-type=\"problem\" id=\"fs-id1167834084432\">\n<p id=\"fs-id1167834395944\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c19f4185aa42f4b918ee90cb3aa949a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835595469\">\n<div data-type=\"problem\" id=\"fs-id1167835595471\">\n<p id=\"fs-id1167835595473\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0d114ca900168936b5c270433aff883_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835381796\"><span data-type=\"media\" id=\"fs-id1167835381800\" data-alt=\"The solution is x is greater than 2. The number line shows a left parenthesis at 2 with shading to its right. The interval notation is 2 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than 2. The number line shows a left parenthesis at 2 with shading to its right. The interval notation is 2 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834340090\">\n<div data-type=\"problem\" id=\"fs-id1167834340092\">\n<p id=\"fs-id1167834340094\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-049599758a2288a6dd98a2ce2cf3d840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#60;&#120;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"88\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831239513\">\n<div data-type=\"problem\" id=\"fs-id1167831239515\">\n<p id=\"fs-id1167831239517\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd8229079595a4a8c96598ebb345c41c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#108;&#101;&#32;&#120;&#60;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830704579\"><span data-type=\"media\" id=\"fs-id1167830704582\" data-alt=\"The solution is negative 5 is less than or equal to x which is less than negative 3. The number line shows a closed circle at negative 5, an open circle at negative 3, and shading between the circles. The interval notation is negative 5 to negative 3 within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_342_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 5 is less than or equal to x which is less than negative 3. The number line shows a closed circle at negative 5, an open circle at negative 3, and shading between the circles. The interval notation is negative 5 to negative 3 within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835358365\">\n<div data-type=\"problem\" id=\"fs-id1167835358367\">\n<p id=\"fs-id1167835358369\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77a24705eec65a8fb0938ebfe1fa705c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#108;&#101;&#32;&#120;&#92;&#108;&#101;&#32;&#51;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831880112\"><strong data-effect=\"bold\">Solve Linear Inequalities<\/strong><\/p>\n<p id=\"fs-id1167831880118\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830699525\">\n<div data-type=\"problem\" id=\"fs-id1167830699528\">\n<p id=\"fs-id1167830699530\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8970cd7bc792010717e6565a1ec28b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#45;&#49;&#50;&#92;&#108;&#101;&#32;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"92\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830868612\"><span data-type=\"media\" id=\"fs-id1167830868615\" data-alt=\"The solution is n is less than or equal to 35. The number line shows a a right bracket at 35 with shading to its left. The interval notation is negative infinity to 35 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_344_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is n is less than or equal to 35. The number line shows a a right bracket at 35 with shading to its left. The interval notation is negative infinity to 35 within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827943826\">\n<div data-type=\"problem\" id=\"fs-id1167827943828\">\n<p id=\"fs-id1167827943830\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8816c00cd9a4c36a5e77f0d83ac77e3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"81\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834186134\">\n<div data-type=\"problem\" id=\"fs-id1167834186136\">\n<p id=\"fs-id1167834186138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4dca73f44de79d28f875ae279ad44daf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#120;&#62;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834119890\"><span data-type=\"media\" id=\"fs-id1167834119893\" data-alt=\"The solution is x is greater than 6. The number line shows a left parenthesis at 6 with shading to its right. The interval notation is 6 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_346_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is greater than 6. The number line shows a left parenthesis at 6 with shading to its right. The interval notation is 6 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831103887\">\n<div data-type=\"problem\" id=\"fs-id1167831103889\">\n<p id=\"fs-id1167831103892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4ab38f2a76ee97ce3751b43742c9a8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#45;&#50;&#125;&#92;&#103;&#101;&#32;&#45;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834432727\">\n<div data-type=\"problem\" id=\"fs-id1167830698724\">\n<p id=\"fs-id1167830698727\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64907f2b4b06cd45e1fdb15f8438c7a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#112;&#62;&#49;&#53;&#112;&#45;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832081955\"><span data-type=\"media\" id=\"fs-id1167832081958\" data-alt=\"The solution is p is less than ten-thirds. The number line shows a right parenthesis at ten-thirds with shading to its left. The interval notation is negative infinity to ten-thirds within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_348_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is p is less than ten-thirds. The number line shows a right parenthesis at ten-thirds with shading to its left. The interval notation is negative infinity to ten-thirds within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835317539\">\n<div data-type=\"problem\" id=\"fs-id1167835317541\">\n<p id=\"fs-id1167835317544\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-999f8fd675f2f3820995ddaf0fbf6f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#104;&#45;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#52;&#104;&#45;&#50;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834376443\">\n<div data-type=\"problem\" id=\"fs-id1167834376445\">\n<p id=\"fs-id1167834376447\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39aaf6b2ab722b977e44ed25d63fcfcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#110;&#45;&#49;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#48;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"267\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826940750\"><span data-type=\"media\" id=\"fs-id1167826940753\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_350_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832096934\">\n<div data-type=\"problem\" id=\"fs-id1167832096937\">\n<p id=\"fs-id1167835213526\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0d82a2bb4c968c283b34c96d0722079_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#97;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#97;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;&#97;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"148\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835216800\"><strong data-effect=\"bold\">Translate Words to an Inequality and Solve<\/strong><\/p>\n<p id=\"fs-id1167835216807\">In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832076547\">\n<div data-type=\"problem\" id=\"fs-id1167832076549\">\n<p id=\"fs-id1167832076551\">Five more than <em data-effect=\"italics\">z<\/em> is at most 19.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834502686\"><span data-type=\"media\" id=\"fs-id1167834502689\" data-alt=\"The inequality is z plus 5 is less than or equal to 19. Its solution is z is less than or equal to 14. The number line shows a right bracket at 14 with shading to its left. The interval notation is negative infinity to 14 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_352_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is z plus 5 is less than or equal to 19. Its solution is z is less than or equal to 14. The number line shows a right bracket at 14 with shading to its left. The interval notation is negative infinity to 14 within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832117857\">\n<div data-type=\"problem\" id=\"fs-id1167832117859\">\n<p id=\"fs-id1167832117861\">Three less than <em data-effect=\"italics\">c<\/em> is at least 360.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834189544\">\n<div data-type=\"problem\" id=\"fs-id1167834189546\">\n<p id=\"fs-id1167834189548\">Nine times <em data-effect=\"italics\">n<\/em> exceeds 42.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834189557\"><span data-type=\"media\" id=\"fs-id1167834130099\" data-alt=\"The inequality is 9 n is greater than 42. Its solution is n is greater than fourteen-thirds. The number line shows a left parentheses at fourteen-thirds with shading to its right. The interval notation is fourteen-thirds to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_354_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is 9 n is greater than 42. Its solution is n is greater than fourteen-thirds. The number line shows a left parentheses at fourteen-thirds with shading to its right. The interval notation is fourteen-thirds to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834060372\">\n<div data-type=\"problem\" id=\"fs-id1167834060374\">\n<p id=\"fs-id1167834060376\">Negative two times <em data-effect=\"italics\">a<\/em> is no more than eight.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835381679\"><strong data-effect=\"bold\">Solve Applications with Linear Inequalities<\/strong><\/p>\n<p id=\"fs-id1167835347503\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835347506\">\n<div data-type=\"problem\" id=\"fs-id1167835347509\">\n<p id=\"fs-id1167835347511\">Julianne has a weekly food budget of ?231 for her family. If she plans to budget the same amount for each of the seven days of the week, what is the maximum amount she can spend on food each day?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826997072\">\n<p id=\"fs-id1167826997074\">?33 per day<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997079\">\n<div data-type=\"problem\" id=\"fs-id1167826997082\">\n<p id=\"fs-id1167826997084\">Rogelio paints watercolors. He got a ?100 gift card to the art supply store and wants to use it to buy 12\u2033 \u00d7 16\u2033 canvases. Each canvas costs ?10.99. What is the maximum number of canvases he can buy with his gift card?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835283185\">\n<div data-type=\"problem\" id=\"fs-id1167835283187\">\n<p id=\"fs-id1167826799367\">Briana has been offered a sales job in another city. The offer was for ?42,500 plus 8% of her total sales. In order to make it worth the move, Briana needs to have an annual salary of at least ?66,500. What would her total sales need to be for her to move?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826799373\">\n<p id=\"fs-id1167826799375\">at least ?300,000<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826799381\">\n<div data-type=\"problem\" id=\"fs-id1167831823842\">\n<p id=\"fs-id1167831823844\">Renee\u2019s car costs her ?195 per month plus ?0.09 per mile. How many miles can Renee drive so that her monthly car expenses are no more than ?250?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835217782\">\n<div data-type=\"problem\" id=\"fs-id1167835217784\">\n<p id=\"fs-id1167835217786\">Costa is an accountant. During tax season, he charges ?125 to do a simple tax return. His expenses for buying software, renting an office, and advertising are ?6,000. How many tax returns must he do if he wants to make a profit of at least ?8,000?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835534279\">\n<p id=\"fs-id1167835534281\">at least 112 jobs<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835534286\">\n<div data-type=\"problem\" id=\"fs-id1167835534288\">\n<p id=\"fs-id1167835534290\">Jenna is planning a five-day resort vacation with three of her friends. It will cost her ?279 for airfare, ?300 for food and entertainment, and ?65 per day for her share of the hotel. She has ?550 saved towards her vacation and can earn ?25 per hour as an assistant in her uncle\u2019s photography studio. How many hours must she work in order to have enough money for her vacation?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834517700\">\n<h4 data-type=\"title\"><a href=\"\/contents\/730fb57f-9832-4993-a96f-39ecb47c371d\" class=\"target-chapter\">Solve Compound Inequalities<\/a><\/h4>\n<p id=\"fs-id1167834252729\"><strong data-effect=\"bold\">Solve Compound Inequalities with \u201cand\u201d<\/strong><\/p>\n<p id=\"fs-id1167834252736\">In each of the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834324650\">\n<div data-type=\"problem\" id=\"fs-id1167834324652\">\n<p id=\"fs-id1167834324654\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b157a9b63bb99ace04045920c37207c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ceded6a2922868f3bbe7e58544df2850_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835254053\"><span data-type=\"media\" id=\"fs-id1167835254056\" data-alt=\"The solution is negative 3 is less than x which is less than or equal to 5. The number line shows an open circle at negative 3 and a closed circle at 5. The interval notation is negative 3 to 5 within a parenthesis and a bracket.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_356_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative 3 is less than x which is less than or equal to 5. The number line shows an open circle at negative 3 and a closed circle at 5. The interval notation is negative 3 to 5 within a parenthesis and a bracket.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834433653\">\n<div data-type=\"problem\" id=\"fs-id1167834433655\">\n<p id=\"fs-id1167835615128\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fdb4fde6d628fa524399a16c10fff04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#50;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\" \/> and<\/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-563ccdaaf0c1f9f034a8a5fb8f6a5fa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#120;&#45;&#49;&#62;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"82\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834194611\">\n<div data-type=\"problem\" id=\"fs-id1167834194614\">\n<p id=\"fs-id1167834194616\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-594549f6ba861961afca3b6bf3603539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -4px;\" \/> and<\/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-55528fd78826afd9fcfbaa14767d46e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834536900\"><span data-type=\"media\" id=\"fs-id1167834536904\" data-alt=\"The solution is negative x is less than negative five-fourths. The number line shows an open circle at negative five-fourths with shading to its left. The interval notation is negative infinity to negative five-fourths within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_358_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is negative x is less than negative five-fourths. The number line shows an open circle at negative five-fourths with shading to its left. The interval notation is negative infinity to negative five-fourths within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835302985\">\n<div data-type=\"problem\" id=\"fs-id1167835302988\">\n<p id=\"fs-id1167835302990\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42d2770607f772d81512c58df285621e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/> and<\/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-8183d31820cacf2f601a97d132635335_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835320407\">\n<div data-type=\"problem\" id=\"fs-id1167835361804\">\n<p id=\"fs-id1167835361806\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac7a99a4906a6d3eeb67b3331ea24f53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#53;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/> and<\/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-1419ad0c339488c86f099a4d6c2f2053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167831920623\"><span data-type=\"media\" id=\"fs-id1167831920626\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_360_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835346616\">\n<div data-type=\"problem\" id=\"fs-id1167835346618\">\n<p id=\"fs-id1167835433642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f36910e5b6a6fe7fd7aa55ac249489bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#108;&#101;&#32;&#52;&#120;&#45;&#49;&#60;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834403456\"><strong data-effect=\"bold\">Solve Compound Inequalities with \u201cor\u201d<\/strong><\/p>\n<p id=\"fs-id1167834403462\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831116499\">\n<div data-type=\"problem\" id=\"fs-id1167831116501\">\n<p id=\"fs-id1167831116503\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0aab3d56014de64c179884097b164391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#50;&#120;&#92;&#108;&#101;&#32;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/> or<\/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-a7a12560cce559dc35112872e459d828_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#43;&#51;&#120;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167831822966\"><span data-type=\"media\" id=\"fs-id1167831822969\" data-alt=\"The solution is x is less than negative two-thirds or x is greater than or equal to 3. The number line shows a closed circle at negative two-thirds with shading to its left and a closed circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative two-thirds within a parenthesis and a bracket and 3 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_362_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative two-thirds or x is greater than or equal to 3. The number line shows a closed circle at negative two-thirds with shading to its left and a closed circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative two-thirds within a parenthesis and a bracket and 3 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835514661\">\n<div data-type=\"problem\" id=\"fs-id1167835514663\">\n<p id=\"fs-id1167831822973\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ceff58bf9f9d2be1c9b4cb3b76a16734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -4px;\" \/> or<\/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-7e8a1f30094af704b193c42f89b7bde1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#49;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835355098\">\n<div data-type=\"problem\" id=\"fs-id1167835355100\">\n<p id=\"fs-id1167835363594\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1f4cfd2417cd35c038bda4044c601e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#50;&#62;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e09e121a0a310b5372e1daa545dca010_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835415809\"><span data-type=\"media\" id=\"fs-id1167835415812\" data-alt=\"The solution is x is less than 2 or x is greater than 8. The number line shows an open circle at 2 with shading to its left and an open circle at 8 with shading to its right. The interval notation is the union of negative infinity to 8 within parentheses and 8 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_364_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than 2 or x is greater than 8. The number line shows an open circle at 2 with shading to its left and an open circle at 8 with shading to its right. The interval notation is the union of negative infinity to 8 within parentheses and 8 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834226120\">\n<div data-type=\"problem\" id=\"fs-id1167834226122\">\n<p id=\"fs-id1167834226124\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0dc64f4a9a1c6608d2114a88b8b3cf02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/> or<\/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-328314acedf3adc978c9005194c78096_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830963955\">\n<div data-type=\"problem\" id=\"fs-id1167830963957\">\n<p id=\"fs-id1167830963959\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd01cb283861fe29e3b1138a51343803_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/> or<\/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-e3af2539959a35abc6f25916ef212316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"112\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834221993\"><span data-type=\"media\" id=\"fs-id1167834221996\" data-alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_366_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167834526656\"><strong data-effect=\"bold\">Solve Applications with Compound Inequalities<\/strong><\/p>\n<p id=\"fs-id1167831894427\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831894430\">\n<div data-type=\"problem\" id=\"fs-id1167831894432\">\n<p id=\"fs-id1167831894434\">Liam is playing a number game with his sister Audry. Liam is thinking of a number and wants Audry to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Liam might be thinking of.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831922049\">\n<div data-type=\"problem\" id=\"fs-id1167831922051\">\n<p id=\"fs-id1167831922054\">Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831922060\">\n<p id=\"fs-id1167831922062\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7b19069cf79c2973312b174e81025f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#32;&#119;&#92;&#108;&#101;&#32;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834306707\">\n<h4 data-type=\"title\"><a href=\"\/contents\/6cfd2376-f945-4ebb-bdb6-92ac7398c9ad\" class=\"target-chapter\">Solve Absolute Value Inequalities<\/a><\/h4>\n<p id=\"fs-id1167834394984\"><strong data-effect=\"bold\">Solve Absolute Value Equations<\/strong><\/p>\n<p id=\"fs-id1167834394990\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826978048\">\n<div data-type=\"problem\" id=\"fs-id1167826978050\">\n<p id=\"fs-id1167826978053\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47fe8cdef5ea7773cc75b95fb9ceae0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831910773\">\n<div data-type=\"problem\" id=\"fs-id1167831910775\">\n<p id=\"fs-id1167831910777\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91bb0ec7b033752d0ac103064dce8153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#121;&#124;&#61;&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835531584\">\n<p id=\"fs-id1167835356798\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835356803\">\n<div data-type=\"problem\" id=\"fs-id1167835356805\">\n<p id=\"fs-id1167835356807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bff1356215b10f5e9799486f4268133d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#122;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834346568\">\n<p id=\"fs-id1167834346570\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5f4eb79ecf4fc464cf210ffb56af574_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#52;&#124;&#43;&#53;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831148746\">\n<p id=\"fs-id1167831148748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6e045a887015316e13017030284df7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834184739\">\n<div data-type=\"problem\" id=\"fs-id1167834184741\">\n<p id=\"fs-id1167834184743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d60d19d57a21fd1716683cb8fe39515e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#124;&#120;&#45;&#49;&#124;&#43;&#50;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831912139\">\n<div data-type=\"problem\" id=\"fs-id1167831912142\">\n<p id=\"fs-id1167831912144\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8294302b40904c68b66260c9ec89563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#124;&#120;&#45;&#51;&#124;&#43;&#56;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830769668\">\n<p id=\"fs-id1167830769670\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fdbe25cf1e69c261d95b0beb8b3b4cc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#57;&#44;&#120;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826875112\">\n<div data-type=\"problem\" id=\"fs-id1167826875114\">\n<p id=\"fs-id1167826875116\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85829cfd33923513f937f2f28531aad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#53;&#124;&#43;&#52;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831115234\">\n<div data-type=\"problem\" id=\"fs-id1167831115237\">\n<p id=\"fs-id1167831115239\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d2b5b8ef4630065a6133ca4d8fb3404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#54;&#120;&#45;&#53;&#124;&#61;&#124;&#50;&#120;&#43;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835302391\">\n<p id=\"fs-id1167835302394\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1e917808fa7ca0f834d2b21b538b4db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835509790\"><strong data-effect=\"bold\">Solve Absolute Value Inequalities with \u201cless than\u201d<\/strong><\/p>\n<p id=\"fs-id1167835509795\">In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835509799\">\n<div data-type=\"problem\" id=\"fs-id1167835509801\">\n<p id=\"fs-id1167835360885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ae5d20c1fa5416e0afdda592d92557c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#92;&#108;&#101;&#32;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835623478\">\n<div data-type=\"problem\" id=\"fs-id1167835623481\">\n<p id=\"fs-id1167835623483\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d95407653924d8de57efed6d87e34d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#45;&#53;&#124;&#92;&#108;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835230321\"><span data-type=\"media\" id=\"fs-id1167835340944\" data-alt=\"The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading in between the circles. The interval notation is 1 to 4 within brackets.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_368_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading in between the circles. The interval notation is 1 to 4 within brackets.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835340958\">\n<div data-type=\"problem\" id=\"fs-id1167835340960\">\n<p id=\"fs-id1167835390176\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-945956595bc7c7058e368c51ac1a3609_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#54;&#120;&#45;&#53;&#124;&#60;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835512107\">\n<div data-type=\"problem\" id=\"fs-id1167835512109\">\n<p id=\"fs-id1167835512111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef50e46a395dac5c85243f776270266f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#53;&#120;&#43;&#49;&#124;&#92;&#108;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835489105\"><span data-type=\"media\" id=\"fs-id1167835489108\" data-alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_370_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167835367695\"><strong data-effect=\"bold\">Solve Absolute Value Inequalities with \u201cgreater than\u201d<\/strong><\/p>\n<p id=\"fs-id1167835367701\">In the following exercises, solve. Graph the solution and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835367705\">\n<div data-type=\"problem\" id=\"fs-id1167834229123\">\n<p id=\"fs-id1167834229125\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0961bd5dd5da1a564a8f65f0e3b3986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#62;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834523740\">\n<div data-type=\"problem\" id=\"fs-id1167834523742\">\n<p id=\"fs-id1167834523744\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb21df9ef3ed1d8c966f4c88056b3406_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#124;&#92;&#103;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832067007\"><span data-type=\"media\" id=\"fs-id1167832067010\" data-alt=\"The solution is x is less than negative 2 or x is greater than 6. The number line shows a closed circle at negative 2 with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_372_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than negative 2 or x is greater than 6. The number line shows a closed circle at negative 2 with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826996917\">\n<div data-type=\"problem\" id=\"fs-id1167826996919\">\n<p id=\"fs-id1167832067013\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9283f32b3d73e63d26f44983b964136f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#45;&#53;&#124;&#62;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835230288\">\n<div data-type=\"problem\" id=\"fs-id1167835230290\">\n<p id=\"fs-id1167835230292\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e0cd4109eb7172b37182f0cafa27e87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#45;&#55;&#124;&#92;&#103;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835281147\"><span data-type=\"media\" id=\"fs-id1167834329824\" data-alt=\"The solution is x is less than or equal to 6 or x is greater than or equal to 8. The number line shows a closed circle at 6 with shading to its left and a closed circle at 8 with shading to its right. The interval notation is the union of negative infinity to negative 6 within a parenthesis and a bracket and 8 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_374_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to 6 or x is greater than or equal to 8. The number line shows a closed circle at 6 with shading to its left and a closed circle at 8 with shading to its right. The interval notation is the union of negative infinity to negative 6 within a parenthesis and a bracket and 8 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835182598\">\n<div data-type=\"problem\" id=\"fs-id1167835182600\">\n<p id=\"fs-id1167835182602\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4afb8b8693a1bfe4792ca65527496bb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#124;&#120;&#124;&#43;&#52;&#92;&#103;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834515323\"><strong data-effect=\"bold\">Solve Applications with Absolute Value<\/strong><\/p>\n<p id=\"fs-id1167826801815\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826801818\">\n<div data-type=\"problem\" id=\"fs-id1167826801820\">\n<p id=\"fs-id1167826801823\">A craft beer brewer needs 215,000 bottle per day. But this total can vary by as much as 5,000 bottles. What is the maximum and minimum expected usage at the bottling company?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826801828\">\n<p id=\"fs-id1167826801831\">The minimum to maximum expected usage is 210,000 to 220,000 bottles<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832056602\">\n<div data-type=\"problem\" id=\"fs-id1167832056604\">\n<p id=\"fs-id1167832056606\">At Fancy Grocery, the ideal weight of a loaf of bread is 16 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167826977992\">\n<h3 data-type=\"title\">Practice Test<\/h3>\n<p id=\"fs-id1167826977999\">In the following exercises, solve each equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826978003\">\n<div data-type=\"problem\" id=\"fs-id1167835534241\">\n<p id=\"fs-id1167835534244\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95e3f6605f7ea24782ccf712ad84f2bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826808724\">\n<p id=\"fs-id1167826808726\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3e213d8d687e32831c24e16c432b60e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826986742\">\n<div data-type=\"problem\" id=\"fs-id1167826986744\">\n<p id=\"fs-id1167826986746\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90588cc7e0e850e298a43500794c367b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#109;&#43;&#50;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#109;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"232\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835360007\">\n<div data-type=\"problem\" id=\"fs-id1167835356019\">\n<p id=\"fs-id1167835356021\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca82d164df344669b7883e54cf545a2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#97;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#97;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#48;&#45;&#51;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"251\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832042649\">\n<p id=\"fs-id1167832042651\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ab0ec038774d605a9169ad5fd82ebed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#52;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834422892\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69cf6b4db99110325cbc20adfc5564df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#49;&#100;&#43;&#48;&#46;&#50;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#46;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"187\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834394761\">\n<div data-type=\"problem\" id=\"fs-id1167834394763\">\n<p id=\"fs-id1167834394765\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c7b60c7dd8751df91a96da432ff0b01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#110;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#110;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#57;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"259\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835297404\">\n<p id=\"fs-id1167835297406\">contradiction; no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835297412\">\n<div data-type=\"problem\" id=\"fs-id1167835297414\">\n<p id=\"fs-id1167835297416\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-865cef03ff9c77ad01dead387b0e8b66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#117;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#91;&#54;&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#61;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"304\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834059385\">\n<div data-type=\"problem\" id=\"fs-id1167835609691\">\n<p id=\"fs-id1167835609693\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb67ffbd891c523323f533ac949114bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834489813\">\n<p id=\"fs-id1167834489815\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-116dfacb4c2a32bdfb5ab9e852f29986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832153843\">\n<div data-type=\"problem\" id=\"fs-id1167832153845\">\n<p id=\"fs-id1167832153847\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a98f8abbc19e8643791273d4c0883c11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#120;&#45;&#52;&#124;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831076534\">\n<div data-type=\"problem\" id=\"fs-id1167831076536\">\n<p id=\"fs-id1167831076538\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-663000fa3b6da56dc077df3676279170_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#50;&#120;&#45;&#49;&#124;&#61;&#124;&#52;&#120;&#43;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835479596\">\n<p id=\"fs-id1167835479599\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ff3575f9ce9e606ff5477e90e4ed0a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835514170\">\n<div data-type=\"problem\" id=\"fs-id1167835514172\">\n<p id=\"fs-id1167835514174\">Solve the formula<\/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-538d43aa6536e497faef2a7897039a14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;&#121;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/> for <em data-effect=\"italics\">y<\/em>.<\/div>\n<\/div>\n<p id=\"fs-id1167835387100\">In the following exercises, graph the inequality on the number line and write in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835387104\">\n<div data-type=\"problem\" id=\"fs-id1167831919540\">\n<p id=\"fs-id1167831919543\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c029d017a90a170d23da2c5619934e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#45;&#51;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831919554\"><span data-type=\"media\" id=\"fs-id1167834464190\" data-alt=\"The inequality is x is greater than or equal to negative 3.5. The number line shows a left bracket at negative 3.5 and shading to the right. The interval notation is negative 3.5 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_376_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is x is greater than or equal to negative 3.5. The number line shows a left bracket at negative 3.5 and shading to the right. The interval notation is negative 3.5 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834464204\">\n<div data-type=\"problem\" id=\"fs-id1167834189855\">\n<p id=\"fs-id1167834189857\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb6e5f8204b9f783b2abd0ff14d793c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835327400\">\n<div data-type=\"problem\" id=\"fs-id1167835327402\">\n<p id=\"fs-id1167835327404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89cbaccf283bde388d8888442cd37a01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#92;&#108;&#101;&#32;&#120;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832052659\"><span data-type=\"media\" id=\"fs-id1167832052662\" data-alt=\"The inequality is negative two is less than or equal to x which is less than 5. The number line shows a closed circle at negative 2 and an open circle at 5 with shading between the circles. The interval notation is negative 2 to 5 within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_378_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The inequality is negative two is less than or equal to x which is less than 5. The number line shows a closed circle at negative 2 and an open circle at 5 with shading between the circles. The interval notation is negative 2 to 5 within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167831883587\">In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831883591\">\n<div data-type=\"problem\" id=\"fs-id1167831883593\">\n<p id=\"fs-id1167831883595\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a7adddb67f9e8877c9896de70047dca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#107;&#92;&#103;&#101;&#32;&#53;&#107;&#45;&#49;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"110\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835376497\">\n<div data-type=\"problem\" id=\"fs-id1167835376499\">\n<p id=\"fs-id1167835376501\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3112f7e2f9abc5274118e27217b2411_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#99;&#45;&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#53;&#99;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"191\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835373843\"><span data-type=\"media\" id=\"fs-id1167835373846\" data-alt=\"The solution is c is greater than one-third. The number line shows a left parenthesis at one-third with shading to its right. The interval notation is one-third to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_380_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is c is greater than one-third. The number line shows a left parenthesis at one-third with shading to its right. The interval notation is one-third to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834132955\">\n<div data-type=\"problem\" id=\"fs-id1167834132957\">\n<p id=\"fs-id1167834132959\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac7a99a4906a6d3eeb67b3331ea24f53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#53;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/> and<\/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-1419ad0c339488c86f099a4d6c2f2053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831023918\">\n<div data-type=\"problem\" id=\"fs-id1167831023920\">\n<p id=\"fs-id1167831023922\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ceff58bf9f9d2be1c9b4cb3b76a16734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -4px;\" \/> or<\/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-7e8a1f30094af704b193c42f89b7bde1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#45;&#49;&#62;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167826978335\"><span data-type=\"media\" id=\"fs-id1167826978338\" data-alt=\"The solution is x is less than two-thirds or x is greater than 1. The number line shows an open circle at two-thirds with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 to infinity within parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_382_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than two-thirds or x is greater than 1. The number line shows an open circle at two-thirds with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 to infinity within parentheses.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834062394\">\n<div data-type=\"problem\" id=\"fs-id1167834062396\">\n<p id=\"fs-id1167834062398\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd01cb283861fe29e3b1138a51343803_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#108;&#101;&#32;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/> or<\/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-e3af2539959a35abc6f25916ef212316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#103;&#101;&#32;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"112\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835352568\">\n<div data-type=\"problem\" id=\"fs-id1167835352570\">\n<p id=\"fs-id1167835352572\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-160cd3df01cfa2c08c79112fff46a014_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#120;&#45;&#51;&#124;&#92;&#103;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832134036\"><span data-type=\"media\" id=\"fs-id1167832134040\" data-alt=\"The solution is x is less than or equal to negative one-half or x is greater than or equal to 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and bracket and 2 to infinity within a bracket and a parenthesis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_384_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The solution is x is less than or equal to negative one-half or x is greater than or equal to 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and bracket and 2 to infinity within a bracket and a parenthesis.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167832152980\">In the following exercises, translate to an equation or inequality and solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832152983\">\n<div data-type=\"problem\" id=\"fs-id1167832152985\">\n<p id=\"fs-id1167832152987\">Four less than twice <em data-effect=\"italics\">x<\/em> is 16.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826967326\">\n<div data-type=\"problem\" id=\"fs-id1167826967328\">\n<p id=\"fs-id1167826967330\">Find the length of the missing side.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167826967334\" data-alt=\"The figure is a right triangle with a base of 6 units and a height of 9 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/05\/CNX_IntAlg_Figure_02_07_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a right triangle with a base of 6 units and a height of 9 units.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835329113\">\n<p id=\"fs-id1167835329115\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7693633ed2bfbf8f357552701025c079_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#46;&#56;\" 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-id1167835329124\">\n<div data-type=\"problem\" id=\"fs-id1167835329126\">\n<p id=\"fs-id1167835329128\">One number is four more than twice another. Their sum is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-788786a72d542e0a7cce51fed827661e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"35\" style=\"vertical-align: -1px;\" \/> Find the numbers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831116947\">\n<div data-type=\"problem\" id=\"fs-id1167831116949\">\n<p id=\"fs-id1167831116951\">The sum of two consecutive odd integers is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a98197ab63a13f152ccea36dfddc07d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: -1px;\" \/> Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826803036\">\n<p id=\"fs-id1167826803038\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec43fda5543401b429e4c473c7e0bbcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#55;&#44;&#45;&#53;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834219646\">\n<div data-type=\"problem\" id=\"fs-id1167834219648\">\n<p id=\"fs-id1167834219650\">Marcus bought a television on sale for ?626.50 The original price of the television was ?895. Find <span class=\"token\">\u24d0<\/span> the amount of discount and <span class=\"token\">\u24d1<\/span> the discount rate.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834532431\">\n<div data-type=\"problem\" id=\"fs-id1167834532433\">\n<p id=\"fs-id1167835281905\">Bonita has ?2.95 in dimes and quarters in her pocket. If she has five more dimes than quarters, how many of each coin does she have?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835281911\">\n<p id=\"fs-id1167835281913\">12 dimes, seven quarters<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835281918\">\n<div data-type=\"problem\" id=\"fs-id1167835281920\">\n<p id=\"fs-id1167835329472\">Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs ?6.04 per gallon and the soda costs ?4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs ?5.71 per gallon?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832075936\">\n<div data-type=\"problem\" id=\"fs-id1167832075938\">\n<p id=\"fs-id1167832075940\">The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832150928\">\n<p id=\"fs-id1167832150930\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15e239cf1e06573729fe06fd1829f65d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;&#44;&#54;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;&#44;&#57;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&deg;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835529713\">\n<div data-type=\"problem\" id=\"fs-id1167835529715\">\n<p id=\"fs-id1167835529717\">The length of a rectangle is five feet more than four times the width. The perimeter is 60 feet. Find the dimensions of the rectangle.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830705577\">\n<div data-type=\"problem\" id=\"fs-id1167830705579\">\n<p id=\"fs-id1167830705581\">Two planes leave Dallas at the same time. One heads east at a speed of 428 miles per hour. The other plane heads west at a speed of 382 miles per hour. How many hours will it take them to be 2,025 miles apart?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831076614\">\n<p id=\"fs-id1167831076616\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01948ea56b434f27f412043e785fe880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> hours<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831076625\">\n<div data-type=\"problem\" id=\"fs-id1167831076628\">\n<p id=\"fs-id1167830925372\">Leon drove from his house in Cincinnati to his sister\u2019s house in Cleveland, a distance of 252 miles. It took him <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1276e8a4b6c5c69d6d1a2bba275f28e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"18\" style=\"vertical-align: -6px;\" \/> hours. For the first half hour, he had heavy traffic, and the rest of the time his speed was five miles per hour less than twice his speed in heavy traffic. What was his speed in heavy traffic?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831887239\">\n<div data-type=\"problem\" id=\"fs-id1167831887241\">\n<p id=\"fs-id1167831887243\">Sara has a budget of ?1,000 for costumes for the 18 members of her musical theater group. What is the maximum she can spend for each costume?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834183595\">\n<p id=\"fs-id1167834183597\">At most ?55.56 per costume.<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":103,"menu_order":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15231","chapter","type-chapter","status-publish","hentry"],"part":991,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15231","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":2,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15231\/revisions"}],"predecessor-version":[{"id":15236,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15231\/revisions\/15236"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/991"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15231\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=15231"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=15231"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=15231"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=15231"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}