{"id":3471,"date":"2018-12-11T13:52:03","date_gmt":"2018-12-11T18:52:03","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-rational-inequalities\/"},"modified":"2018-12-11T13:52:03","modified_gmt":"2018-12-11T18:52:03","slug":"solve-rational-inequalities","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-rational-inequalities\/","title":{"raw":"Solve Rational Inequalities","rendered":"Solve Rational 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 rational inequalities<\/li><li>Solve an inequality with rational functions<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167832010437\" class=\"be-prepared\"><p id=\"fs-id1167834535475\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167834463709\" type=\"1\"><li>Find the value of \\(x-5\\) when <span class=\"token\">\u24d0<\/span> \\(x=6\\) <span class=\"token\">\u24d1<\/span> \\(x=-3\\) <span class=\"token\">\u24d2<\/span> \\(x=5.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve: \\(8-2x&lt;12.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835324646\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Write in interval notation: \\(-3\\le x&lt;5.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835524181\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167826998368\"><h3 data-type=\"title\">Solve Rational Inequalities<\/h3><p id=\"fs-id1167834061460\">We learned to solve linear inequalities after learning to solve linear equations. The techniques were very much the same with one major exception. When we multiplied or divided by a negative number, the inequality sign reversed.<\/p><p>Having just learned to solve rational equations we are now ready to solve rational inequalities. A <span data-type=\"term\">rational inequality<\/span> is an inequality that contains a rational expression.<\/p><div data-type=\"note\" id=\"fs-id1167835390175\"><div data-type=\"title\">Rational Inequality<\/div><p>A <strong data-effect=\"bold\">rational inequality<\/strong> is an inequality that contains a rational expression.<\/p><\/div><p>Inequalities such as \\(\\frac{3}{2x}&gt;1,\\phantom{\\rule{0.5em}{0ex}}\\frac{2x}{x-3}&lt;4,\\phantom{\\rule{0.5em}{0ex}}\\frac{2x-3}{x-6}\\ge x,\\) and \\(\\frac{1}{4}-\\frac{2}{{x}^{2}}\\le \\frac{3}{x}\\) are rational inequalities as they each contain a rational expression.<\/p><p id=\"fs-id1167835341662\">When we solve a rational inequality, we will use many of the techniques we used solving linear inequalities. We especially must remember that when we multiply or divide by a negative number, the inequality sign must reverse.<\/p><p id=\"fs-id1167835354144\">Another difference is that we must carefully consider what value might make the rational expression undefined and so must be excluded.<\/p><p id=\"fs-id1167834191238\">When we solve an equation and the result is \\(x=3,\\) we know there is one solution, which is 3.<\/p><p id=\"fs-id1167835515515\">When we solve an inequality and the result is \\(x&gt;3,\\) we know there are many solutions. We graph the result to better help show all the solutions, and we start with 3. Three becomes a <span data-type=\"term\">critical point<\/span> and then we decide whether to shade to the left or right of it. The numbers to the right of 3 are larger than 3, so we shade to the right.<\/p><span data-type=\"media\" id=\"fs-id1167831882511\" data-alt=\"This figure shows the solution, the interval 3 to infinity, of the inequality x is greater than 3 on a number line. The values range from negative 5 to 5 on the number line. The inequality is modeled by an open parenthesis at the critical point 3 and shading the right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the solution, the interval 3 to infinity, of the inequality x is greater than 3 on a number line. The values range from negative 5 to 5 on the number line. The inequality is modeled by an open parenthesis at the critical point 3 and shading the right.\"><\/span><p>To solve a rational inequality, we first must write the inequality with only one quotient on the left and 0 on the right.<\/p><p id=\"fs-id1167835277282\">Next we determine the critical points to use to divide the number line into intervals. A <strong data-effect=\"bold\">critical point<\/strong> is a number which make the rational expression zero or undefined.<\/p><p id=\"fs-id1167831883353\">We then will evaluate the factors of the numerator and denominator, and find the quotient in each interval. This will identify the interval, or intervals, that contains all the solutions of the rational inequality.<\/p><p id=\"fs-id1167834186143\">We write the solution in interval notation being careful to determine whether the endpoints are included.<\/p><div data-type=\"example\" id=\"fs-id1167835200478\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834194894\"><div data-type=\"problem\" id=\"fs-id1167834095300\"><p id=\"fs-id1167831872191\">Solve and write the solution in interval notation: \\(\\frac{x-1}{x+3}\\ge 0.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835423426\"><strong data-effect=\"bold\">Step 1.<\/strong> Write the inequality as one quotient on the left and zero on the right.<\/p><p id=\"fs-id1167826799368\">Our inequality is in this form. \\(\\phantom{\\rule{5em}{0ex}}\\frac{x-1}{x+3}\\ge 0\\)<\/p><p id=\"fs-id1167834525164\"><strong data-effect=\"bold\">Step 2.<\/strong> Determine the critical points\u2014the points where the rational expression will be zero or undefined.<\/p><p id=\"fs-id1167831838506\">The rational expression will be zero when the numerator is zero. Since \\(x-1=0\\) when \\(x=1,\\) then \\(1\\) is a critical point.<\/p><p id=\"fs-id1167835341311\">The rational expression will be undefined when the denominator is zero. Since \\(x+3=0\\) when \\(x=-3,\\) then \\(-3\\) is a critical point.<\/p><p id=\"fs-id1167826880318\">The critical points are 1 and \\(-3.\\)<\/p><p id=\"fs-id1167834505421\"><strong data-effect=\"bold\">Step 3.<\/strong> Use the critical points to divide the number line into intervals.<\/p><span data-type=\"media\" id=\"fs-id1167834190549\" data-alt=\"This figure shows a number line divided into three intervals by its critical points marked at negative 3 and 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a number line divided into three intervals by its critical points marked at negative 3 and 0.\"><\/span><p id=\"fs-id1167835370621\">The number line is divided into three intervals:<\/p><p id=\"fs-id1165927646021\">\\(\\phantom{\\rule{10.3em}{0ex}}\\left(\\text{\u2212}\\infty ,-3\\right)\\phantom{\\rule{5em}{0ex}}\\left(-3,1\\right)\\phantom{\\rule{5.5em}{0ex}}\\left(1,\\infty \\right)\\)<\/p><p id=\"fs-id1167834196002\"><strong data-effect=\"bold\">Step 4.<\/strong> Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient.<\/p><p id=\"fs-id1167835254103\">To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.<\/p><div data-type=\"equation\" id=\"fs-id1167835320815\" class=\"unnumbered\" data-label=\"\">\\(\\mathbf{\\text{Interval}}\\phantom{\\rule{0.2em}{0ex}}\\left(\\text{\u2212}\\infty ,-3\\right)\\)<\/div><p id=\"fs-id1167832116048\">The number \\(-4\\) is in the interval \\(\\left(\\text{\u2212}\\infty ,-3\\right).\\) Test \\(x=-4\\) in the expression in the numerator and the denominator.<\/p><span data-type=\"media\" id=\"fs-id1167835376324\" data-alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when negative 4 is substituted into the expression for x, the result is negative 5. It labels the result as \u201cnegative\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when negative 4 is substituted into the expression for x, the result is negative 1. It labels the result \u201cnegative\u201d.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when negative 4 is substituted into the expression for x, the result is negative 5. It labels the result as \u201cnegative\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when negative 4 is substituted into the expression for x, the result is negative 1. It labels the result \u201cnegative\u201d.\"><\/span><p id=\"fs-id1167835362637\">Above the number line, mark the factor \\(x-1\\) negative and mark the factor \\(x+3\\) negative.<\/p><p id=\"fs-id1167835524209\">Since a negative divided by a negative is positive, mark the quotient positive in the interval \\(\\left(\\text{\u2212}\\infty ,-3\\right).\\)<\/p><span data-type=\"media\" data-alt=\"This figure shows the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is negative, which is positive. It shows a number line divided into three intervals by its critical points marked at negative 3 and 0. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is negative, which is positive. It shows a number line divided into three intervals by its critical points marked at negative 3 and 0. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3.\"><\/span><div data-type=\"equation\" id=\"fs-id1167835239820\" class=\"unnumbered\" data-label=\"\">\\(\\mathbf{\\text{Interval}}\\phantom{\\rule{0.2em}{0ex}}\\left(-3,1\\right)\\)<\/div><p id=\"fs-id1167828401707\">The number 0 is in the interval \\(\\left(-3,1\\right).\\) Test \\(x=0.\\)<\/p><span data-type=\"media\" id=\"fs-id1167835421416\" data-alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when 0 is substituted into the expression for x, the result is negative 1. It labels the result as \u201cnegative\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when 0 is substituted into the expression for x, the result is 3. It labels the result \u201cpositive\u201d.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when 0 is substituted into the expression for x, the result is negative 1. It labels the result as \u201cnegative\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when 0 is substituted into the expression for x, the result is 3. It labels the result \u201cpositive\u201d.\"><\/span><p id=\"fs-id1167834300208\">Above the number line, mark the factor \\(x-1\\) negative and mark \\(x+3\\) positive.<\/p><p id=\"fs-id1167834448595\">Since a negative divided by a positive is negative, the quotient is marked negative in the interval \\(\\left(-3,1\\right).\\)<\/p><span data-type=\"media\" id=\"fs-id1167835362793\" data-alt=\"This figure shows a shows the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is positive, which is negative. It shows a number line divided into three intervals by its critical points marked at negative 3 and 0. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a shows the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is positive, which is negative. It shows a number line divided into three intervals by its critical points marked at negative 3 and 0. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1.\"><\/span><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(\\mathbf{\\text{Interval}}\\phantom{\\rule{0.2em}{0ex}}\\left(1,\\infty \\right)\\)<\/div><p>The number 2 is in the interval \\(\\left(1,\\infty \\right).\\) Test \\(x=2.\\)<\/p><span data-type=\"media\" id=\"fs-id1167835384330\" data-alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when 2 is substituted into the expression for x, the result is 1. It labels the result as \u201cpositive\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when 2 is substituted into the expression for x, the result is 5. It labels the result \u201cpositive\u201d.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when 2 is substituted into the expression for x, the result is 1. It labels the result as \u201cpositive\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when 2 is substituted into the expression for x, the result is 5. It labels the result \u201cpositive\u201d.\"><\/span><p>Above the number line, mark the factor \\(x-1\\) positive and mark \\(x+3\\) positive.<\/p><p id=\"fs-id1167835326166\">Since a positive divided by a positive is positive, mark the quotient positive in the interval \\(\\left(1,\\infty \\right).\\)<\/p><span data-type=\"media\" id=\"fs-id1167835303485\" data-alt=\"The figure shows that in the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is positive, which is negative. It shows a number line is divided into intervals by critical points at negative 3 and 1. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1. The factors x minus 1 and x plus 3 are marked as positive above the number line for the interval 1 to infinity. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative 1 to infinity.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows that in the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is positive, which is negative. It shows a number line is divided into intervals by critical points at negative 3 and 1. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1. The factors x minus 1 and x plus 3 are marked as positive above the number line for the interval 1 to infinity. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative 1 to infinity.\"><\/span><p id=\"fs-id1167834099272\"><strong data-effect=\"bold\">Step 5.<\/strong> Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/p><p id=\"fs-id1167834059152\">We want the quotient to be greater than or equal to zero, so the numbers in the intervals \\(\\left(\\text{\u2212}\\infty ,-3\\right)\\) and \\(\\left(1,\\infty \\right)\\) are solutions.<\/p><p id=\"fs-id1167831920853\">But what about the critical points?<\/p><p id=\"fs-id1167830865437\">The critical point \\(x=-3\\) makes the denominator 0, so it must be excluded from the solution and we mark it with a parenthesis.<\/p><p id=\"fs-id1167834536833\">The critical point \\(x=1\\) makes the whole rational expression 0. The inequality requires that the rational expression be greater than or equal to. So, 1 is part of the solution and we will mark it with a bracket.<\/p><span data-type=\"media\" id=\"fs-id1167831823765\" data-alt=\"The number line is divided into intervals by critical points at negative 3 and 1. A closed parenthesis is used at 3 and an open bracket is used at 1. The number is shaded to the left of 3 and to the right of 1. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1. The factors x minus 1 and x plus 3 are marked as positive above the number line for the interval 1 to infinity. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative 1 to infinity.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The number line is divided into intervals by critical points at negative 3 and 1. A closed parenthesis is used at 3 and an open bracket is used at 1. The number is shaded to the left of 3 and to the right of 1. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1. The factors x minus 1 and x plus 3 are marked as positive above the number line for the interval 1 to infinity. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative 1 to infinity.\"><\/span><p id=\"fs-id1167828426637\">Recall that when we have a solution made up of more than one interval we use the union symbol, \\(\\cup ,\\) to connect the two intervals. The solution in interval notation is \\(\\left(\\text{\u2212}\\infty ,-3\\right)\\cup \\left[1,\\infty \\right).\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835342857\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831112014\"><div data-type=\"problem\"><p>Solve and write the solution in interval notation: \\(\\frac{x-2}{x+4}\\ge 0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831847167\"><p id=\"fs-id1167835318289\">\\(\\left(\\text{\u2212}\\infty ,-4\\right)\\cup \\left[2,\\infty \\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835319437\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834309629\"><div data-type=\"problem\" id=\"fs-id1167834084806\"><p id=\"fs-id1167826798819\">Solve and write the solution in interval notation: \\(\\frac{x+2}{x-4}\\ge 0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826987807\"><p id=\"fs-id1167835343175\">\\(\\left(\\text{\u2212}\\infty ,-2\\right]\\cup \\left(4,\\infty \\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835317235\">We summarize the steps for easy reference.<\/p><div data-type=\"note\" id=\"fs-id1167835233307\" class=\"howto\"><div data-type=\"title\">Solve a rational inequality.<\/div><ol id=\"fs-id1167826995382\" type=\"1\" class=\"stepwise\"><li>Write the inequality as one quotient on the left and zero on the right.<\/li><li>Determine the critical points\u2013the points where the rational expression will be zero or undefined.<\/li><li>Use the critical points to divide the number line into intervals.<\/li><li>Test a value in each interval. Above the number line show the sign of each factor of the numerator and denominator in each interval. Below the number line show the sign of the quotient.<\/li><li>Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/li><\/ol><\/div><p id=\"fs-id1167834294544\">The next example requires that we first get the rational inequality into the correct form.<\/p><div data-type=\"example\" id=\"fs-id1167835358203\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167830963564\"><div data-type=\"problem\"><p id=\"fs-id1167835379650\">Solve and write the solution in interval notation: \\(\\frac{4x}{x-6}&lt;1.\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167826804582\" class=\"unnumbered unstyled can-break\" summary=\"Solve the inequality 4 x divided by the quantity x minus 6 is less than 1. Subtract 1 to get zero on the right. The result is the difference between 4 x divided by the quantity x minus 6 and 1 is less than 0. Rewrite 1 as a fraction using the least common denominator. The result is the difference between the quotient of 4 x and the quantity x minus 6 and the quotient of the quantity x minus 6 and x minus 6 is less than 0. Subtract the numerators and place the difference over the common denominator. The result is 4 x minus the quantity x minus 6 all divided by the quantity x minus 6 is less than 0. Simplify the numerator. The result is the quotient of the quantity 3 x plus 6 and the quantity x minus 6 is less than 0. Factor the numerator to show all factors. The result is 3 times the quantity x plus 2 all divided by the quantity x minus 6 is less than 0. Find the critical points. The quotient will be 0 when the numerator is 0. The quotient is undefined when the numerator is 0. That is 3 times the quantity x plus 2 is equal to 0 and x minus 6 is equal to 0. So, 3 is not equal to 0, x is equal to negative 2, and x is equal to 6. Use the critical points, negative 2 and 6, to divide the number line into intervals. The intervals are negative infinity to negative 2, negative 2 to 6, and 6 to infinity. Test a value in each interval using a chart. The chart has four columns and three rows. The first row is a header row and it labels the second column the interval negative infinity to negative 2, the third column the interval negative 2 to 6, and the fourth column the interval 6 to infinity. The first column is a header column and it labels the first row the factor x plus 2 and the second row the factor x minus 6. Test a value in each interval. The factor, x plus 2, is negative when negative 3 is substituted for x in the interval negative infinity to negative 2. The factor, x minus 6, is negative when negative 3 is substituted for x in the interval negative infinity to negative 2. The factor, x plus 2, is positive when 0 is substituted for x in the interval negative 2 to 6. The factor, x minus 6, is negative when 0 is substituted for x in the interval negative 2 to 6. The factor, x plus 2, is positive when 7 is substituted for x in the interval 6 to infinity. The factor, x minus 6, is positive when 0 is substituted for x in the interval 6 to infinity. Above the number line, show the sign of each factor of the rational expression in each interval. Below the number line, show the sign of the quotient. The number line is divided into intervals by critical points at negative 2 and 6. An open parenthesis is used at negative 2 and a closed parenthesis is used at 6. The number line is shaded between negative 2 and 6. The factors x plus 2 and x minus 6 are marked as negative above the number line for the interval negative infinity to negative 2. The quotient of the quantity x plus 2 and the quantity x minus 6 is marked as positive below the number line for the interval negative infinity to negative 2. The factor x plus 2 is marked as positive and the factor x minus 6 is marked as negative above the number line for the interval negative 2 to 6. The quotient of the quantity x plus 2 and the quantity x minus 6 is marked as negative below the number line for the interval negative 2 to 6. The factors x plus 2 and x minus 6 are marked as positive above the number line for the interval 6 to infinity. The quotient of the quantity x plus 2 and the quantity x minus 6 is marked as positive below the number line for the interval 6 to infinity. Determine the intervals where the inequality is correct. Write the solution in interval notation. The solution is the interval negative 2 and 6 with negative 2 and 6 not included.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{4x}{x-6}&lt;1\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract 1 to get zero on the right.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{4x}{x-6}-1&lt;0\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite 1 as a fraction using the LCD.<\/td><td data-align=\"right\">\\(\\frac{4x}{x-6}-\\frac{x-6}{x-6}&lt;0\\)<\/td><td><\/td><\/tr><tr><td>Subtract the numerators and place the<div data-type=\"newline\"><br><\/div>difference over the common denominator.<\/td><td data-align=\"right\">\\(\\frac{4x-\\left(x-6\\right)}{x-6}&lt;0\\)<\/td><td><\/td><\/tr><tr><td>Simplify.<\/td><td data-align=\"right\">\\(\\frac{3x+6}{x-6}&lt;0\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the numerator to show all factors.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{3\\left(x+2\\right)}{x-6}&lt;0\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the critical points.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The quotient will be zero when the numerator is zero.<div data-type=\"newline\"><br><\/div>The quotient is undefined when the denominator is zero.<\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{cccccccc}\\hfill x+2&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; \\hfill x-6&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; \\text{\u2212}2\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; 6\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Use the critical points to divide the number line into intervals.<\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Test a value in each interval.<\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826804542\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Above the number line show the sign of each factor of the rational expression in each interval.<div data-type=\"newline\"><br><\/div>Below the number line show the sign of the quotient.<\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Determine the intervals where the inequality is correct. We want the quotient to be negative, so the solution includes the points between \u22122 and 6. Since the inequality is strictly less than, the endpoints are not included.<\/td><\/tr><tr valign=\"top\"><td colspan=\"3\" data-valign=\"top\" data-align=\"left\">We write the solution in interval notation as (\u22122, 6).<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834234265\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835353129\"><div data-type=\"problem\" id=\"fs-id1167832128828\"><p>Solve and write the solution in interval notation: \\(\\frac{3x}{x-3}&lt;1.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834300261\">\\(\\left(-\\frac{3}{2},3\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835257383\"><div data-type=\"problem\" id=\"fs-id1167835595165\"><p>Solve and write the solution in interval notation: \\(\\frac{3x}{x-4}&lt;2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835269471\"><p id=\"fs-id1167835329101\">\\(\\left(-8,4\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165927605011\">In the next example, the numerator is always positive, so the sign of the rational expression depends on the sign of the denominator.<\/p><div data-type=\"example\" id=\"fs-id1167835534361\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167831884237\">Solve and write the solution in interval notation: \\(\\frac{5}{{x}^{2}-2x-15}&gt;0.\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167834162030\" class=\"unnumbered unstyled can-break\" summary=\"The inequality 5 divided by the quantity x squared minus 2 x minus 15 is greater than 0 is already in the correct form. Factor the denominator. The result is 5 divided by the product of the quantity x plus 3 and the quantity x minus 5 is greater than 0. Find the critical points. The quotient is 0 when the numerator is 0. Since the numerator is always 5, the quotient cannot be 0. The quotient will be undefined when the denominator is 0. That is the product of the quantity x plus 3 and the quantity x minus 5 is equal to 0, which is x is equal to negative 3 and x is equal to 5. Use the critical points, negative 3 and 5, to divide the number line into intervals. Test values in each interval. Above the number line, show the sign of each factor of the rational expression in each interval. Below the number line, show the sign of the quotient. The number line is divided into intervals by critical points at negative 3 and 5. A closed parenthesis is used at negative 3 and an open parenthesis is used at 5. The number line is shaded to the left of 3 and to the right of 5. The factors x plus 3 and x minus 5 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of 4 and the product of the quantity x plus 3 and the quantity x minus 5 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x plus 3 is marked as positive and the factor x minus 5 is marked as negative above the number line for the interval negative 3 to 5. The quotient of 4 and the product of the quantity x plus 3 and the quantity x minus 5 is marked as negative below the number line for the interval negative 3 to 5. The factors x plus 3 and x minus 5 are marked as positive above the number line for the interval 5 to infinity. The quotient of 4 and the product of the quantity x plus 3 and the quantity x minus 5 is marked as positive below the number line for the interval 5 to infinity. Write the solution in interval notation. The solution is the union of the interval negative infinity to negative 3 and the interval 5 to infinity, with 3 and 5 not included\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The inequality is in the correct form.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{5}{{x}^{2}-2x-15}&gt;0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the denominator.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{5}{\\left(x+3\\right)\\left(x-5\\right)}&gt;0\\)<\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Find the critical points.<div data-type=\"newline\"><br><\/div>The quotient is 0 when the numerator is 0.<div data-type=\"newline\"><br><\/div>Since the numerator is always 5, the quotient cannot be 0.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The quotient will be undefined when the<div data-type=\"newline\"><br><\/div>denominator is zero.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{c}\\left(x+3\\right)\\left(x-5\\right)=0\\hfill \\\\ x=-3,\\text{\\hspace{0.17em}}x=5\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the critical points to divide the number line into intervals.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834327372\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Test values in each interval.<div data-type=\"newline\"><br><\/div>Above the number line show the sign of each<div data-type=\"newline\"><br><\/div>factor of the denominator in each interval.<div data-type=\"newline\"><br><\/div>Below the number line, show the sign of the quotient.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the solution in interval notation.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(\\text{\u2212}\\infty ,-3\\right)\\cup \\left(5,\\infty \\right)\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835421722\"><div data-type=\"problem\" id=\"fs-id1167835338178\"><p id=\"fs-id1167834423187\">Solve and write the solution in interval notation: \\(\\frac{1}{{x}^{2}+2x-8}&gt;0.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167831197123\">\\(\\left(\\text{\u2212}\\infty ,-4\\right)\\cup \\left(2,\\infty \\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835344975\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835332644\"><div data-type=\"problem\" id=\"fs-id1167835358618\"><p id=\"fs-id1167835259002\">Solve and write the solution in interval notation: \\(\\frac{3}{{x}^{2}+x-12}&gt;0.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834247068\">\\(\\left(\\text{\u2212}\\infty ,-4\\right)\\cup \\left(3,\\infty \\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835489103\">The next example requires some work to get it into the needed form.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834309742\"><p id=\"fs-id1167835370918\">Solve and write the solution in interval notation: \\(\\frac{1}{3}-\\frac{2}{{x}^{2}}&lt;\\frac{5}{3x}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835434192\"><table class=\"unnumbered unstyled can-break\" summary=\"Solve the difference between one-third and the quantity 2 divided by x squared is less than the quantity 5 divided by 3 x. Subtract 5 divided by 3 x to get 0 on the right side. The result is one-third minus the quantity 2 divided by x squared minus the quantity 5 divided by 3 x is less than 0. Rewrite the inequality to get each fraction with the least common denominator, 3 x squared. The result is 1 times x squared all divided by 3 times x squared minus 2 times 6 all divided by x squared times 3 minus 5 times x all divided by 3 x times x is less than 0. Simplify. The result is the quantity x squared divided by 3 x squared minus the quantity 6 divided by 3 x squared minus the quantity 5 x divided by 3 x squared is less than 0. Subtract the numerators and place the difference over the common denominator. The result is the quantity x squared minus 5 x minus 6 divided by the quantity 3 x squared is less than 0. Factor the numerator. The result is the product of the quantity x minus 6 and the quantity x plus divided by 3 x squared is less than 0. Find the critical points using 3 x squared is equal to 0, x minus 6 is equal to 0, and x plus 1 is equal to 0. The critical points are x is equal to 0, x is equal to 6, and x is equal to negative 1. Use the critical points to divide the number line into intervals. Above the number line, show the sign of each factor in each interval. Below the number line, show the sign of the quotient. The number line is divided into intervals by critical points at negative 1, 0, and 6. An open parenthesis is used at negative 1, an open and closed parenthesis is used at 0, and a closed parenthesis is used at 6. The number line is shaded between negative 1 and 0 and between 0 and 6. The factors x minus 6 and x plus 1 are marked as negative above the number line for the interval negative infinity to negative 1. The factor x squared is marked as positive above the number line for the interval negative infinity to negative 1. The quotient of the product of the quantity x minus 6 and the quantity x plus 1 all divided by 3 x squared is marked as positive below the number line for the interval negative infinity to negative 1. The factor x minus 6 is marked as negative above the number line for the interval negative 1 to 0. The factors x plus 1 and x squared are marked as positive above the number line for the interval negative 1 and 0. The quotient of the product of the quantity x minus 6 and the quantity x plus 1 all divided by 3 x squared is marked as negative below the number line on the interval negative 1 to 0. The factor x minus 6 is marked as negative above the number line on the interval 0 to 6. The factors x plus 1 and x squared are marked as positive above the number line on the interval 0 to 6. The quotient of the product of the quantity x minus 6 and the quantity x plus 1 all divided by 3 x squared is marked as negative below the number line on the interval 0 to 6. The factors x minus 6, x plus 1 and x squared are marked positive above the number line on the interval 6 to infinity. The quotient of the product of the quantity x minus 6 and the quantity x plus 1 all divided by 3 x squared is marked as positive below the number line on the interval 6 to infinity. Since 0 is excluded, the solution is the two intervals negative 1 to 0 and 0 to 6, or the union of the intervals negative 1 to 0 and 0 to 6.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{1}{3}-\\frac{2}{{x}^{2}}&lt;\\frac{5}{3x}\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract \\(\\frac{5}{3x}\\) to get zero on the right.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{1}{3}-\\frac{2}{{x}^{2}}-\\frac{5}{3x}&lt;0\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite to get each fraction with the LCD \\(3{x}^{2}.\\)<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{1\\cdot {x}^{2}}{3\\cdot {x}^{2}}-\\frac{2\\cdot 3}{{x}^{2}\\cdot 3}-\\frac{5\\cdot x}{3x\\cdot x}&lt;0\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-align=\"right\">\\(\\frac{{x}^{2}}{3{x}^{2}}-\\frac{6}{3{x}^{2}}-\\frac{5x}{3{x}^{2}}&lt;0\\)<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Subtract the numerators and place the<div data-type=\"newline\"><br><\/div>difference over the common denominator.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{{x}^{2}-5x-6}{3{x}^{2}}&lt;0\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the numerator.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\frac{\\left(x-6\\right)\\left(x+1\\right)}{3{x}^{2}}&lt;0\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the critical points.<\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{}\\\\ \\hfill 3{x}^{2}&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; \\hfill x-6&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; \\hfill x+1&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; 6\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; \\text{\u2212}1\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the critical points to divide the number<div data-type=\"newline\"><br><\/div>line into intervals.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835342893\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Above the number line show the sign of each<div data-type=\"newline\"><br><\/div>factor in each interval. Below the number line, show the sign of the quotient.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since, 0 is excluded, the solution is the two<div data-type=\"newline\"><br><\/div>intervals, \\(\\left(-1,0\\right)\\) and \\(\\left(0,6\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(-1,0\\right)\\cup \\left(0,6\\right)\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834301158\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167835306283\">Solve and write the solution in interval notation: \\(\\frac{1}{2}+\\frac{4}{{x}^{2}}&lt;\\frac{3}{x}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835355496\"><p>\\(\\left(2,4\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835164921\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835369147\"><div data-type=\"problem\" id=\"fs-id1167834526570\"><p id=\"fs-id1167834593562\">Solve and write the solution in interval notation: \\(\\frac{1}{3}+\\frac{6}{{x}^{2}}&lt;\\frac{3}{x}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831871519\"><p id=\"fs-id1167830702763\">\\(\\left(3,6\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835309086\"><h3 data-type=\"title\">Solve an Inequality with Rational Functions<\/h3><p id=\"fs-id1167832152764\">When working with rational functions, it is sometimes useful to know when the function is greater than or less than a particular value. This leads to a rational inequality.<\/p><div data-type=\"example\" id=\"fs-id1167835511373\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p>Given the function \\(R\\left(x\\right)=\\frac{x+3}{x-5},\\) find the values of <em data-effect=\"italics\">x<\/em> that make the function less than or equal to 0.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832195491\"><p id=\"fs-id1167834190517\">We want the function to be less than or equal to 0.<\/p><table class=\"unnumbered unstyled\" summary=\"The function R is less than or equal to 0. Substitute the rational expression, the quotient of the quantity x plus 3 and the quantity x minus 5, for the function R. The result is the quotient of the quantity x plus 3 and the quantity x minus 5 is less than or equal to 0, where x is not equal to 5. Find the critical points using x plus 3 is equal to 3 and x minus 5 is equal to 0. The critical points are x is equal to negative 3 and x is equal to 5. Use the critical points to divide the number line into intervals. Test values in each interval. Above the number line show the sign of each factor in each interval. Below the number line, show the sign of the quotient. The number line is divided into intervals by critical points at negative 3 and 5. An open bracket is used at negative 3 and a closed parenthesis is used at 5. The number line is shaded between negative 3 and 5. The factors x plus 3 and x minus 5 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x plus 3 and the quantity x minus 5 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x plus 3 is marked as positive above the number line for the interval negative 3 to 5. The factor x minus 5 is marked as negative above the number line for the interval negative 3 to 5. The quotient of the quantity x plus 3 and the quantity x minus 5 is marked as negative below the number line on the interval negative 3 to 5. The factors x plus 3 and x minus 5 are marked as positive above the number line on the interval 5 to infinity. The quotient of the quantity x plus 3 and the quantity x minus 5 is marked as positive below the number line on the interval 5 to infinity. Write the solution in interval notation. Since 5 is excluded, we do not include it in the interval. The solution is the interval negative 3 to 5, with negative 3 included.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(R\\left(x\\right)\\le 0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute the rational expression for \\(R\\left(x\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\frac{x+3}{x-5}\\le 0\\phantom{\\rule{3.5em}{0ex}}x\\ne 5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the critical points.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{cccccccc}\\hfill x+3&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; \\hfill x-5&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; -3\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; 5\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the critical points to divide the number line into intervals.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Test values in each interval. Above the<div data-type=\"newline\"><br><\/div>number line, show the sign of each factor<div data-type=\"newline\"><br><\/div>in each interval. Below the number line,<div data-type=\"newline\"><br><\/div>show the sign of the quotient<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the solution in interval notation. Since<div data-type=\"newline\"><br><\/div>5 is excluded we, do not include it in the interval.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left[-3,5\\right)\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832054727\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831239696\"><div data-type=\"problem\" id=\"fs-id1167831117331\"><p id=\"fs-id1167835369032\">Given the function \\(R\\left(x\\right)=\\frac{x-2}{x+4},\\) find the values of <em data-effect=\"italics\">x<\/em> that make the function less than or equal to 0.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835337641\"><p id=\"fs-id1167834584349\">\\(\\left(-4,2\\right]\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834534707\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167828421075\"><div data-type=\"problem\" id=\"fs-id1167834156746\"><p>Given the function \\(R\\left(x\\right)=\\frac{x+1}{x-4},\\) find the values of <em data-effect=\"italics\">x<\/em> that make the function less than or equal to 0.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835369961\"><p id=\"fs-id1167835318799\">\\(\\left[-1,4\\right)\\)<\/p><\/div><\/div><\/div><p>In economics, the function \\(C\\left(x\\right)\\) is used to represent the cost of producing <em data-effect=\"italics\">x<\/em> units of a commodity. The average cost per unit can be found by dividing \\(C\\left(x\\right)\\) by the number of items \\(x.\\) Then, the average cost per unit is \\(c\\left(x\\right)=\\frac{C\\left(x\\right)}{x}.\\)<\/p><div data-type=\"example\" id=\"fs-id1167835283342\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835355662\"><div data-type=\"problem\"><p>The function \\(C\\left(x\\right)=10x+3000\\) represents the cost to produce \\(x,\\) number of items. Find <span class=\"token\">\u24d0<\/span> the average cost function, \\(c\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> how many items should be produced so that the average cost is less than ?40.<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835232228\"><span class=\"token\">\u24d0<\/span><\/p><p id=\"fs-id1167835308313\">\\(\\begin{array}{cccc}&amp; &amp; &amp; C\\left(x\\right)=10x+3000\\hfill \\\\ \\text{The average cost function is}\\phantom{\\rule{0.2em}{0ex}}c\\left(x\\right)=\\frac{C\\left(x\\right)}{x}.\\hfill &amp; &amp; &amp; \\\\ \\begin{array}{c}\\text{To find the average cost function, divide the}\\hfill \\\\ \\text{cost function by}\\phantom{\\rule{0.2em}{0ex}}x.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{c}c\\left(x\\right)=\\frac{C\\left(x\\right)}{x}\\hfill \\\\ c\\left(x\\right)=\\frac{10x+3000}{x}\\hfill \\end{array}\\hfill \\\\ &amp; &amp; &amp; \\text{The average cost function is}\\phantom{\\rule{0.2em}{0ex}}c\\left(x\\right)=\\frac{10x+3000}{x}.\\hfill \\end{array}\\)<\/p><p><span class=\"token\">\u24d1<\/span><\/p><p>\\(\\begin{array}{cccc}\\text{We want the function}\\phantom{\\rule{0.2em}{0ex}}c\\left(x\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{to be less than}\\phantom{\\rule{0.2em}{0ex}}40.\\hfill &amp; &amp; &amp; \\phantom{\\rule{6.4em}{0ex}}c\\left(x\\right)&lt;40\\hfill \\\\ \\text{Substitute the rational expression for}\\phantom{\\rule{0.2em}{0ex}}c\\left(x\\right).\\hfill &amp; &amp; &amp; \\phantom{\\rule{3.3em}{0ex}}\\frac{10x+3000}{x}&lt;40\\phantom{\\rule{0.5em}{0ex}}x\\ne 0\\hfill \\\\ \\text{Subtract 40 to get 0 on the right.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{1.11em}{0ex}}\\frac{10x+3000}{x}-40&lt;0\\hfill \\\\ \\begin{array}{c}\\text{Rewrite the left side as one quotient by finding}\\hfill \\\\ \\text{the LCD and performing the subtraction.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\frac{10x+3000}{x}-40\\left(\\frac{x}{x}\\right)&lt;0\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{0.5em}{0ex}}\\frac{10x+3000}{x}-\\frac{40x}{x}&lt;0\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{0.6em}{0ex}}\\frac{10x+3000-40x}{x}&lt;0\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{2.55em}{0ex}}\\frac{-30x+3000}{x}&lt;0\\hfill \\\\ \\text{Factor the numerator to show all factors.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2.4em}{0ex}}\\frac{-30\\left(x-100\\right)}{x}&lt;0\\hfill \\\\ \\text{Find the critical points.}\\hfill &amp; &amp; &amp; \\begin{array}{cccc}\\hfill -30\\left(x-100\\right)&amp; =\\hfill &amp; 0\\hfill &amp; \\phantom{\\rule{1em}{0ex}}x=0\\hfill \\\\ \\hfill -30\\ne 0\\phantom{\\rule{1em}{0ex}}x-100&amp; =\\hfill &amp; 0\\hfill &amp; \\\\ \\hfill x&amp; =\\hfill &amp; 100\\hfill &amp; \\end{array}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167835349647\">More than 100 items must be produced to keep the average cost below ?40 per item.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835321953\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167827967245\"><div data-type=\"problem\" id=\"fs-id1167834534467\"><p id=\"fs-id1167835357225\">The function \\(C\\left(x\\right)=20x+6000\\) represents the cost to produce \\(x,\\) number of items. Find <span class=\"token\">\u24d0<\/span> the average cost function, \\(c\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> how many items should be produced so that the average cost is less than ?60?<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167831883113\"><span class=\"token\">\u24d0<\/span>\\(c\\left(x\\right)=\\frac{20x+6000}{x}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> More than 150 items must be produced to keep the average cost below ?60 per item.<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826807809\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834084459\"><div data-type=\"problem\" id=\"fs-id1167834473717\"><p id=\"fs-id1167835479012\">The function \\(C\\left(x\\right)=5x+900\\) represents the cost to produce \\(x,\\) number of items. Find <span class=\"token\">\u24d0<\/span> the average cost function, \\(c\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> how many items should be produced so that the average cost is less than ?20?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834059188\"><p id=\"fs-id1167835419423\"><span class=\"token\">\u24d0<\/span>\\(c\\left(x\\right)=\\frac{5x+900}{x}\\)<span class=\"token\">\u24d1<\/span> More than 60 items must be produced to keep the average cost below ?20 per item.<\/p><\/div><\/div><\/div><\/div><div class=\"textbox\" data-depth=\"1\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167834189870\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Solve a rational inequality.<\/strong><ol id=\"fs-id1167835338241\" type=\"1\" class=\"stepwise\"><li>Write the inequality as one quotient on the left and zero on the right.<\/li><li>Determine the critical points\u2013the points where the rational expression will be zero or undefined.<\/li><li>Use the critical points to divide the number line into intervals.<\/li><li>Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient.<\/li><li>Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835325987\"><h3 data-type=\"title\">Section Exercises<\/h3><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167835531810\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167832060113\"><strong data-effect=\"bold\">Solve Rational Inequalities<\/strong><\/p><p id=\"fs-id1167834221970\">In the following exercises, solve each rational inequality and write the solution in interval notation.<\/p><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835329701\"><p>\\(\\frac{x-3}{x+4}\\ge 0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835345830\"><p>\\(\\left(\\text{\u2212}\\infty ,-4\\right)\\cup \\left[3,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835302959\"><div data-type=\"problem\"><p id=\"fs-id1167830704699\">\\(\\frac{x+6}{x-5}\\ge 0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834161495\"><div data-type=\"problem\"><p id=\"fs-id1167830693464\">\\(\\frac{x+1}{x-3}\\le 0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831846980\"><p id=\"fs-id1167835238679\">\\(\\left[-1,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830700952\"><div data-type=\"problem\"><p id=\"fs-id1167835362648\">\\(\\frac{x-4}{x+2}\\le 0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834058872\"><div data-type=\"problem\" id=\"fs-id1167835357903\"><p id=\"fs-id1167834473611\">\\(\\frac{x-7}{x-1}&gt;0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835420364\"><p id=\"fs-id1167832134098\">\\(\\left(\\text{\u2212}\\infty ,1\\right)\\cup \\left(7,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826780686\"><div data-type=\"problem\" id=\"fs-id1167834458685\"><p id=\"fs-id1167835387103\">\\(\\frac{x+8}{x+3}&gt;0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834192313\"><div data-type=\"problem\" id=\"fs-id1167834464330\"><p id=\"fs-id1167831872115\">\\(\\frac{x-6}{x+5}&lt;0\\)<\/p><\/div><div data-type=\"solution\"><p>\\(\\left(-5,6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832053800\"><div data-type=\"problem\" id=\"fs-id1167834473138\"><p id=\"fs-id1167831117326\">\\(\\frac{x+5}{x-2}&lt;0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827967212\"><div data-type=\"problem\" id=\"fs-id1167834111781\"><p>\\(\\frac{3x}{x-5}&lt;1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834151983\"><p id=\"fs-id1167835352372\">\\(\\left(-\\frac{5}{2},5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834397407\"><div data-type=\"problem\" id=\"fs-id1167835422785\"><p id=\"fs-id1167834432058\">\\(\\frac{5x}{x-2}&lt;1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826801738\"><div data-type=\"problem\"><p id=\"fs-id1167835358530\">\\(\\frac{6x}{x-6}&gt;2\\)<\/p><\/div><div data-type=\"solution\"><p>\\(\\left(\\text{\u2212}\\infty ,-3\\right)\\cup \\left(6,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830961889\"><div data-type=\"problem\" id=\"fs-id1167834141728\"><p id=\"fs-id1167830961550\">\\(\\frac{3x}{x-4}&gt;2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835269015\"><div data-type=\"problem\" id=\"fs-id1167834473828\"><p id=\"fs-id1167831892858\">\\(\\frac{2x+3}{x-6}\\le 1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835318143\"><p id=\"fs-id1167834184599\">\\(\\left[-9,6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834429013\"><div data-type=\"problem\" id=\"fs-id1167826997429\"><p id=\"fs-id1167831148828\">\\(\\frac{4x-1}{x-4}\\le 1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827943096\"><div data-type=\"problem\" id=\"fs-id1167835609306\"><p>\\(\\frac{3x-2}{x-4}\\ge 2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835468463\"><p id=\"fs-id1167835309545\">\\(\\left(\\text{\u2212}\\infty ,-6\\right]\\cup \\left(4,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831895060\"><div data-type=\"problem\" id=\"fs-id1167834501963\"><p id=\"fs-id1167834501965\">\\(\\frac{4x-3}{x-3}\\ge 2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835317202\"><div data-type=\"problem\" id=\"fs-id1167835317204\"><p id=\"fs-id1167835317206\">\\(\\frac{1}{{x}^{2}+7x+12}&gt;0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832043548\"><p id=\"fs-id1167832043550\">\\(\\left(\\text{\u2212}\\infty ,-4\\right)\\cup \\left(-3,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834523673\"><div data-type=\"problem\" id=\"fs-id1167834523675\"><p id=\"fs-id1167834523677\">\\(\\frac{1}{{x}^{2}-4x-12}&gt;0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830696702\"><div data-type=\"problem\" id=\"fs-id1167830696704\"><p id=\"fs-id1167830696707\">\\(\\frac{3}{{x}^{2}-5x+4}&lt;0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835367356\"><p id=\"fs-id1167835367358\">\\(\\left(1,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832042101\"><div data-type=\"problem\" id=\"fs-id1167832042103\"><p id=\"fs-id1167832042105\">\\(\\frac{4}{{x}^{2}+7x+12}&lt;0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832042614\"><div data-type=\"problem\" id=\"fs-id1167832042616\"><p id=\"fs-id1167831833074\">\\(\\frac{2}{2{x}^{2}+x-15}\\ge 0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831833104\"><p id=\"fs-id1167831833106\">\\(\\left(\\text{\u2212}\\infty ,-3\\right)\\cup \\left(\\frac{5}{2},\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835509903\"><div data-type=\"problem\" id=\"fs-id1167835509905\"><p id=\"fs-id1167835509907\">\\(\\frac{6}{3{x}^{2}-2x-5}\\ge 0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830960568\"><div data-type=\"problem\" id=\"fs-id1167830960570\"><p id=\"fs-id1167830960572\">\\(\\frac{-2}{6{x}^{2}-13x+6}\\le 0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830697144\"><p id=\"fs-id1167830697146\">\\(\\left(\\text{\u2212}\\infty ,\\frac{2}{3}\\right)\\cup \\left(\\frac{3}{2},\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834420250\"><div data-type=\"problem\" id=\"fs-id1167834420253\"><p id=\"fs-id1167834420255\">\\(\\frac{-1}{10{x}^{2}+11x-6}\\le 0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835534032\"><div data-type=\"problem\" id=\"fs-id1167835534034\"><p id=\"fs-id1167835534037\">\\(\\frac{1}{2}+\\frac{12}{{x}^{2}}&gt;\\frac{5}{x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831894260\"><p id=\"fs-id1167831894262\">\\(\\left(\\text{\u2212}\\infty ,0\\right)\\cup \\left(0,4\\right)\\cup \\left(6,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834527613\"><div data-type=\"problem\" id=\"fs-id1167834527615\"><p id=\"fs-id1167834527617\">\\(\\frac{1}{3}+\\frac{1}{{x}^{2}}&gt;\\frac{4}{3x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830963228\"><div data-type=\"problem\" id=\"fs-id1167834426228\"><p id=\"fs-id1167834426231\">\\(\\frac{1}{2}-\\frac{4}{{x}^{2}}\\le \\frac{1}{x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834426260\"><p id=\"fs-id1167834426263\">\\(\\left[-2,0\\right)\\cup \\left(0,4\\right]\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835533528\"><div data-type=\"problem\" id=\"fs-id1167835533530\"><p id=\"fs-id1167835533532\">\\(\\frac{1}{2}-\\frac{3}{2{x}^{2}}\\ge \\frac{1}{x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831892961\"><div data-type=\"problem\" id=\"fs-id1167831892963\"><p id=\"fs-id1167831892965\">\\(\\frac{1}{{x}^{2}-16}&lt;0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834432280\"><p id=\"fs-id1167834432282\">\\(\\left(-4,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834432302\"><div data-type=\"problem\" id=\"fs-id1167834432304\"><p id=\"fs-id1167834432306\">\\(\\frac{4}{{x}^{2}-25}&gt;0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831895237\"><div data-type=\"problem\" id=\"fs-id1167831895240\"><p id=\"fs-id1167831895242\">\\(\\frac{4}{x-2}\\ge \\frac{3}{x+1}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835332716\"><p id=\"fs-id1167835332718\">\\(\\left[-10,-1\\right)\\cup \\left(2,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835332752\"><div data-type=\"problem\" id=\"fs-id1167835332754\"><p id=\"fs-id1167830885209\">\\(\\frac{5}{x-1}\\le \\frac{4}{x+2}\\)<\/p><\/div><\/div><p id=\"fs-id1167831191351\"><strong data-effect=\"bold\">Solve an Inequality with Rational Functions<\/strong><\/p><p id=\"fs-id1167831191356\">In the following exercises, solve each rational function inequality and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167831191361\"><div data-type=\"problem\" id=\"fs-id1167831191363\"><p id=\"fs-id1167831191365\">Given the function \\(R\\left(x\\right)=\\frac{x-5}{x-2},\\) find the values of \\(x\\) that make the function less than or equal to 0.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830991631\"><p id=\"fs-id1167830991633\">\\(\\left(2,5\\right]\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834194935\"><div data-type=\"problem\" id=\"fs-id1167834194937\"><p id=\"fs-id1167834194939\">Given the function \\(R\\left(x\\right)=\\frac{x+1}{x+3},\\) find the values of \\(x\\) that make the function less than or equal to 0.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832119260\"><div data-type=\"problem\" id=\"fs-id1167832119262\"><p id=\"fs-id1167832119264\">Given the function \\(R\\left(x\\right)=\\frac{x-6}{x+2}\\), find the values of x that make the function less than or equal to 0.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831116246\"><p id=\"fs-id1167831116248\">\\(\\left(\\text{\u2212}\\infty ,-2\\right)\\cup \\left[6,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835299648\"><div data-type=\"problem\" id=\"fs-id1167835299650\"><p id=\"fs-id1167835299653\">Given the function \\(R\\left(x\\right)=\\frac{x+1}{x-4},\\) find the values of x that make the function less than or equal to 0.<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834463928\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167834463935\"><div data-type=\"problem\" id=\"fs-id1167834463937\"><p id=\"fs-id1167834463939\">Write the steps you would use to explain solving rational inequalities to your little brother.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835529307\"><p id=\"fs-id1167835529309\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835529315\"><div data-type=\"problem\" id=\"fs-id1167835529317\"><p id=\"fs-id1167835529319\">Create a rational inequality whose solution is \\(\\left(\\text{\u2212}\\infty ,-2\\right]\\cup \\left[4,\\infty \\right).\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167831893326\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167831893331\"><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-id1167831893343\" data-alt=\"This table has four columns and three 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 rational inequalities. In row 3, the I can was solve an inequality with rational functions.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and three 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 rational inequalities. In row 3, the I can was solve an inequality with rational functions.\"><\/span><p id=\"fs-id1167832151391\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p><\/div><\/div><div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167832151400\"><h3 data-type=\"title\">Chapter Review Exercises<\/h3><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832151403\"><h4 data-type=\"title\"><a href=\"\/contents\/c392efcb-8505-423b-9356-890700515e3b\" class=\"target-chapter\">Simplify, Multiply, and Divide Rational Expressions<\/a><\/h4><p id=\"fs-id1167832151411\"><strong data-effect=\"bold\">Determine the Values for Which a Rational Expression is Undefined<\/strong><\/p><p id=\"fs-id1167832151417\">In the following exercises, determine the values for which the rational expression is undefined.<\/p><div data-type=\"exercise\" id=\"fs-id1167832151420\"><div data-type=\"problem\" id=\"fs-id1167832151422\"><p id=\"fs-id1167832151424\">\\(\\frac{5a+3}{3a-2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834489514\"><p id=\"fs-id1167834489516\">\\(a\\ne \\frac{2}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834489532\"><div data-type=\"problem\" id=\"fs-id1167834489535\"><p id=\"fs-id1167834489537\">\\(\\frac{b-7}{{b}^{2}-25}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826780637\"><div data-type=\"problem\" id=\"fs-id1167826780639\"><p id=\"fs-id1167826780641\">\\(\\frac{5{x}^{2}{y}^{2}}{8y}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826780667\"><p id=\"fs-id1167826780669\">\\(y\\ne 0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834473956\"><div data-type=\"problem\" id=\"fs-id1167834473958\"><p id=\"fs-id1167834473960\">\\(\\frac{x-3}{{x}^{2}-x-30}\\)<\/p><\/div><\/div><p id=\"fs-id1167835361743\"><strong data-effect=\"bold\">Simplify Rational Expressions<\/strong><\/p><p id=\"fs-id1167835361748\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167835361752\"><div data-type=\"problem\" id=\"fs-id1167835361754\"><p id=\"fs-id1167835361756\">\\(\\frac{18}{24}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830705625\"><p id=\"fs-id1167830705627\">\\(\\frac{3}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830705638\"><div data-type=\"problem\" id=\"fs-id1167830705641\"><p id=\"fs-id1167830705643\">\\(\\frac{9{m}^{4}}{18m{n}^{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831923697\"><div data-type=\"problem\" id=\"fs-id1167831923700\"><p id=\"fs-id1167831923702\">\\(\\frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834219693\"><p id=\"fs-id1167834219695\">\\(\\frac{x+3}{x+4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835390012\"><div data-type=\"problem\" id=\"fs-id1167835390014\"><p id=\"fs-id1167835390016\">\\(\\frac{7v-35}{25-{v}^{2}}\\)<\/p><\/div><\/div><p id=\"fs-id1167834473291\"><strong data-effect=\"bold\">Multiply Rational Expressions<\/strong><\/p><p id=\"fs-id1167834473296\">In the following exercises, multiply.<\/p><div data-type=\"exercise\" id=\"fs-id1167834473299\"><div data-type=\"problem\" id=\"fs-id1167834473301\"><p id=\"fs-id1167834473304\">\\(\\frac{5}{8}\u00b7\\frac{4}{15}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835281051\"><p id=\"fs-id1167835281053\">\\(\\frac{1}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835281065\"><div data-type=\"problem\" id=\"fs-id1167835281067\"><p id=\"fs-id1167835281069\">\\(\\frac{3x{y}^{2}}{8{y}^{3}}\u00b7\\frac{16{y}^{2}}{24x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831881468\"><div data-type=\"problem\" id=\"fs-id1167831086784\"><p id=\"fs-id1167831086786\">\\(\\frac{72x-12{x}^{2}}{8x+32}\u00b7\\frac{{x}^{2}+10x+24}{{x}^{2}-36}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835415767\"><p id=\"fs-id1167835415769\">\\(\\frac{-3x}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835415784\"><div data-type=\"problem\" id=\"fs-id1167835415786\"><p id=\"fs-id1167835415788\">\\(\\frac{6{y}^{2}-2y-10}{9-{y}^{2}}\u00b7\\frac{{y}^{2}-6y+9}{6{y}^{2}+29y-20}\\)<\/p><\/div><\/div><p id=\"fs-id1167835389850\"><strong data-effect=\"bold\">Divide Rational Expressions<\/strong><\/p><p id=\"fs-id1167835389856\">In the following exercises, divide.<\/p><div data-type=\"exercise\" id=\"fs-id1167835389859\"><div data-type=\"problem\" id=\"fs-id1167835389861\"><p id=\"fs-id1167835389863\">\\(\\frac{{x}^{2}-4x+12}{{x}^{2}+8x+12}\u00f7\\frac{{x}^{2}-36}{3x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831995222\"><p id=\"fs-id1167831995224\">\\(\\frac{3x}{\\left(x+6\\right)\\left(x+6\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835530095\"><div data-type=\"problem\" id=\"fs-id1167835530097\"><p id=\"fs-id1167835530099\">\\(\\frac{{y}^{2}-16}{4}\u00f7\\frac{{y}^{3}-64}{2{y}^{2}+8y+32}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834517331\"><div data-type=\"problem\" id=\"fs-id1167834517333\"><p id=\"fs-id1167834517336\">\\(\\frac{11+w}{w-9}\u00f7\\frac{121-{w}^{2}}{9-w}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834282574\"><p id=\"fs-id1167834282576\">\\(\\frac{1}{11+w}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832026070\"><div data-type=\"problem\" id=\"fs-id1167832026072\"><p id=\"fs-id1167832026074\">\\(\\frac{3{y}^{2}-12y-63}{4y+3}\u00f7\\left(6{y}^{2}-42y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834554815\"><div data-type=\"problem\" id=\"fs-id1167834554817\"><p id=\"fs-id1167834554819\">\\(\\frac{\\frac{{c}^{2}-64}{3{c}^{2}+26c+16}}{\\frac{{c}^{2}-4c-32}{15c+10}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828235423\"><p id=\"fs-id1167830700855\">\\(\\frac{5}{c+4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830700872\"><div data-type=\"problem\" id=\"fs-id1167830700875\"><p id=\"fs-id1167830700877\">\\(\\frac{8{a}^{2}+16a}{a-4}\u00b7\\frac{{a}^{2}+2a-24}{{a}^{2}+7a+10}\u00f7\\frac{2{a}^{2}-6a}{a+5}\\)<\/p><\/div><\/div><p id=\"fs-id1167834282615\"><strong data-effect=\"bold\">Multiply and Divide Rational Functions<\/strong><\/p><div data-type=\"exercise\" id=\"fs-id1167834282621\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834282623\"><p id=\"fs-id1167834282625\">Find \\(R\\left(x\\right)=f\\left(x\\right)\u00b7g\\left(x\\right)\\) where \\(f\\left(x\\right)=\\frac{9{x}^{2}+9x}{{x}^{2}-3x-4}\\) and \\(g\\left(x\\right)=\\frac{{x}^{2}-16}{3{x}^{2}+12x}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834141892\"><p id=\"fs-id1167834141894\">\\(R\\left(x\\right)=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831911673\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831911675\"><p id=\"fs-id1167831911677\">Find \\(R\\left(x\\right)=\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\) where \\(f\\left(x\\right)=\\frac{27{x}^{2}}{3x-21}\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(g\\left(x\\right)=\\frac{9{x}^{2}+54x}{{x}^{2}-x-42}.\\)<\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835609846\"><h4 data-type=\"title\"><a href=\"\/contents\/6c5aa31f-890a-445a-9a9c-44a7941355f7\" class=\"target-chapter\">Add and Subtract Rational Expressions<\/a><\/h4><p id=\"fs-id1167835609854\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions with a Common Denominator<\/strong><\/p><p id=\"fs-id1167835609860\">In the following exercises, perform the indicated operations.<\/p><div data-type=\"exercise\" id=\"fs-id1167835609863\"><div data-type=\"problem\" id=\"fs-id1167835609865\"><p id=\"fs-id1167835609867\">\\(\\frac{7}{15}+\\frac{8}{15}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834533452\"><p id=\"fs-id1167834533454\">\\(1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834533462\"><div data-type=\"problem\" id=\"fs-id1167834533464\"><p id=\"fs-id1167834533466\">\\(\\frac{4{a}^{2}}{2a-1}-\\frac{1}{2a-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831884309\"><div data-type=\"problem\" id=\"fs-id1167831884311\"><p id=\"fs-id1167831892787\">\\(\\frac{{y}^{2}+10y}{y+5}+\\frac{25}{y+5}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167831892830\">\\(y+5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835509750\"><div data-type=\"problem\" id=\"fs-id1167835509752\"><p id=\"fs-id1167835509754\">\\(\\frac{7{x}^{2}}{{x}^{2}-9}+\\frac{21x}{{x}^{2}-9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831919688\"><div data-type=\"problem\" id=\"fs-id1167831919690\"><p id=\"fs-id1167831919692\">\\(\\frac{{x}^{2}}{x-7}-\\frac{3x+28}{x-7}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835359033\"><p id=\"fs-id1167835359036\">\\(x+4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831883771\"><div data-type=\"problem\" id=\"fs-id1167831883773\"><p id=\"fs-id1167831883775\">\\(\\frac{{y}^{2}}{y+11}-\\frac{121}{y+11}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834111847\"><div data-type=\"problem\" id=\"fs-id1167834111849\"><p id=\"fs-id1167834111851\">\\(\\frac{4{q}^{2}-q+3}{{q}^{2}+6q+5}-\\frac{3{q}^{2}-q-6}{{q}^{2}+6q+5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832074864\"><p id=\"fs-id1167832074866\">\\(\\frac{q-3}{q+5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832074888\"><div data-type=\"problem\" id=\"fs-id1167832074890\"><p id=\"fs-id1167832074892\">\\(\\frac{5t+4t+3}{{t}^{2}-25}-\\frac{4{t}^{2}-8t-32}{{t}^{2}-25}\\)<\/p><\/div><\/div><p id=\"fs-id1167831862571\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions Whose Denominators Are Opposites<\/strong><\/p><p id=\"fs-id1167831862576\">In the following exercises, add and subtract.<\/p><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834079478\"><p id=\"fs-id1167834079480\">\\(\\frac{18w}{6w-1}+\\frac{3w-2}{1-6w}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834133902\"><p>\\(\\frac{15w+2}{6w-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834133931\"><div data-type=\"problem\" id=\"fs-id1167834133933\"><p id=\"fs-id1167834133935\">\\(\\frac{{a}^{2}+3a}{{a}^{2}-4}-\\frac{3a-8}{4-{a}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835414934\"><div data-type=\"problem\" id=\"fs-id1167835414936\"><p id=\"fs-id1167835414938\">\\(\\frac{2{b}^{2}+3b-15}{{b}^{2}-49}-\\frac{{b}^{2}+16b-1}{49-{b}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826857038\"><p id=\"fs-id1167826857040\">\\(\\frac{3b-2}{b+7}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826857064\"><div data-type=\"problem\" id=\"fs-id1167826857066\"><p id=\"fs-id1167826857068\">\\(\\frac{8{y}^{2}-10y+7}{2y-5}+\\frac{2{y}^{2}+7y+2}{5-2y}\\)<\/p><\/div><\/div><p id=\"fs-id1167834454265\"><strong data-effect=\"bold\">Find the Least Common Denominator of Rational Expressions<\/strong><\/p><p id=\"fs-id1167834454270\">In the following exercises, find the LCD.<\/p><div data-type=\"exercise\" id=\"fs-id1167834454273\"><div data-type=\"problem\" id=\"fs-id1167834454276\"><p id=\"fs-id1167834454278\">\\(\\frac{7}{{a}^{2}-3a-10},\\frac{3a}{{a}^{2}-a-20}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831920685\"><p id=\"fs-id1167831920687\">\\(\\left(a+2\\right)\\left(a-5\\right)\\left(a+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835320099\"><div data-type=\"problem\" id=\"fs-id1167835320101\"><p id=\"fs-id1167835320103\">\\(\\frac{6}{{n}^{2}-4},\\frac{2n}{{n}^{2}-4n+4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831822005\"><div data-type=\"problem\" id=\"fs-id1167831822007\"><p id=\"fs-id1167835622856\">\\(\\frac{5}{3{p}^{2}+17p-6},\\frac{2m}{3{p}^{2}-23p-8}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834376197\"><p id=\"fs-id1167834376200\">\\(\\left(3p+1\\right)\\left(p+6\\right)\\left(p+8\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167834327388\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions with Unlike Denominators<\/strong><\/p><p id=\"fs-id1167834327394\">In the following exercises, perform the indicated operations.<\/p><div data-type=\"exercise\" id=\"fs-id1167834327397\"><div data-type=\"problem\" id=\"fs-id1167834327399\"><p id=\"fs-id1167834327401\">\\(\\frac{7}{5a}+\\frac{3}{2b}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835301259\"><div data-type=\"problem\" id=\"fs-id1167835301261\"><p id=\"fs-id1167835301264\">\\(\\frac{2}{c-2}+\\frac{9}{c+3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834299889\"><p id=\"fs-id1167834299891\">\\(\\frac{11c-12}{\\left(c-2\\right)\\left(c+3\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826995141\"><div data-type=\"problem\" id=\"fs-id1167826995143\"><p id=\"fs-id1167826995145\">\\(\\frac{3x}{{x}^{2}-9}+\\frac{5}{{x}^{2}+6x+9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835215385\"><div data-type=\"problem\" id=\"fs-id1167835215387\"><p id=\"fs-id1167835215389\">\\(\\frac{2x}{{x}^{2}+10x+24}+\\frac{3x}{{x}^{2}+8x+16}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835417204\"><p id=\"fs-id1167835417206\">\\(\\frac{5{x}^{2}+26x}{\\left(x+4\\right)\\left(x+4\\right)\\left(x+6\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826808680\"><div data-type=\"problem\" id=\"fs-id1167826808682\"><p id=\"fs-id1167826808684\">\\(\\frac{5q}{{p}^{2}q-{p}^{2}}+\\frac{4q}{{q}^{2}-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835410632\"><div data-type=\"problem\" id=\"fs-id1167835410635\"><p id=\"fs-id1167830698010\">\\(\\frac{3y}{y+2}-\\frac{y+2}{y+8}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830698047\"><p id=\"fs-id1167830698050\">\\(\\frac{2\\left({y}^{2}+10y-2\\right)}{\\left(y+2\\right)\\left(y+8\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835550842\"><div data-type=\"problem\" id=\"fs-id1167835550844\"><p id=\"fs-id1167835550846\">\\(\\frac{-3w-15}{{w}^{2}+w-20}-\\frac{w+2}{4-w}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834525788\"><div data-type=\"problem\" id=\"fs-id1167834525790\"><p id=\"fs-id1167834525792\">\\(\\frac{7m+3}{m+2}-5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835302571\"><p id=\"fs-id1167835302573\">\\(\\frac{2m-7}{m+2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835302598\"><div data-type=\"problem\" id=\"fs-id1167835302600\"><p id=\"fs-id1167835320304\">\\(\\frac{n}{n+3}+\\frac{2}{n-3}-\\frac{n-9}{{n}^{2}-9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535865\"><div data-type=\"problem\" id=\"fs-id1167834535867\"><p id=\"fs-id1167834535869\">\\(\\frac{8a}{{a}^{2}-64}-\\frac{4}{a+8}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826863901\"><p id=\"fs-id1167826863903\">\\(\\frac{4}{a-8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826863920\"><div data-type=\"problem\" id=\"fs-id1167826863923\"><p id=\"fs-id1167826863925\">\\(\\frac{5}{12{x}^{2}y}+\\frac{7}{20x{y}^{3}}\\)<\/p><\/div><\/div><p id=\"fs-id1167832042426\"><strong data-effect=\"bold\">Add and Subtract Rational Functions<\/strong><\/p><p id=\"fs-id1167832042431\">In the following exercises, find \\(R\\left(x\\right)=f\\left(x\\right)+g\\left(x\\right)\\) where \\(f\\left(x\\right)\\) and \\(g\\left(x\\right)\\) are given.<\/p><div data-type=\"exercise\" id=\"fs-id1167834397163\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834397165\"><p id=\"fs-id1167834397167\">\\(f\\left(x\\right)=\\frac{2{x}^{2}+12x-11}{{x}^{2}+3x-10},\\)\\(g\\left(x\\right)=\\frac{x+1}{2-x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826779457\"><p id=\"fs-id1167826779459\">\\(R\\left(x\\right)=\\frac{x+8}{x+5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830698105\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830698107\"><p id=\"fs-id1167830698109\">\\(f\\left(x\\right)=\\frac{-4x+31}{{x}^{2}+x-30},\\)\\(g\\left(x\\right)=\\frac{5}{x+6}\\)<\/p><\/div><\/div><p id=\"fs-id1167835533765\">In the following exercises, find \\(R\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)\\) where \\(f\\left(x\\right)\\) and \\(g\\left(x\\right)\\) are given.<\/p><div data-type=\"exercise\" id=\"fs-id1167827966859\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167827966861\"><p id=\"fs-id1167827966863\">\\(f\\left(x\\right)=\\frac{4x}{{x}^{2}-121},\\)\\(g\\left(x\\right)=\\frac{2}{x-11}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831031167\"><p id=\"fs-id1167831031169\">\\(R\\left(x\\right)=\\frac{2}{x+11}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831031198\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831031200\"><p id=\"fs-id1167831031202\">\\(f\\left(x\\right)=\\frac{7}{x+6},\\)\\(g\\left(x\\right)=\\frac{14x}{{x}^{2}-36}\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834183710\"><h4 data-type=\"title\"><a href=\"\/contents\/9c6e36cd-6eb0-4457-8db5-0da97afca2ac\" class=\"target-chapter\">Simplify Complex Rational Expressions<\/a><\/h4><p id=\"fs-id1167834183718\"><strong data-effect=\"bold\">Simplify a Complex Rational Expression by Writing It as Division<\/strong><\/p><p id=\"fs-id1167834525808\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167834525811\"><div data-type=\"problem\" id=\"fs-id1167834525813\"><p id=\"fs-id1167834525815\">\\(\\frac{\\frac{7x}{x+2}}{\\frac{14{x}^{2}}{{x}^{2}-4}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831825653\"><p id=\"fs-id1167831825655\">\\(\\frac{x-2}{2x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831825675\"><div data-type=\"problem\" id=\"fs-id1167831825678\"><p id=\"fs-id1167831825680\">\\(\\frac{\\frac{2}{5}+\\frac{5}{6}}{\\frac{1}{3}+\\frac{1}{4}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831115347\"><div data-type=\"problem\" id=\"fs-id1167831115349\"><p id=\"fs-id1167831115351\">\\(\\frac{x-\\frac{3x}{x+5}}{\\frac{1}{x+5}+\\frac{1}{x-5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826864294\"><p id=\"fs-id1167826864296\">\\(\\frac{\\left(x-8\\right)\\left(x-5\\right)}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826932489\"><div data-type=\"problem\" id=\"fs-id1167826932492\"><p id=\"fs-id1167826932494\">\\(\\frac{\\frac{2}{m}+\\frac{m}{n}}{\\frac{n}{m}-\\frac{1}{n}}\\)<\/p><\/div><\/div><p id=\"fs-id1167835378244\"><strong data-effect=\"bold\">Simplify a Complex Rational Expression by Using the LCD<\/strong><\/p><p id=\"fs-id1167835378250\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167835378253\"><div data-type=\"problem\" id=\"fs-id1167835368010\"><p id=\"fs-id1167835368013\">\\(\\frac{\\frac{1}{3}+\\frac{1}{8}}{\\frac{1}{4}+\\frac{1}{12}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835368048\"><p id=\"fs-id1167835368050\">\\(\\frac{11}{8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831881490\"><div data-type=\"problem\" id=\"fs-id1167831881493\"><p id=\"fs-id1167831881495\">\\(\\frac{\\frac{3}{{a}^{2}}-\\frac{1}{b}}{\\frac{1}{a}+\\frac{1}{{b}^{2}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835180539\"><div data-type=\"problem\" id=\"fs-id1167835180542\"><p id=\"fs-id1167835180544\">\\(\\frac{\\frac{2}{{z}^{2}-49}+\\frac{1}{z+7}}{\\frac{9}{z+7}+\\frac{12}{z-7}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831116404\"><p id=\"fs-id1167831116407\">\\(\\frac{z-5}{23z+21}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831116431\"><div data-type=\"problem\" id=\"fs-id1167831116433\"><p id=\"fs-id1167831116435\">\\(\\frac{\\frac{3}{{y}^{2}-4y-32}}{\\frac{2}{y-8}+\\frac{1}{y+4}}\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834130696\"><h4 data-type=\"title\"><a href=\"\/contents\/114b6c20-ac2e-4d26-8ede-f6f4a0bce191\" class=\"target-chapter\">7.4 Solve Rational Equations<\/a><\/h4><p id=\"fs-id1167831887997\"><strong data-effect=\"bold\">Solve Rational Equations<\/strong><\/p><p id=\"fs-id1167831888003\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167831888006\"><div data-type=\"problem\" id=\"fs-id1167831888008\"><p id=\"fs-id1167831888010\">\\(\\frac{1}{2}+\\frac{2}{3}=\\frac{1}{x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831888036\"><p id=\"fs-id1167831888038\">\\(x=\\frac{6}{7}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831103003\"><div data-type=\"problem\" id=\"fs-id1167831103005\"><p id=\"fs-id1167831103007\">\\(1-\\frac{2}{m}=\\frac{8}{{m}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831922363\"><div data-type=\"problem\" id=\"fs-id1167831922365\"><p id=\"fs-id1167831922367\">\\(\\frac{1}{b-2}+\\frac{1}{b+2}=\\frac{3}{{b}^{2}-4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835531491\"><p id=\"fs-id1167835531493\">\\(b=\\frac{3}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835531509\"><div data-type=\"problem\" id=\"fs-id1167835531511\"><p id=\"fs-id1167835531513\">\\(\\frac{3}{q+8}-\\frac{2}{q-2}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826940578\"><div data-type=\"problem\" id=\"fs-id1167826940580\"><p id=\"fs-id1167826940582\">\\(\\frac{v-15}{{v}^{2}-9v+18}=\\frac{4}{v-3}+\\frac{2}{v-6}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835621206\"><p id=\"fs-id1167835621208\">no solution <\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835621214\"><div data-type=\"problem\" id=\"fs-id1167835621216\"><p id=\"fs-id1167835621218\">\\(\\frac{z}{12}+\\frac{z+3}{3z}=\\frac{1}{z}\\)<\/p><\/div><\/div><p id=\"fs-id1167835414568\"><strong data-effect=\"bold\">Solve Rational Equations that Involve Functions<\/strong><\/p><div data-type=\"exercise\" id=\"fs-id1167835414573\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834429412\"><p id=\"fs-id1167834429414\">For rational function, \\(f\\left(x\\right)=\\frac{x+2}{{x}^{2}-6x+8},\\) <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> solve \\(f\\left(x\\right)=1\\) <span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835621277\"><p id=\"fs-id1167835621279\"><span class=\"token\">\u24d0<\/span> The domain is all real numbers except \\(x\\ne 2\\) and \\(x\\ne 4.\\) <span class=\"token\">\u24d1<\/span> \\(x=1,x=6\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> \\(\\left(1,1\\right),\\left(6,1\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832153353\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832153356\"><p id=\"fs-id1167832153358\">For rational function, \\(f\\left(x\\right)=\\frac{2-x}{{x}^{2}+7x+10},\\) <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> solve \\(f\\left(x\\right)=2\\) <span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/p><\/div><\/div><p id=\"fs-id1167834539223\"><strong data-effect=\"bold\">Solve a Rational Equation for a Specific Variable<\/strong><\/p><p id=\"fs-id1167834539228\">In the following exercises, solve for the indicated variable.<\/p><div data-type=\"exercise\" id=\"fs-id1167834539232\"><div data-type=\"problem\" id=\"fs-id1167834539234\"><p id=\"fs-id1167834539236\">\\(\\frac{V}{l}=hw\\) for \\(l.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830702695\"><p id=\"fs-id1167830702697\">\\(l=\\frac{V}{hw}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830702716\"><div data-type=\"problem\" id=\"fs-id1167830702719\"><p id=\"fs-id1167830702721\">\\(\\frac{1}{x}-\\frac{2}{y}=5\\) for \\(y.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834191397\"><div data-type=\"problem\" id=\"fs-id1167834191399\"><p id=\"fs-id1167834191401\">\\(x=\\frac{y+5}{z-7}\\) for \\(z.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834309683\"><p id=\"fs-id1167834309685\">\\(z=\\frac{y+5+7x}{x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835353171\"><div data-type=\"problem\" id=\"fs-id1167835353173\"><p id=\"fs-id1167835353175\">\\(P=\\frac{k}{V}\\) for \\(V.\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835353217\"><h4 data-type=\"title\"><a href=\"\/contents\/b694470e-ede5-454b-8052-34b9bc92ae11\" class=\"target-chapter\">Solve Applications with Rational Equations<\/a><\/h4><p id=\"fs-id1167834239074\"><strong data-effect=\"bold\">Solve Proportions<\/strong><\/p><p id=\"fs-id1167834239080\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167834239083\"><div data-type=\"problem\" id=\"fs-id1167834239085\"><p id=\"fs-id1167834239087\">\\(\\frac{x}{4}=\\frac{3}{5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834239105\"><p id=\"fs-id1167834239107\">\\(\\frac{12}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831909917\"><div data-type=\"problem\" id=\"fs-id1167831909920\"><p id=\"fs-id1167831909922\">\\(\\frac{3}{y}=\\frac{9}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831909954\"><div data-type=\"problem\" id=\"fs-id1167831909956\"><p id=\"fs-id1167831909958\">\\(\\frac{s}{s+20}=\\frac{3}{7}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830698152\"><p id=\"fs-id1167830698154\">\\(15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830698162\"><div data-type=\"problem\" id=\"fs-id1167830698164\"><p id=\"fs-id1167830698167\">\\(\\frac{t-3}{5}=\\frac{t+2}{9}\\)<\/p><\/div><\/div><p id=\"fs-id1167835621389\"><strong data-effect=\"bold\">Solve Using Proportions<\/strong><\/p><p id=\"fs-id1167835621395\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167835621398\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835621400\"><p id=\"fs-id1167835621402\">Rachael had a 21-ounce strawberry shake that has 739 calories. How many calories are there in a 32-ounce shake?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835621408\"><p id=\"fs-id1167835621410\">\\(1161\\) calories<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835621419\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835621421\"><p id=\"fs-id1167835621424\">Leo went to Mexico over Christmas break and changed ?525 dollars into Mexican pesos. At that time, the exchange rate had ?1 US is equal to 16.25 Mexican pesos. How many Mexican pesos did he get for his trip?<\/p><\/div><\/div><p id=\"fs-id1167832215360\"><strong data-effect=\"bold\">Solve Similar Figure Applications<\/strong><\/p><p id=\"fs-id1171790307463\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167832215366\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832215368\"><p id=\"fs-id1167832215371\">\\(\\text{\u0394}ABC\\) is similar to \\(\\text{\u0394}XYZ.\\) The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.<\/p><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167834192033\" data-alt=\"The first figure is triangle A B C with side A B 8 units long, side B C 7 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 2 and two-thirds units long, side Y Z x units long, and side X Z 3 units long.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B 8 units long, side B C 7 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 2 and two-thirds units long, side Y Z x units long, and side X Z 3 units long.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167834192003\"><p id=\"fs-id1167834192005\">\\(b=9;x=2\\frac{1}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834192043\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826803599\"><p id=\"fs-id1167826803602\">On a map of Europe, Paris, Rome, and Vienna form a triangle whose sides are shown in the figure below. If the actual distance from Rome to Vienna is 700 miles, find the distance from<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> Paris to Rome<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Paris to Vienna<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167826803633\" data-alt=\"The figure is a triangle formed by Paris, Vienna, and Rome. The distance between Paris and Vienna is 7.7 centimeters. The distance between Vienna and Rome is 7 centimeters. The distance between Rome and Paris is 8.9 centimeters.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by Paris, Vienna, and Rome. The distance between Paris and Vienna is 7.7 centimeters. The distance between Vienna and Rome is 7 centimeters. The distance between Rome and Paris is 8.9 centimeters.\"><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826803645\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826803647\"><p id=\"fs-id1167826803649\">Francesca is 5.75 feet tall. Late one afternoon, her shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835317998\"><p id=\"fs-id1167835318000\">23 feet<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835318006\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835318008\"><p id=\"fs-id1167835318010\">The height of a lighthouse in Pensacola, Florida is 150 feet. Standing next to the statue, 5.5-foot-tall Natasha cast a 1.1-foot shadow. How long would the shadow of the lighthouse be?<\/p><\/div><\/div><p id=\"fs-id1167835318023\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p><p id=\"fs-id1167835318029\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167835318032\"><div data-type=\"problem\" id=\"fs-id1167835318034\"><p id=\"fs-id1167835318036\">When making the 5-hour drive home from visiting her parents, Lolo ran into bad weather. She was able to drive 176 miles while the weather was good, but then driving 10 mph slower, went 81 miles when it turned bad. How fast did she drive when the weather was bad?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835318043\"><p id=\"fs-id1167835318045\">\\(45\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835621321\"><div data-type=\"problem\" id=\"fs-id1167835621323\"><p id=\"fs-id1167835621325\">Mark is riding on a plane that can fly 490 miles with a tailwind of 20 mph in the same time that it can fly 350 miles against a tailwind of 20 mph. What is the speed of the plane?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835621343\"><div data-type=\"problem\" id=\"fs-id1167835621345\"><p id=\"fs-id1167835621347\">Josue can ride his bicycle 8 mph faster than Arjun can ride his bike. It takes Luke 3 hours longer than Josue to ride 48 miles. How fast can John ride his bike?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835621353\"><p id=\"fs-id1167835621355\">\\(16\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835621364\"><div data-type=\"problem\" id=\"fs-id1167835621366\"><p id=\"fs-id1167835621369\">Curtis was training for a triathlon. He ran 8 kilometers and biked 32 kilometers in a total of 3 hours. His running speed was 8 kilometers per hour less than his biking speed. What was his running speed?<\/p><\/div><\/div><p id=\"fs-id1167831949094\"><strong data-effect=\"bold\">Solve Work Applications<\/strong><\/p><p id=\"fs-id1167831949099\">In the following exercises, solve.<\/p><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167831949104\"><p id=\"fs-id1167831949106\">Brandy can frame a room in 1 hour, while Jake takes 4 hours. How long could they frame a room working together?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831949112\"><p id=\"fs-id1167831949114\">\\(\\frac{4}{5}\\) hour<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831949127\"><div data-type=\"problem\" id=\"fs-id1167831949129\"><p id=\"fs-id1167831949131\">Prem takes 3 hours to mow the lawn while her cousin, Barb, takes 2 hours. How long will it take them working together?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832116068\"><div data-type=\"problem\" id=\"fs-id1167832116071\"><p id=\"fs-id1167832116073\">Jeffrey can paint a house in 6 days, but if he gets a helper he can do it in 4 days. How long would it take the helper to paint the house alone?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832116078\"><p id=\"fs-id1167832116080\">\\(12\\) days<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832116090\"><div data-type=\"problem\" id=\"fs-id1167832116092\"><p id=\"fs-id1167832116094\">Marta and Deb work together writing a book that takes them 90 days. If Sue worked alone it would take her 120 days. How long would it take Deb to write the book alone?<\/p><\/div><\/div><p id=\"fs-id1167830703231\"><strong data-effect=\"bold\">Solve Direct Variation Problems<\/strong><\/p><p id=\"fs-id1167830703236\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167830703239\"><div data-type=\"problem\" id=\"fs-id1167830703241\"><p id=\"fs-id1167830703243\">If \\(y\\) varies directly as \\(x\\) when \\(y=9\\) and \\(x=3,\\) find \\(x\\) when \\(y=21.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830809300\"><p id=\"fs-id1167830809303\">\\(7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830809310\"><div data-type=\"problem\" id=\"fs-id1167830809312\"><p id=\"fs-id1167830809314\">If \\(y\\) varies inversely as \\(x\\) when \\(y=20\\) and \\(x=2,\\) find \\(y\\) when \\(x=4.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830697548\"><div data-type=\"problem\" id=\"fs-id1167830697550\"><p id=\"fs-id1167830697552\">Vanessa is traveling to see her fianc\u00e9. The distance, \\(d,\\) varies directly with the speed, \\(v,\\) she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835414678\"><p id=\"fs-id1167835414680\">\\(301\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835414690\"><div data-type=\"problem\" id=\"fs-id1167835414692\"><p id=\"fs-id1167835414694\">If the cost of a pizza varies directly with its diameter, and if an 8\u201d diameter pizza costs ?12, how much would a 6\u201d diameter pizza cost?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835517931\"><div data-type=\"problem\" id=\"fs-id1167835517933\"><p id=\"fs-id1167835517935\">The distance to stop a car varies directly with the square of its speed. It takes 200 feet to stop a car going 50 mph. How many feet would it take to stop a car going 60 mph?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835517941\"><p id=\"fs-id1167835517943\">\\(288\\) feet<\/p><\/div><\/div><p id=\"fs-id1167835517952\"><strong data-effect=\"bold\">Solve Inverse Variation Problems<\/strong><\/p><p id=\"fs-id1167835517958\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167835517961\"><div data-type=\"problem\" id=\"fs-id1167835517963\"><p id=\"fs-id1167835517965\">If \\(m\\) varies inversely with the square of \\(n,\\) when \\(m=4\\) and \\(n=6\\) find \\(m\\) when \\(n=2.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831920975\"><div data-type=\"problem\" id=\"fs-id1167831920978\"><p id=\"fs-id1167831920980\">The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase 12 tickets for ?6, how many tickets can Madelyn afford to buy if the price increased to ?8?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835377879\"><p id=\"fs-id1167835377881\">\\(97\\) tickets<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835377890\"><div data-type=\"problem\" id=\"fs-id1167835377893\"><p id=\"fs-id1167835377895\">On a string instrument, the length of a string varies inversely with the frequency of its vibrations. If an 11-inch string on a violin has a frequency of 360 cycles per second, what frequency does a 12-inch string have?<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835377914\"><h4 data-type=\"title\"><a href=\"\/contents\/a68b06f6-2833-4512-b24f-c0da889a8759\" class=\"target-chapter\">Solve Rational Inequalities<\/a><\/h4><p id=\"fs-id1167834327303\"><strong data-effect=\"bold\">Solve Rational Inequalities<\/strong><\/p><p id=\"fs-id1167834327308\">In the following exercises, solve each rational inequality and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167834327312\"><div data-type=\"problem\" id=\"fs-id1167834327314\"><p id=\"fs-id1167834327316\">\\(\\frac{x-3}{x+4}\\le 0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834327342\"><p id=\"fs-id1167834327344\">\\(\\left(-4,3\\right]\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832117897\"><div data-type=\"problem\" id=\"fs-id1167832117899\"><p id=\"fs-id1167832117901\">\\(\\frac{5x}{x-2}&gt;1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826987657\"><div data-type=\"problem\" id=\"fs-id1167826987659\"><p id=\"fs-id1167826987662\">\\(\\frac{3x-2}{x-4}\\le 2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826987689\"><p id=\"fs-id1167835497978\">\\(\\left[-6,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835497997\"><div data-type=\"problem\" id=\"fs-id1167835497999\"><p id=\"fs-id1167835498002\">\\(\\frac{1}{{x}^{2}-4x-12}&lt;0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834525225\"><div data-type=\"problem\" id=\"fs-id1167834525227\"><p id=\"fs-id1167834525229\">\\(\\frac{1}{2}-\\frac{4}{{x}^{2}}\\ge \\frac{1}{x}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834525261\">\\(\\left(\\text{\u2212}\\infty ,-2\\right]\\cup \\left[4,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835414782\"><div data-type=\"problem\" id=\"fs-id1167835414784\"><p id=\"fs-id1167835414786\">\\(\\frac{4}{x-2}&lt;\\frac{3}{x+1}\\)<\/p><\/div><\/div><p id=\"fs-id1167832068272\"><strong data-effect=\"bold\">Solve an Inequality with Rational Functions<\/strong><\/p><p id=\"fs-id1167832068277\">In the following exercises, solve each rational function inequality and write the solution in interval notation<\/p><div data-type=\"exercise\" id=\"fs-id1167828441724\"><div data-type=\"problem\" id=\"fs-id1167828441726\"><p id=\"fs-id1167828441729\">Given the function, \\(R\\left(x\\right)=\\frac{x-5}{x-2},\\) find the values of \\(x\\) that make the function greater than or equal to 0.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828441772\"><p id=\"fs-id1167828441774\">\\(\\left(\\text{\u2212}\\infty ,2\\right)\\cup \\left[5,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827957228\"><div data-type=\"problem\" id=\"fs-id1167827957230\"><p id=\"fs-id1167827957232\">Given the function, \\(R\\left(x\\right)=\\frac{x+1}{x+3},\\) find the values of \\(x\\) that make the function less than or equal to 0.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827957618\"><div data-type=\"problem\" id=\"fs-id1167827957620\"><p id=\"fs-id1167827957622\">The function<\/p><div data-type=\"newline\"><br><\/div>\\(C\\left(x\\right)=150x+100,000\\) represents the cost to produce \\(x,\\) number of items. Find <span class=\"token\">\u24d0<\/span> the average cost function, \\(c\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> how many items should be produced so that the average cost is less than ?160.<\/div><div data-type=\"solution\" id=\"fs-id1167834463827\"><p id=\"fs-id1167834463829\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(c\\left(x\\right)=\\frac{150x+100000}{x}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> More than 10,000 items must be produced to keep the average cost below \\(\\text{?}160\\) per item.<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835354781\"><div data-type=\"problem\" id=\"fs-id1167835354783\"><p id=\"fs-id1167835354785\">Tillman is starting his own business by selling tacos at the beach. Accounting for the cost of his food truck and ingredients for the tacos, the function \\(C\\left(x\\right)=2x+6,000\\) represents the cost for Tillman to produce \\(x,\\) tacos. Find <span class=\"token\">\u24d0<\/span> the average cost function, \\(c\\left(x\\right)\\) for Tillman\u2019s Tacos <span class=\"token\">\u24d1<\/span> how many tacos should Tillman produce so that the average cost is less than ?4.<\/p><\/div><\/div><\/div><\/div><div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167835575399\"><h3 data-type=\"title\">Practice Test<\/h3><p id=\"fs-id1167835575406\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167835575409\"><div data-type=\"problem\" id=\"fs-id1167835575411\"><p id=\"fs-id1167835575413\">\\(\\frac{4{a}^{2}b}{12a{b}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835575441\"><p id=\"fs-id1167835575443\">\\(\\frac{a}{3b}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826783185\"><div data-type=\"problem\" id=\"fs-id1167826783187\"><p id=\"fs-id1167826783189\">\\(\\frac{6x-18}{{x}^{2}-9}\\)<\/p><\/div><\/div><p id=\"fs-id1167826783235\">In the following exercises, perform the indicated operation and simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167831030461\"><div data-type=\"problem\" id=\"fs-id1167831030463\"><p id=\"fs-id1167831030466\">\\(\\frac{4x}{x+2}\u00b7\\frac{{x}^{2}+5x+6}{12{x}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831030514\"><p id=\"fs-id1167831030516\">\\(\\frac{x+3}{3x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834402831\"><div data-type=\"problem\" id=\"fs-id1167834402834\"><p id=\"fs-id1167834402836\">\\(\\frac{2{y}^{2}}{{y}^{2}-1}\u00f7\\frac{{y}^{3}-{y}^{2}+y}{{y}^{3}-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832215290\"><div data-type=\"problem\" id=\"fs-id1167832215292\"><p id=\"fs-id1167832215294\">\\(\\frac{6{x}^{2}-x+20}{{x}^{2}-81}-\\frac{5{x}^{2}+11x-7}{{x}^{2}-81}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830699035\"><p id=\"fs-id1167830923399\">\\(\\frac{x-3}{x+9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830923422\"><div data-type=\"problem\" id=\"fs-id1167830923424\"><p id=\"fs-id1167830923426\">\\(\\frac{-3a}{3a-3}+\\frac{5a}{{a}^{2}+3a-4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826849514\"><div data-type=\"problem\" id=\"fs-id1167826849517\"><p id=\"fs-id1167826849519\">\\(\\frac{2{n}^{2}+8n-1}{{n}^{2}-1}-\\frac{{n}^{2}-7n-1}{1-{n}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826828378\"><p id=\"fs-id1167826828380\">\\(\\frac{3n-2}{n-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834562469\"><div data-type=\"problem\" id=\"fs-id1167834562471\"><p id=\"fs-id1167834562473\">\\(\\frac{10{x}^{2}+16x-7}{8x-3}+\\frac{2{x}^{2}+3x-1}{3-8x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831822896\"><div data-type=\"problem\" id=\"fs-id1167831822899\"><p id=\"fs-id1167831822901\">\\(\\frac{\\frac{1}{m}-\\frac{1}{n}}{\\frac{1}{n}+\\frac{1}{m}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832059767\"><p id=\"fs-id1167832059769\">\\(\\frac{n-m}{m+n}\\)<\/p><\/div><\/div><p id=\"fs-id1167832059792\">In the following exercises, solve each equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167832059795\"><div data-type=\"problem\" id=\"fs-id1167832059797\"><p id=\"fs-id1167832059799\">\\(\\frac{1}{x}+\\frac{3}{4}=\\frac{5}{8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830705543\"><div data-type=\"problem\" id=\"fs-id1167830705545\"><p id=\"fs-id1167830705547\">\\(\\frac{1}{z-5}+\\frac{1}{z+5}=\\frac{1}{{z}^{2}-25}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835496307\"><p id=\"fs-id1167835496309\">\\(z=\\frac{1}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835496325\"><div data-type=\"problem\" id=\"fs-id1167835496327\"><p id=\"fs-id1167835496329\">\\(\\frac{z}{2z+8}-\\frac{3}{4z-8}=\\frac{3{z}^{2}-16z-16}{8{z}^{2}+2z-64}\\)<\/p><\/div><\/div><p id=\"fs-id1167826819292\">In the following exercises, solve each rational inequality and write the solution in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167826819296\"><div data-type=\"problem\" id=\"fs-id1167826819298\"><p id=\"fs-id1167826819300\">\\(\\frac{6x}{x-6}\\le 2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826819323\"><p id=\"fs-id1167826819325\">\\(\\left[-3,6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830700720\"><div data-type=\"problem\" id=\"fs-id1167830700722\"><p id=\"fs-id1167830700725\">\\(\\frac{2x+3}{x-6}&gt;1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835531841\"><div data-type=\"problem\" id=\"fs-id1167835531843\"><p id=\"fs-id1167835531845\">\\(\\frac{1}{2}+\\frac{12}{{x}^{2}}\\ge \\frac{5}{x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835531876\"><p id=\"fs-id1167835531878\">\\(\\left(\\text{\u2212}\\infty ,0\\right)\\cup \\left(0,4\\right]\\cup \\left[6,\\infty \\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167834494827\">In the following exercises, find \\(R\\left(x\\right)\\) given \\(f\\left(x\\right)=\\frac{x-4}{{x}^{2}-3x-10}\\) and \\(g\\left(x\\right)=\\frac{x-5}{{x}^{2}-2x-8}.\\)<\/p><div data-type=\"exercise\" id=\"fs-id1167830961400\"><div data-type=\"problem\" id=\"fs-id1167830961402\"><p id=\"fs-id1167830961404\">\\(R\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834346852\"><div data-type=\"problem\" id=\"fs-id1167834346854\"><p id=\"fs-id1167834346856\">\\(R\\left(x\\right)=f\\left(x\\right)\u00b7g\\left(x\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834346898\"><p id=\"fs-id1167834120083\">\\(R\\left(x\\right)=\\frac{1}{\\left(x+2\\right)\\left(x+2\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834120131\"><div data-type=\"problem\" id=\"fs-id1167834120134\"><p id=\"fs-id1167834120136\">\\(R\\left(x\\right)=f\\left(x\\right)\u00f7g\\left(x\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834346478\"><div data-type=\"problem\" id=\"fs-id1167834346480\"><p id=\"fs-id1167834346483\">Given the function,<\/p><div data-type=\"newline\"><br><\/div>\\(R\\left(x\\right)=\\frac{2}{2{x}^{2}+x-15},\\) find the values of \\(x\\) that make the function less than or equal to 0.<\/div><div data-type=\"solution\" id=\"fs-id1167835414618\"><p id=\"fs-id1167835414620\">\\(\\left(2,5\\right]\\)<\/p><\/div><\/div><p id=\"fs-id1167835414640\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167835414643\"><div data-type=\"problem\" id=\"fs-id1167835414645\"><p id=\"fs-id1167835414647\">If \\(y\\) varies directly with \\(x\\), and \\(x=5\\) when \\(y=30,\\) find \\(x\\) when \\(y=42.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834123576\"><div data-type=\"problem\" id=\"fs-id1167834123578\"><p id=\"fs-id1167834123580\">If \\(y\\) varies inversely with the square of \\(x\\) and \\(x=3\\) when \\(y=9,\\) find \\(y\\) when \\(x=4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832212105\"><p id=\"fs-id1167832212107\">\\(y=\\frac{81}{16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834224702\"><div data-type=\"problem\" id=\"fs-id1167834224704\"><p id=\"fs-id1167834224706\">Matheus can ride his bike for 30 miles with the wind in the same amount of time that he can go 21 miles against the wind. If the wind\u2019s speed is 6 mph, what is Matheus\u2019 speed on his bike?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834224726\"><div data-type=\"problem\" id=\"fs-id1167834224728\"><p id=\"fs-id1167834224730\">Oliver can split a truckload of logs in 8 hours, but working with his dad they can get it done in 3 hours. How long would it take Oliver\u2019s dad working alone to split the logs?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834224737\"><p id=\"fs-id1167834224739\">Oliver\u2019s dad would take \\(4\\frac{4}{5}\\) hours to split the logs himself.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834224755\"><div data-type=\"problem\" id=\"fs-id1167834224757\"><p id=\"fs-id1167830964041\">The volume of a gas in a container varies inversely with the pressure on the gas. If a container of nitrogen has a volume of 29.5 liters with 2000 psi, what is the volume if the tank has a 14.7 psi rating? Round to the nearest whole number.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830964060\"><div data-type=\"problem\" id=\"fs-id1167830964062\"><p id=\"fs-id1167830964064\">The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio. The diagram gives the map distances between these cities in inches.<\/p><span data-type=\"media\" id=\"fs-id1167830964071\" data-alt=\"The figure is a triangle formed by Cincinnati, Dayton, and Columbus. The distance between Cincinnati and Dayton is 2.4 inches. The distance between Dayton and Columbus is 3.2 inches. The distance between Columbus and Cincinnati is 5.3 inches.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by Cincinnati, Dayton, and Columbus. The distance between Cincinnati and Dayton is 2.4 inches. The distance between Dayton and Columbus is 3.2 inches. The distance between Columbus and Cincinnati is 5.3 inches.\"><\/span><p id=\"fs-id1167830964083\">The actual distance from Dayton to Cincinnati is 48 miles. What is the actual distance between Dayton and Columbus?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830964088\"><p id=\"fs-id1167830964090\">The distance between Dayton and Columbus is 64 miles.<\/p><\/div><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167830964100\"><dt>critical point of a rational inequality<\/dt><dd id=\"fs-id1167828420827\">The critical point of a rational inequality is a number which makes the rational expression zero or undefined.<\/dd><\/dl><dl id=\"fs-id1167828420833\"><dt>rational inequality<\/dt><dd id=\"fs-id1167828420838\">A rational inequality is an inequality that contains a rational expression.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Solve rational inequalities<\/li>\n<li>Solve an inequality with rational functions<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832010437\" class=\"be-prepared\">\n<p id=\"fs-id1167834535475\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167834463709\" type=\"1\">\n<li>Find the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c307cdabd1259e3f52730a26a4e77eaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/> when <span class=\"token\">\u24d0<\/span> <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;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e135cd6350a4c21195c621240f7aee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-293cae06997efc99f11b7f0e51bfa8ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9be17d560cc26403fde13c7469360b91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#50;&#120;&#60;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"95\" style=\"vertical-align: -1px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835324646\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Write in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61520f8e14b1f0abc6104b40b4da504d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#32;&#120;&#60;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -3px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/27141d73-3006-49fa-98eb-5744b312aea7#fs-id1167835524181\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167826998368\">\n<h3 data-type=\"title\">Solve Rational Inequalities<\/h3>\n<p id=\"fs-id1167834061460\">We learned to solve linear inequalities after learning to solve linear equations. The techniques were very much the same with one major exception. When we multiplied or divided by a negative number, the inequality sign reversed.<\/p>\n<p>Having just learned to solve rational equations we are now ready to solve rational inequalities. A <span data-type=\"term\">rational inequality<\/span> is an inequality that contains a rational expression.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835390175\">\n<div data-type=\"title\">Rational Inequality<\/div>\n<p>A <strong data-effect=\"bold\">rational inequality<\/strong> is an inequality that contains a rational expression.<\/p>\n<\/div>\n<p>Inequalities such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6c283ce6775b6af91e233c6becb07ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#120;&#125;&#62;&#49;&#44;&#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;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#125;&#123;&#120;&#45;&#51;&#125;&#60;&#52;&#44;&#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;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#45;&#51;&#125;&#123;&#120;&#45;&#54;&#125;&#92;&#103;&#101;&#32;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"218\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e8500204549af93affbc5ad5bbdbe97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"82\" style=\"vertical-align: -7px;\" \/> are rational inequalities as they each contain a rational expression.<\/p>\n<p id=\"fs-id1167835341662\">When we solve a rational inequality, we will use many of the techniques we used solving linear inequalities. We especially must remember that when we multiply or divide by a negative number, the inequality sign must reverse.<\/p>\n<p id=\"fs-id1167835354144\">Another difference is that we must carefully consider what value might make the rational expression undefined and so must be excluded.<\/p>\n<p id=\"fs-id1167834191238\">When we solve an equation and the result is <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;\" \/> we know there is one solution, which is 3.<\/p>\n<p id=\"fs-id1167835515515\">When we solve an inequality and the result is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fc1c4141a1431f2f09732dccd35dced_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> we know there are many solutions. We graph the result to better help show all the solutions, and we start with 3. Three becomes a <span data-type=\"term\">critical point<\/span> and then we decide whether to shade to the left or right of it. The numbers to the right of 3 are larger than 3, so we shade to the right.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831882511\" data-alt=\"This figure shows the solution, the interval 3 to infinity, of the inequality x is greater than 3 on a number line. The values range from negative 5 to 5 on the number line. The inequality is modeled by an open parenthesis at the critical point 3 and shading the right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the solution, the interval 3 to infinity, of the inequality x is greater than 3 on a number line. The values range from negative 5 to 5 on the number line. The inequality is modeled by an open parenthesis at the critical point 3 and shading the right.\" \/><\/span><\/p>\n<p>To solve a rational inequality, we first must write the inequality with only one quotient on the left and 0 on the right.<\/p>\n<p id=\"fs-id1167835277282\">Next we determine the critical points to use to divide the number line into intervals. A <strong data-effect=\"bold\">critical point<\/strong> is a number which make the rational expression zero or undefined.<\/p>\n<p id=\"fs-id1167831883353\">We then will evaluate the factors of the numerator and denominator, and find the quotient in each interval. This will identify the interval, or intervals, that contains all the solutions of the rational inequality.<\/p>\n<p id=\"fs-id1167834186143\">We write the solution in interval notation being careful to determine whether the endpoints are included.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835200478\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834194894\">\n<div data-type=\"problem\" id=\"fs-id1167834095300\">\n<p id=\"fs-id1167831872191\">Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78dad6e118c13895a9d0d9c951ef1de7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#49;&#125;&#123;&#120;&#43;&#51;&#125;&#92;&#103;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"64\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835423426\"><strong data-effect=\"bold\">Step 1.<\/strong> Write the inequality as one quotient on the left and zero on the right.<\/p>\n<p id=\"fs-id1167826799368\">Our inequality is in this form. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d898e366d1755a68a1ef10a644357810_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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#49;&#125;&#123;&#120;&#43;&#51;&#125;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"59\" style=\"vertical-align: -8px;\" \/><\/p>\n<p id=\"fs-id1167834525164\"><strong data-effect=\"bold\">Step 2.<\/strong> Determine the critical points\u2014the points where the rational expression will be zero or undefined.<\/p>\n<p id=\"fs-id1167831838506\">The rational expression will be zero when the numerator is zero. Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a57ca6c48b6f646aeb64eb7f05e4840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"73\" style=\"vertical-align: -1px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2883f0b53c531552fde7ff189f83165_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/> is a critical point.<\/p>\n<p id=\"fs-id1167835341311\">The rational expression will be undefined when the denominator is zero. Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95b5c974a49dc79ee80a18760fa315f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"73\" style=\"vertical-align: -2px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09bee320bf56e1f1abb3aeb986256f0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> is a critical point.<\/p>\n<p id=\"fs-id1167826880318\">The critical points are 1 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfa36400e576c82fd4847d2da37b1d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167834505421\"><strong data-effect=\"bold\">Step 3.<\/strong> Use the critical points to divide the number line into intervals.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834190549\" data-alt=\"This figure shows a number line divided into three intervals by its critical points marked at negative 3 and 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a number line divided into three intervals by its critical points marked at negative 3 and 0.\" \/><\/span><\/p>\n<p id=\"fs-id1167835370621\">The number line is divided into three intervals:<\/p>\n<p id=\"fs-id1165927646021\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d0a5895f985736f404dd1859a9e8b64_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;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"359\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167834196002\"><strong data-effect=\"bold\">Step 4.<\/strong> Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient.<\/p>\n<p id=\"fs-id1167835254103\">To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835320815\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67963aa79bf2982438f85989ca9904f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#116;&#101;&#114;&#118;&#97;&#108;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167832116048\">The number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> is in the interval <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a7f439f766dad1d86060735a0844e2e_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;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/> Test <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/> in the expression in the numerator and the denominator.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835376324\" data-alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when negative 4 is substituted into the expression for x, the result is negative 5. It labels the result as \u201cnegative\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when negative 4 is substituted into the expression for x, the result is negative 1. It labels the result \u201cnegative\u201d.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when negative 4 is substituted into the expression for x, the result is negative 5. It labels the result as \u201cnegative\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when negative 4 is substituted into the expression for x, the result is negative 1. It labels the result \u201cnegative\u201d.\" \/><\/span><\/p>\n<p id=\"fs-id1167835362637\">Above the number line, mark the factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce48b2b21d45b1802bcce97645acc819_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/> negative and mark the factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e012de22392b339bc76f22e4ddf59f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/> negative.<\/p>\n<p id=\"fs-id1167835524209\">Since a negative divided by a negative is positive, mark the quotient positive in the interval <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a7f439f766dad1d86060735a0844e2e_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;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" data-alt=\"This figure shows the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is negative, which is positive. It shows a number line divided into three intervals by its critical points marked at negative 3 and 0. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is negative, which is positive. It shows a number line divided into three intervals by its critical points marked at negative 3 and 0. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3.\" \/><\/span><\/p>\n<div data-type=\"equation\" id=\"fs-id1167835239820\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c36e425db702bc2afcbb8ffaf6252d2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#116;&#101;&#114;&#118;&#97;&#108;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167828401707\">The number 0 is in the interval <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3560f7b8b2e183613bae675c40413816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> Test <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835421416\" data-alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when 0 is substituted into the expression for x, the result is negative 1. It labels the result as \u201cnegative\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when 0 is substituted into the expression for x, the result is 3. It labels the result \u201cpositive\u201d.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when 0 is substituted into the expression for x, the result is negative 1. It labels the result as \u201cnegative\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when 0 is substituted into the expression for x, the result is 3. It labels the result \u201cpositive\u201d.\" \/><\/span><\/p>\n<p id=\"fs-id1167834300208\">Above the number line, mark the factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce48b2b21d45b1802bcce97645acc819_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/> negative and mark <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e012de22392b339bc76f22e4ddf59f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/> positive.<\/p>\n<p id=\"fs-id1167834448595\">Since a negative divided by a positive is negative, the quotient is marked negative in the interval <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3560f7b8b2e183613bae675c40413816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835362793\" data-alt=\"This figure shows a shows the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is positive, which is negative. It shows a number line divided into three intervals by its critical points marked at negative 3 and 0. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a shows the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is positive, which is negative. It shows a number line divided into three intervals by its critical points marked at negative 3 and 0. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1.\" \/><\/span><\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7cf1774e08478c87c8c6aadff683ca30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#116;&#101;&#114;&#118;&#97;&#108;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>The number 2 is in the interval <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa5fa4581591b8aab1c976a4b2d691a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> Test <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d11400eca30b4e5ab418bd61ede75fa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835384330\" data-alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when 2 is substituted into the expression for x, the result is 1. It labels the result as \u201cpositive\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when 2 is substituted into the expression for x, the result is 5. It labels the result \u201cpositive\u201d.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure labels the expression, x minus 1, as the \u201cnumerator\u201d. It shows that when 2 is substituted into the expression for x, the result is 1. It labels the result as \u201cpositive\u201d. It also labels the expression, x plus 3, as \u201cthe denominator\u201d. It shows that when 2 is substituted into the expression for x, the result is 5. It labels the result \u201cpositive\u201d.\" \/><\/span><\/p>\n<p>Above the number line, mark the factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce48b2b21d45b1802bcce97645acc819_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/> positive and mark <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e012de22392b339bc76f22e4ddf59f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/> positive.<\/p>\n<p id=\"fs-id1167835326166\">Since a positive divided by a positive is positive, mark the quotient positive in the interval <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa5fa4581591b8aab1c976a4b2d691a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835303485\" data-alt=\"The figure shows that in the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is positive, which is negative. It shows a number line is divided into intervals by critical points at negative 3 and 1. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1. The factors x minus 1 and x plus 3 are marked as positive above the number line for the interval 1 to infinity. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative 1 to infinity.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_012_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows that in the quotient of the quantity x minus 1 and the quantity x plus 3, the numerator is negative and the denominator is positive, which is negative. It shows a number line is divided into intervals by critical points at negative 3 and 1. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1. The factors x minus 1 and x plus 3 are marked as positive above the number line for the interval 1 to infinity. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative 1 to infinity.\" \/><\/span><\/p>\n<p id=\"fs-id1167834099272\"><strong data-effect=\"bold\">Step 5.<\/strong> Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/p>\n<p id=\"fs-id1167834059152\">We want the quotient to be greater than or equal to zero, so the numbers in the intervals <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-248cef0524586c6b6fb1e1cf94c191f0_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;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9f21629da5fb031dd41e08c778ccb28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> are solutions.<\/p>\n<p id=\"fs-id1167831920853\">But what about the critical points?<\/p>\n<p id=\"fs-id1167830865437\">The critical point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e135cd6350a4c21195c621240f7aee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/> makes the denominator 0, so it must be excluded from the solution and we mark it with a parenthesis.<\/p>\n<p id=\"fs-id1167834536833\">The critical point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/> makes the whole rational expression 0. The inequality requires that the rational expression be greater than or equal to. So, 1 is part of the solution and we will mark it with a bracket.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831823765\" data-alt=\"The number line is divided into intervals by critical points at negative 3 and 1. A closed parenthesis is used at 3 and an open bracket is used at 1. The number is shaded to the left of 3 and to the right of 1. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1. The factors x minus 1 and x plus 3 are marked as positive above the number line for the interval 1 to infinity. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative 1 to infinity.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The number line is divided into intervals by critical points at negative 3 and 1. A closed parenthesis is used at 3 and an open bracket is used at 1. The number is shaded to the left of 3 and to the right of 1. The factors x minus 1 and x plus 3 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x minus 1 is marked as negative and the factor x plus 3 is marked as positive above the number line for the interval negative 3 to 1. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as negative below the number line for the interval negative 3 to 1. The factors x minus 1 and x plus 3 are marked as positive above the number line for the interval 1 to infinity. The quotient of the quantity x minus 1 and the quantity x plus 3 is marked as positive below the number line for the interval negative 1 to infinity.\" \/><\/span><\/p>\n<p id=\"fs-id1167828426637\">Recall that when we have a solution made up of more than one interval we use the union symbol, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95b36c405736204df10868714fb6d043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#99;&#117;&#112;&#32;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -4px;\" \/> to connect the two intervals. The solution in interval notation is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63db07460551a6e6343c831fa3b38444_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;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#49;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835342857\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831112014\">\n<div data-type=\"problem\">\n<p>Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f2583e3efff8187fdb947a565656dee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#50;&#125;&#123;&#120;&#43;&#52;&#125;&#92;&#103;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"64\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831847167\">\n<p id=\"fs-id1167835318289\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba744731f341a6053c40d08af38d4231_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;&#91;&#50;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835319437\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834309629\">\n<div data-type=\"problem\" id=\"fs-id1167834084806\">\n<p id=\"fs-id1167826798819\">Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb71eb5f5198285d92e1046df263ccf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#50;&#125;&#123;&#120;&#45;&#52;&#125;&#92;&#103;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826987807\">\n<p id=\"fs-id1167835343175\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-830e440e463be4a596ce2001a65f5efe_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#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=\"19\" width=\"126\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835317235\">We summarize the steps for easy reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835233307\" class=\"howto\">\n<div data-type=\"title\">Solve a rational inequality.<\/div>\n<ol id=\"fs-id1167826995382\" type=\"1\" class=\"stepwise\">\n<li>Write the inequality as one quotient on the left and zero on the right.<\/li>\n<li>Determine the critical points\u2013the points where the rational expression will be zero or undefined.<\/li>\n<li>Use the critical points to divide the number line into intervals.<\/li>\n<li>Test a value in each interval. Above the number line show the sign of each factor of the numerator and denominator in each interval. Below the number line show the sign of the quotient.<\/li>\n<li>Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167834294544\">The next example requires that we first get the rational inequality into the correct form.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835358203\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167830963564\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835379650\">Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98a22670ff1dc93041407c7465345cbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#125;&#123;&#120;&#45;&#54;&#125;&#60;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167826804582\" class=\"unnumbered unstyled can-break\" summary=\"Solve the inequality 4 x divided by the quantity x minus 6 is less than 1. Subtract 1 to get zero on the right. The result is the difference between 4 x divided by the quantity x minus 6 and 1 is less than 0. Rewrite 1 as a fraction using the least common denominator. The result is the difference between the quotient of 4 x and the quantity x minus 6 and the quotient of the quantity x minus 6 and x minus 6 is less than 0. Subtract the numerators and place the difference over the common denominator. The result is 4 x minus the quantity x minus 6 all divided by the quantity x minus 6 is less than 0. Simplify the numerator. The result is the quotient of the quantity 3 x plus 6 and the quantity x minus 6 is less than 0. Factor the numerator to show all factors. The result is 3 times the quantity x plus 2 all divided by the quantity x minus 6 is less than 0. Find the critical points. The quotient will be 0 when the numerator is 0. The quotient is undefined when the numerator is 0. That is 3 times the quantity x plus 2 is equal to 0 and x minus 6 is equal to 0. So, 3 is not equal to 0, x is equal to negative 2, and x is equal to 6. Use the critical points, negative 2 and 6, to divide the number line into intervals. The intervals are negative infinity to negative 2, negative 2 to 6, and 6 to infinity. Test a value in each interval using a chart. The chart has four columns and three rows. The first row is a header row and it labels the second column the interval negative infinity to negative 2, the third column the interval negative 2 to 6, and the fourth column the interval 6 to infinity. The first column is a header column and it labels the first row the factor x plus 2 and the second row the factor x minus 6. Test a value in each interval. The factor, x plus 2, is negative when negative 3 is substituted for x in the interval negative infinity to negative 2. The factor, x minus 6, is negative when negative 3 is substituted for x in the interval negative infinity to negative 2. The factor, x plus 2, is positive when 0 is substituted for x in the interval negative 2 to 6. The factor, x minus 6, is negative when 0 is substituted for x in the interval negative 2 to 6. The factor, x plus 2, is positive when 7 is substituted for x in the interval 6 to infinity. The factor, x minus 6, is positive when 0 is substituted for x in the interval 6 to infinity. Above the number line, show the sign of each factor of the rational expression in each interval. Below the number line, show the sign of the quotient. The number line is divided into intervals by critical points at negative 2 and 6. An open parenthesis is used at negative 2 and a closed parenthesis is used at 6. The number line is shaded between negative 2 and 6. The factors x plus 2 and x minus 6 are marked as negative above the number line for the interval negative infinity to negative 2. The quotient of the quantity x plus 2 and the quantity x minus 6 is marked as positive below the number line for the interval negative infinity to negative 2. The factor x plus 2 is marked as positive and the factor x minus 6 is marked as negative above the number line for the interval negative 2 to 6. The quotient of the quantity x plus 2 and the quantity x minus 6 is marked as negative below the number line for the interval negative 2 to 6. The factors x plus 2 and x minus 6 are marked as positive above the number line for the interval 6 to infinity. The quotient of the quantity x plus 2 and the quantity x minus 6 is marked as positive below the number line for the interval 6 to infinity. Determine the intervals where the inequality is correct. Write the solution in interval notation. The solution is the interval negative 2 and 6 with negative 2 and 6 not included.\" 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-bf164a2a32f17eb27706ded7e92f2edb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#125;&#123;&#120;&#45;&#54;&#125;&#60;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract 1 to get zero on the right.<\/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-fde46b9d3951fc9f930dabf95e30c58d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#125;&#123;&#120;&#45;&#54;&#125;&#45;&#49;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"90\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite 1 as a fraction using the LCD.<\/td>\n<td data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73aa2a0914599fb580fc2f1cf98272b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#125;&#123;&#120;&#45;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#54;&#125;&#123;&#120;&#45;&#54;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"111\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Subtract the numerators and place the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>difference over the common denominator.<\/td>\n<td data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-027d1dee14c65c59ebdd17d70a0d28da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#45;&#54;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"96\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d646cb50ea8c5779ca6fda787928cd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#43;&#54;&#125;&#123;&#120;&#45;&#54;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"67\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the numerator to show all factors.<\/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-594ec8ce4cf6d684f068bf820e06bfd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#45;&#54;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"77\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the critical points.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The quotient will be zero when the numerator is zero.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The quotient is undefined when the denominator is zero.<\/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-dca55fc03b049fe175a82bfcf4c27e26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#54;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#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;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"240\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Use the critical points to divide the number line into intervals.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" 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\/2018\/12\/CNX_IntAlg_Figure_07_06_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Test a value in each interval.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826804542\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Above the number line show the sign of each factor of the rational expression in each interval.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Below the number line show the sign of the quotient.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" 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\/2018\/12\/CNX_IntAlg_Figure_07_06_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\">Determine the intervals where the inequality is correct. We want the quotient to be negative, so the solution includes the points between \u22122 and 6. Since the inequality is strictly less than, the endpoints are not included.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"3\" data-valign=\"top\" data-align=\"left\">We write the solution in interval notation as (\u22122, 6).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834234265\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835353129\">\n<div data-type=\"problem\" id=\"fs-id1167832128828\">\n<p>Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a957e23371170450ecad7bd8b7977123_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#120;&#45;&#51;&#125;&#60;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834300261\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1e4d2676f505ccb645ebceb223e3014_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835257383\">\n<div data-type=\"problem\" id=\"fs-id1167835595165\">\n<p>Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5a3287ba77a68d51562c3b505053385_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#120;&#45;&#52;&#125;&#60;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835269471\">\n<p id=\"fs-id1167835329101\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d882d4e7603228eeebec275d33f68b22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165927605011\">In the next example, the numerator is always positive, so the sign of the rational expression depends on the sign of the denominator.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835534361\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831884237\">Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-233bf8ba4c2954962f5d6a36a749d54d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#53;&#125;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"103\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167834162030\" class=\"unnumbered unstyled can-break\" summary=\"The inequality 5 divided by the quantity x squared minus 2 x minus 15 is greater than 0 is already in the correct form. Factor the denominator. The result is 5 divided by the product of the quantity x plus 3 and the quantity x minus 5 is greater than 0. Find the critical points. The quotient is 0 when the numerator is 0. Since the numerator is always 5, the quotient cannot be 0. The quotient will be undefined when the denominator is 0. That is the product of the quantity x plus 3 and the quantity x minus 5 is equal to 0, which is x is equal to negative 3 and x is equal to 5. Use the critical points, negative 3 and 5, to divide the number line into intervals. Test values in each interval. Above the number line, show the sign of each factor of the rational expression in each interval. Below the number line, show the sign of the quotient. The number line is divided into intervals by critical points at negative 3 and 5. A closed parenthesis is used at negative 3 and an open parenthesis is used at 5. The number line is shaded to the left of 3 and to the right of 5. The factors x plus 3 and x minus 5 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of 4 and the product of the quantity x plus 3 and the quantity x minus 5 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x plus 3 is marked as positive and the factor x minus 5 is marked as negative above the number line for the interval negative 3 to 5. The quotient of 4 and the product of the quantity x plus 3 and the quantity x minus 5 is marked as negative below the number line for the interval negative 3 to 5. The factors x plus 3 and x minus 5 are marked as positive above the number line for the interval 5 to infinity. The quotient of 4 and the product of the quantity x plus 3 and the quantity x minus 5 is marked as positive below the number line for the interval 5 to infinity. Write the solution in interval notation. The solution is the union of the interval negative infinity to negative 3 and the interval 5 to infinity, with 3 and 5 not included\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The inequality is in the correct form.<\/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-8db6f563826e6a20c1a27af5fb4c441e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#53;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"99\" style=\"vertical-align: -8px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the denominator.<\/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-0fa1c249d0a63ab92f8b3e80ab998595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"107\" style=\"vertical-align: -9px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Find the critical points.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The quotient is 0 when the numerator is 0.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since the numerator is always 5, the quotient cannot be 0.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The quotient will be undefined when the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>denominator is zero.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d92352a62f9eb78190d001971d13fd7_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;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#61;&#45;&#51;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#104;&#115;&#112;&#97;&#99;&#101;&#123;&#48;&#46;&#49;&#55;&#101;&#109;&#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=\"40\" width=\"145\" style=\"vertical-align: -15px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the critical points to divide the number line into intervals.<\/td>\n<\/tr>\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-id1167834327372\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Test values in each interval.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Above the number line show the sign of each<\/p>\n<div data-type=\"newline\"><\/div>\n<p>factor of the denominator in each interval.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Below the number line, show the sign of the quotient.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the solution in 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-d72761bdb833777ccdae0767c0dd436d_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;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#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<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835421722\">\n<div data-type=\"problem\" id=\"fs-id1167835338178\">\n<p id=\"fs-id1167834423187\">Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9993553308949f52157fd147427802d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#56;&#125;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"96\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167831197123\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a3d6f9da8759b6cd1d980b27926ea16_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;&#50;&#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;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835344975\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835332644\">\n<div data-type=\"problem\" id=\"fs-id1167835358618\">\n<p id=\"fs-id1167835259002\">Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9961601c4b52fad9a7c3cdf82428f6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;&#50;&#125;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"96\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834247068\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd057c82a8a6bff456e435b4fa3b8f52_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;&#51;&#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;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835489103\">The next example requires some work to get it into the needed form.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834309742\">\n<p id=\"fs-id1167835370918\">Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d0da7f540f07519b95378a527671603_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"95\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835434192\">\n<table class=\"unnumbered unstyled can-break\" summary=\"Solve the difference between one-third and the quantity 2 divided by x squared is less than the quantity 5 divided by 3 x. Subtract 5 divided by 3 x to get 0 on the right side. The result is one-third minus the quantity 2 divided by x squared minus the quantity 5 divided by 3 x is less than 0. Rewrite the inequality to get each fraction with the least common denominator, 3 x squared. The result is 1 times x squared all divided by 3 times x squared minus 2 times 6 all divided by x squared times 3 minus 5 times x all divided by 3 x times x is less than 0. Simplify. The result is the quantity x squared divided by 3 x squared minus the quantity 6 divided by 3 x squared minus the quantity 5 x divided by 3 x squared is less than 0. Subtract the numerators and place the difference over the common denominator. The result is the quantity x squared minus 5 x minus 6 divided by the quantity 3 x squared is less than 0. Factor the numerator. The result is the product of the quantity x minus 6 and the quantity x plus divided by 3 x squared is less than 0. Find the critical points using 3 x squared is equal to 0, x minus 6 is equal to 0, and x plus 1 is equal to 0. The critical points are x is equal to 0, x is equal to 6, and x is equal to negative 1. Use the critical points to divide the number line into intervals. Above the number line, show the sign of each factor in each interval. Below the number line, show the sign of the quotient. The number line is divided into intervals by critical points at negative 1, 0, and 6. An open parenthesis is used at negative 1, an open and closed parenthesis is used at 0, and a closed parenthesis is used at 6. The number line is shaded between negative 1 and 0 and between 0 and 6. The factors x minus 6 and x plus 1 are marked as negative above the number line for the interval negative infinity to negative 1. The factor x squared is marked as positive above the number line for the interval negative infinity to negative 1. The quotient of the product of the quantity x minus 6 and the quantity x plus 1 all divided by 3 x squared is marked as positive below the number line for the interval negative infinity to negative 1. The factor x minus 6 is marked as negative above the number line for the interval negative 1 to 0. The factors x plus 1 and x squared are marked as positive above the number line for the interval negative 1 and 0. The quotient of the product of the quantity x minus 6 and the quantity x plus 1 all divided by 3 x squared is marked as negative below the number line on the interval negative 1 to 0. The factor x minus 6 is marked as negative above the number line on the interval 0 to 6. The factors x plus 1 and x squared are marked as positive above the number line on the interval 0 to 6. The quotient of the product of the quantity x minus 6 and the quantity x plus 1 all divided by 3 x squared is marked as negative below the number line on the interval 0 to 6. The factors x minus 6, x plus 1 and x squared are marked positive above the number line on the interval 6 to infinity. The quotient of the product of the quantity x minus 6 and the quantity x plus 1 all divided by 3 x squared is marked as positive below the number line on the interval 6 to infinity. Since 0 is excluded, the solution is the two intervals negative 1 to 0 and 0 to 6, or the union of the intervals negative 1 to 0 and 0 to 6.\" 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-3412db9965f7cb4b94894db8f5bedced_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"89\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7df7ee398dfa8017581801713e859fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"15\" style=\"vertical-align: -6px;\" \/> to get zero on the right.<\/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-b4b0b2d2749409bce404721dad58863b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#120;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"122\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite to get each fraction with the LCD <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f5504a5293c734491a54d004cbff27b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/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-3ed47f1bf0a3b3376eae01186cfa0f0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#92;&#99;&#100;&#111;&#116;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#92;&#99;&#100;&#111;&#116;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#92;&#99;&#100;&#111;&#116;&#32;&#120;&#125;&#123;&#51;&#120;&#92;&#99;&#100;&#111;&#116;&#32;&#120;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"162\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2a533a13f6c578ab7fce97a6562fc94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"150\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Subtract the numerators and place the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>difference over the common denominator.<\/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-0db7307535478e5d7ef9e340be50bfb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#54;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"92\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the numerator.<\/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-e7e1ca082c9ec1a9e5a62dcacddb328c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"107\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the critical points.<\/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-a95df82b2afa3ec4403edbd6ecc10d54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#54;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#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;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#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;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"567\" width=\"33\" style=\"vertical-align: -287px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the critical points to divide the number<\/p>\n<div data-type=\"newline\"><\/div>\n<p>line into intervals.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835342893\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Above the number line show the sign of each<\/p>\n<div data-type=\"newline\"><\/div>\n<p>factor in each interval. Below the number line, show the sign of the quotient.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since, 0 is excluded, the solution is the two<\/p>\n<div data-type=\"newline\"><\/div>\n<p>intervals, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebe7ea89f522e94da67a0a0622127bab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c03c52c49fb36d775ced102857b7b58e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/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-49793f49e56eb964e208b6e6a5a17476_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834301158\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835306283\">Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1efeff26299dd71fdeaf223d089c435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835355496\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65c84a7e1d884e51e9ff8e8338318a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835164921\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835369147\">\n<div data-type=\"problem\" id=\"fs-id1167834526570\">\n<p id=\"fs-id1167834593562\">Solve and write the solution in interval notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08b3fff6c7315b45e52f908e640385ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831871519\">\n<p id=\"fs-id1167830702763\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1fa2be7ed8e95fd6a934ca178fab1d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835309086\">\n<h3 data-type=\"title\">Solve an Inequality with Rational Functions<\/h3>\n<p id=\"fs-id1167832152764\">When working with rational functions, it is sometimes useful to know when the function is greater than or less than a particular value. This leads to a rational inequality.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835511373\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p>Given the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d79b7fe1cd8b4087b3ae3fcd81ff5fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#51;&#125;&#123;&#120;&#45;&#53;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/> find the values of <em data-effect=\"italics\">x<\/em> that make the function less than or equal to 0.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832195491\">\n<p id=\"fs-id1167834190517\">We want the function to be less than or equal to 0.<\/p>\n<table class=\"unnumbered unstyled\" summary=\"The function R is less than or equal to 0. Substitute the rational expression, the quotient of the quantity x plus 3 and the quantity x minus 5, for the function R. The result is the quotient of the quantity x plus 3 and the quantity x minus 5 is less than or equal to 0, where x is not equal to 5. Find the critical points using x plus 3 is equal to 3 and x minus 5 is equal to 0. The critical points are x is equal to negative 3 and x is equal to 5. Use the critical points to divide the number line into intervals. Test values in each interval. Above the number line show the sign of each factor in each interval. Below the number line, show the sign of the quotient. The number line is divided into intervals by critical points at negative 3 and 5. An open bracket is used at negative 3 and a closed parenthesis is used at 5. The number line is shaded between negative 3 and 5. The factors x plus 3 and x minus 5 are marked as negative above the number line for the interval negative infinity to negative 3. The quotient of the quantity x plus 3 and the quantity x minus 5 is marked as positive below the number line for the interval negative infinity to negative 3. The factor x plus 3 is marked as positive above the number line for the interval negative 3 to 5. The factor x minus 5 is marked as negative above the number line for the interval negative 3 to 5. The quotient of the quantity x plus 3 and the quantity x minus 5 is marked as negative below the number line on the interval negative 3 to 5. The factors x plus 3 and x minus 5 are marked as positive above the number line on the interval 5 to infinity. The quotient of the quantity x plus 3 and the quantity x minus 5 is marked as positive below the number line on the interval 5 to infinity. Write the solution in interval notation. Since 5 is excluded, we do not include it in the interval. The solution is the interval negative 3 to 5, with negative 3 included.\" 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-d54b012a0e3eb398d15c8781d5f56845_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute the rational expression for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1438ab1028aa4deeb10291585747a1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/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-7da64dec53e673b929f47700cdee07c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#51;&#125;&#123;&#120;&#45;&#53;&#125;&#92;&#108;&#101;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#110;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"164\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the critical points.<\/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-7341a1788b1bb4eb7e065e35ec88d9dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#53;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#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;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"254\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the critical points to divide the number line into intervals.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Test values in each interval. Above the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>number line, show the sign of each factor<\/p>\n<div data-type=\"newline\"><\/div>\n<p>in each interval. Below the number line,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>show the sign of the quotient<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the solution in interval notation. Since<\/p>\n<div data-type=\"newline\"><\/div>\n<p>5 is excluded we, do not include it in the interval.<\/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-5e1e1d118dfb54b51d5ccf2984a836cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#51;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832054727\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831239696\">\n<div data-type=\"problem\" id=\"fs-id1167831117331\">\n<p id=\"fs-id1167835369032\">Given the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28b8f24a2d77c74bb00441fa601243f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#50;&#125;&#123;&#120;&#43;&#52;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"97\" style=\"vertical-align: -8px;\" \/> find the values of <em data-effect=\"italics\">x<\/em> that make the function less than or equal to 0.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835337641\">\n<p id=\"fs-id1167834584349\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7ad2bc4e12c3517448e5076dbea9f34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834534707\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167828421075\">\n<div data-type=\"problem\" id=\"fs-id1167834156746\">\n<p>Given the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39e472a8cc04119214c9042ed1a8fe01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#120;&#45;&#52;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/> find the values of <em data-effect=\"italics\">x<\/em> that make the function less than or equal to 0.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835369961\">\n<p id=\"fs-id1167835318799\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fda1a365cf7e83b5dc517ebb750c75d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#49;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In economics, the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-574e0c4bf0d6708c2c77394ebfc62f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/> is used to represent the cost of producing <em data-effect=\"italics\">x<\/em> units of a commodity. The average cost per unit can be found by dividing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-574e0c4bf0d6708c2c77394ebfc62f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/> by the number of items <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/> Then, the average cost per unit is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce38e0034ac27c39c1f12dc0d01cdcb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"95\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1167835283342\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835355662\">\n<div data-type=\"problem\">\n<p>The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6de49e4e0e501db21bd8f2ce13b80ff0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#48;&#120;&#43;&#51;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -4px;\" \/> represents the cost to produce <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> number of items. Find <span class=\"token\">\u24d0<\/span> the average cost function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc1adfaa08ed832cb0751363955bdd21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> how many items should be produced so that the average cost is less than ?40.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835232228\"><span class=\"token\">\u24d0<\/span><\/p>\n<p id=\"fs-id1167835308313\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f63195e37aed64075637990269f5884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#48;&#120;&#43;&#51;&#48;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#97;&#118;&#101;&#114;&#97;&#103;&#101;&#32;&#99;&#111;&#115;&#116;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#97;&#118;&#101;&#114;&#97;&#103;&#101;&#32;&#99;&#111;&#115;&#116;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#44;&#32;&#100;&#105;&#118;&#105;&#100;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#116;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#32;&#98;&#121;&#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;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#120;&#43;&#51;&#48;&#48;&#48;&#125;&#123;&#120;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#97;&#118;&#101;&#114;&#97;&#103;&#101;&#32;&#99;&#111;&#115;&#116;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#120;&#43;&#51;&#48;&#48;&#48;&#125;&#123;&#120;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"114\" width=\"756\" style=\"vertical-align: -53px;\" \/><\/p>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1a1902be6c6abef88b357bca8e17704_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#119;&#97;&#110;&#116;&#32;&#116;&#104;&#101;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#116;&#111;&#32;&#98;&#101;&#32;&#108;&#101;&#115;&#115;&#32;&#116;&#104;&#97;&#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;&#52;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#52;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#114;&#97;&#116;&#105;&#111;&#110;&#97;&#108;&#32;&#101;&#120;&#112;&#114;&#101;&#115;&#115;&#105;&#111;&#110;&#32;&#102;&#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;&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#120;&#43;&#51;&#48;&#48;&#48;&#125;&#123;&#120;&#125;&#60;&#52;&#48;&#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;&#110;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#32;&#52;&#48;&#32;&#116;&#111;&#32;&#103;&#101;&#116;&#32;&#48;&#32;&#111;&#110;&#32;&#116;&#104;&#101;&#32;&#114;&#105;&#103;&#104;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#49;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#120;&#43;&#51;&#48;&#48;&#48;&#125;&#123;&#120;&#125;&#45;&#52;&#48;&#60;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#108;&#101;&#102;&#116;&#32;&#115;&#105;&#100;&#101;&#32;&#97;&#115;&#32;&#111;&#110;&#101;&#32;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#98;&#121;&#32;&#102;&#105;&#110;&#100;&#105;&#110;&#103;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#76;&#67;&#68;&#32;&#97;&#110;&#100;&#32;&#112;&#101;&#114;&#102;&#111;&#114;&#109;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#120;&#43;&#51;&#48;&#48;&#48;&#125;&#123;&#120;&#125;&#45;&#52;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#60;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#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;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#120;&#43;&#51;&#48;&#48;&#48;&#125;&#123;&#120;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#48;&#120;&#125;&#123;&#120;&#125;&#60;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#120;&#43;&#51;&#48;&#48;&#48;&#45;&#52;&#48;&#120;&#125;&#123;&#120;&#125;&#60;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#48;&#120;&#43;&#51;&#48;&#48;&#48;&#125;&#123;&#120;&#125;&#60;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#32;&#116;&#111;&#32;&#115;&#104;&#111;&#119;&#32;&#97;&#108;&#108;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#125;&#60;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#99;&#114;&#105;&#116;&#105;&#99;&#97;&#108;&#32;&#112;&#111;&#105;&#110;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#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;&#120;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#48;&#92;&#110;&#101;&#32;&#48;&#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;&#120;&#45;&#49;&#48;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#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=\"261\" width=\"728\" style=\"vertical-align: -124px;\" \/><\/p>\n<p id=\"fs-id1167835349647\">More than 100 items must be produced to keep the average cost below ?40 per item.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835321953\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167827967245\">\n<div data-type=\"problem\" id=\"fs-id1167834534467\">\n<p id=\"fs-id1167835357225\">The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3160602bde660f1827a5e1bf2b6b7280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#48;&#120;&#43;&#54;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -4px;\" \/> represents the cost to produce <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> number of items. Find <span class=\"token\">\u24d0<\/span> the average cost function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc1adfaa08ed832cb0751363955bdd21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> how many items should be produced so that the average cost is less than ?60?<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167831883113\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78d35dcc6abc8c5942738deca77619c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#120;&#43;&#54;&#48;&#48;&#48;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> More than 150 items must be produced to keep the average cost below ?60 per item.<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826807809\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834084459\">\n<div data-type=\"problem\" id=\"fs-id1167834473717\">\n<p id=\"fs-id1167835479012\">The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30455b99536ac706a9b768ce0514375e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#120;&#43;&#57;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -4px;\" \/> represents the cost to produce <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> number of items. Find <span class=\"token\">\u24d0<\/span> the average cost function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc1adfaa08ed832cb0751363955bdd21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> how many items should be produced so that the average cost is less than ?20?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834059188\">\n<p id=\"fs-id1167835419423\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bffadb580cd55f0e9e238ea888684485_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#43;&#57;&#48;&#48;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span> More than 60 items must be produced to keep the average cost below ?20 per item.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167834189870\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Solve a rational inequality.<\/strong>\n<ol id=\"fs-id1167835338241\" type=\"1\" class=\"stepwise\">\n<li>Write the inequality as one quotient on the left and zero on the right.<\/li>\n<li>Determine the critical points\u2013the points where the rational expression will be zero or undefined.<\/li>\n<li>Use the critical points to divide the number line into intervals.<\/li>\n<li>Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient.<\/li>\n<li>Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835325987\">\n<h3 data-type=\"title\">Section Exercises<\/h3>\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167835531810\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167832060113\"><strong data-effect=\"bold\">Solve Rational Inequalities<\/strong><\/p>\n<p id=\"fs-id1167834221970\">In the following exercises, solve each rational inequality and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835329701\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cba05fa4ef5a8de63aabff983bb2c20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#51;&#125;&#123;&#120;&#43;&#52;&#125;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"60\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835345830\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35328ccc1e1fa9682a33fcbab72fa08d_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;&#91;&#51;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835302959\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167830704699\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38d21dd7269a08857180bc63dc69cdfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#54;&#125;&#123;&#120;&#45;&#53;&#125;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834161495\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167830693464\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba44a6d3bfd052e0bd018065d0784cf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#120;&#45;&#51;&#125;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831846980\">\n<p id=\"fs-id1167835238679\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e9bad0ef88eb8cfad6fc32d8ea58560_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#49;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830700952\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835362648\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26b7ccff2a6daf862bd4b31c815684fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#52;&#125;&#123;&#120;&#43;&#50;&#125;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"60\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834058872\">\n<div data-type=\"problem\" id=\"fs-id1167835357903\">\n<p id=\"fs-id1167834473611\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1696abe144a9bc7801ada019e4d0dfc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#55;&#125;&#123;&#120;&#45;&#49;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"60\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835420364\">\n<p id=\"fs-id1167832134098\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42bbf083775c570568d0ed9feb297e00_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;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826780686\">\n<div data-type=\"problem\" id=\"fs-id1167834458685\">\n<p id=\"fs-id1167835387103\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cfacb9f1d81da5a218ebf9049c18655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#56;&#125;&#123;&#120;&#43;&#51;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"60\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834192313\">\n<div data-type=\"problem\" id=\"fs-id1167834464330\">\n<p id=\"fs-id1167831872115\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79988e6e2c9ba2f6652ef32958a1e1ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#54;&#125;&#123;&#120;&#43;&#53;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"60\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee47278eba3e60a9253d35f5859bb2e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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-id1167832053800\">\n<div data-type=\"problem\" id=\"fs-id1167834473138\">\n<p id=\"fs-id1167831117326\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5fca9853986ced2e1a602611599e313_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#53;&#125;&#123;&#120;&#45;&#50;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827967212\">\n<div data-type=\"problem\" id=\"fs-id1167834111781\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14a53a3a3647e451be3e439e5811c991_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#120;&#45;&#53;&#125;&#60;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834151983\">\n<p id=\"fs-id1167835352372\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e659604229240afd8cea6e3039eb588_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834397407\">\n<div data-type=\"problem\" id=\"fs-id1167835422785\">\n<p id=\"fs-id1167834432058\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1cfd5d2add27a8ee8eaceeef377acd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#125;&#123;&#120;&#45;&#50;&#125;&#60;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826801738\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835358530\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf45466005a08aeea9c2ef2ad7910332_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#120;&#125;&#123;&#120;&#45;&#54;&#125;&#62;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa02b5492c83a362dcbaa26af6b94988_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;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830961889\">\n<div data-type=\"problem\" id=\"fs-id1167834141728\">\n<p id=\"fs-id1167830961550\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f380e7f0e9222f6bfbe7f1fe5b578057_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#120;&#45;&#52;&#125;&#62;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835269015\">\n<div data-type=\"problem\" id=\"fs-id1167834473828\">\n<p id=\"fs-id1167831892858\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a963b67ddc714df11c272fad1f72449c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#43;&#51;&#125;&#123;&#120;&#45;&#54;&#125;&#92;&#108;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835318143\">\n<p id=\"fs-id1167834184599\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-148c98b1a709f3a87ba2e6fd01e5ee7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#57;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834429013\">\n<div data-type=\"problem\" id=\"fs-id1167826997429\">\n<p id=\"fs-id1167831148828\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe251cee97fb4e90f9cd57c2735511ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#45;&#49;&#125;&#123;&#120;&#45;&#52;&#125;&#92;&#108;&#101;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827943096\">\n<div data-type=\"problem\" id=\"fs-id1167835609306\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f82b69d72c727711320711ce1eeb4af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#45;&#50;&#125;&#123;&#120;&#45;&#52;&#125;&#92;&#103;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835468463\">\n<p id=\"fs-id1167835309545\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69a10f98aa5f0796a63da3f2d1a84d78_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;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#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=\"19\" width=\"126\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831895060\">\n<div data-type=\"problem\" id=\"fs-id1167834501963\">\n<p id=\"fs-id1167834501965\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfff384a8d2b6c2d08c80336fcec869a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#45;&#51;&#125;&#123;&#120;&#45;&#51;&#125;&#92;&#103;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835317202\">\n<div data-type=\"problem\" id=\"fs-id1167835317204\">\n<p id=\"fs-id1167835317206\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc5a7d86f44a5c473e6c4a7b15a36fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#49;&#50;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"99\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832043548\">\n<p id=\"fs-id1167832043550\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30cd5c146248c545e84dea2d6e439c1c_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;&#45;&#51;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834523673\">\n<div data-type=\"problem\" id=\"fs-id1167834523675\">\n<p id=\"fs-id1167834523677\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b6585d639c71df59ac60f0b6a34b0fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#49;&#50;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"99\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830696702\">\n<div data-type=\"problem\" id=\"fs-id1167830696704\">\n<p id=\"fs-id1167830696707\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cf565cd35476c7343d4bcc77e1ec593_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#52;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"92\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835367356\">\n<p id=\"fs-id1167835367358\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ab73a57c5e039ffb22ed1a8e29747bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832042101\">\n<div data-type=\"problem\" id=\"fs-id1167832042103\">\n<p id=\"fs-id1167832042105\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cb2b2f2d4b3028e80bb236a145a920e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#49;&#50;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"99\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832042614\">\n<div data-type=\"problem\" id=\"fs-id1167832042616\">\n<p id=\"fs-id1167831833074\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ab5718d0bf194ebd678ce34ad3b1892_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;&#53;&#125;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"99\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831833104\">\n<p id=\"fs-id1167831833106\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55b19f892d5b716a9141a6dff7d496b3_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;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"132\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835509903\">\n<div data-type=\"problem\" id=\"fs-id1167835509905\">\n<p id=\"fs-id1167835509907\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c22beb0bb07b49fd98b3bfe1f98a9c1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#53;&#125;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"99\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830960568\">\n<div data-type=\"problem\" id=\"fs-id1167830960570\">\n<p id=\"fs-id1167830960572\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e16357d0f08e9111ccf4e03f734bfa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#125;&#123;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#120;&#43;&#54;&#125;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"106\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830697144\">\n<p id=\"fs-id1167830697146\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62f3b3dc79843a67d19f80432127ff24_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;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"121\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834420250\">\n<div data-type=\"problem\" id=\"fs-id1167834420253\">\n<p id=\"fs-id1167834420255\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ccdaf434132c615159e90390e3b7a94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#125;&#123;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#45;&#54;&#125;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"113\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835534032\">\n<div data-type=\"problem\" id=\"fs-id1167835534034\">\n<p id=\"fs-id1167835534037\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b61a263fd321e43fa497e1cba652d17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831894260\">\n<p id=\"fs-id1167831894262\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b60cda57e22751ebcaf35424d61c542_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;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"174\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834527613\">\n<div data-type=\"problem\" id=\"fs-id1167834527615\">\n<p id=\"fs-id1167834527617\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80fde675153ed32d224a4b29b48caf31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#62;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"89\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830963228\">\n<div data-type=\"problem\" id=\"fs-id1167834426228\">\n<p id=\"fs-id1167834426231\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c358d16187bca43eb3fa079aa843a6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834426260\">\n<p id=\"fs-id1167834426263\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82a440e4c675fd7d8d4253462bef8fcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835533528\">\n<div data-type=\"problem\" id=\"fs-id1167835533530\">\n<p id=\"fs-id1167835533532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5dbf0897cff24752548ab9ef06b36d30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"89\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831892961\">\n<div data-type=\"problem\" id=\"fs-id1167831892963\">\n<p id=\"fs-id1167831892965\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7888f285a44f5d9a654d6075943a3830_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"73\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834432280\">\n<p id=\"fs-id1167834432282\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bd03bb09a34e49dd9f3ebf548e3dab1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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-id1167834432302\">\n<div data-type=\"problem\" id=\"fs-id1167834432304\">\n<p id=\"fs-id1167834432306\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef6deeaff6099e4ba507cd785e805a28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"73\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831895237\">\n<div data-type=\"problem\" id=\"fs-id1167831895240\">\n<p id=\"fs-id1167831895242\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02775efd7cde1eb2f89fcd20f1b570c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#45;&#50;&#125;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"79\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835332716\">\n<p id=\"fs-id1167835332718\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-959a3ce227c070f25db7eb9dc80c95c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#49;&#48;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835332752\">\n<div data-type=\"problem\" id=\"fs-id1167835332754\">\n<p id=\"fs-id1167830885209\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a60fd677f3a34d4976671059f371ff3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#120;&#45;&#49;&#125;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"79\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831191351\"><strong data-effect=\"bold\">Solve an Inequality with Rational Functions<\/strong><\/p>\n<p id=\"fs-id1167831191356\">In the following exercises, solve each rational function inequality and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831191361\">\n<div data-type=\"problem\" id=\"fs-id1167831191363\">\n<p id=\"fs-id1167831191365\">Given the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c89c95b97b496f5d506c17e0449cbe26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#53;&#125;&#123;&#120;&#45;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/> find the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> that make the function less than or equal to 0.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830991631\">\n<p id=\"fs-id1167830991633\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-745ab7d13c2bd8ff8df4c39424bf735b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"35\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834194935\">\n<div data-type=\"problem\" id=\"fs-id1167834194937\">\n<p id=\"fs-id1167834194939\">Given the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-817b40056de4430f3a4361e4a612616c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#120;&#43;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"97\" style=\"vertical-align: -8px;\" \/> find the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> that make the function less than or equal to 0.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832119260\">\n<div data-type=\"problem\" id=\"fs-id1167832119262\">\n<p id=\"fs-id1167832119264\">Given the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7136d4a5e65880c7678ae85c5f8b1cbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#54;&#125;&#123;&#120;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"92\" style=\"vertical-align: -8px;\" \/>, find the values of x that make the function less than or equal to 0.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831116246\">\n<p id=\"fs-id1167831116248\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4985693dc9097d22e8ad61bbf9e75f67_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#54;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835299648\">\n<div data-type=\"problem\" id=\"fs-id1167835299650\">\n<p id=\"fs-id1167835299653\">Given the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39e472a8cc04119214c9042ed1a8fe01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#120;&#45;&#52;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/> find the values of x that make the function less than or equal to 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834463928\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167834463935\">\n<div data-type=\"problem\" id=\"fs-id1167834463937\">\n<p id=\"fs-id1167834463939\">Write the steps you would use to explain solving rational inequalities to your little brother.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835529307\">\n<p id=\"fs-id1167835529309\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835529315\">\n<div data-type=\"problem\" id=\"fs-id1167835529317\">\n<p id=\"fs-id1167835529319\">Create a rational inequality whose solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5384829c93eb9b59e062120574400e93_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;&#50;&#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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167831893326\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167831893331\"><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-id1167831893343\" data-alt=\"This table has four columns and three 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 rational inequalities. In row 3, the I can was solve an inequality with rational functions.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and three 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 rational inequalities. In row 3, the I can was solve an inequality with rational functions.\" \/><\/span><\/p>\n<p id=\"fs-id1167832151391\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n<\/div>\n<\/div>\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167832151400\">\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832151403\">\n<h4 data-type=\"title\"><a href=\"\/contents\/c392efcb-8505-423b-9356-890700515e3b\" class=\"target-chapter\">Simplify, Multiply, and Divide Rational Expressions<\/a><\/h4>\n<p id=\"fs-id1167832151411\"><strong data-effect=\"bold\">Determine the Values for Which a Rational Expression is Undefined<\/strong><\/p>\n<p id=\"fs-id1167832151417\">In the following exercises, determine the values for which the rational expression is undefined.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832151420\">\n<div data-type=\"problem\" id=\"fs-id1167832151422\">\n<p id=\"fs-id1167832151424\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b7c9bac6e3b3ee2283d6b8e38f529d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#97;&#43;&#51;&#125;&#123;&#51;&#97;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834489514\">\n<p id=\"fs-id1167834489516\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53783c832e51e9fa3611d0b9532c7a8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834489532\">\n<div data-type=\"problem\" id=\"fs-id1167834489535\">\n<p id=\"fs-id1167834489537\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4adad9b892cbc15201708c4c42b95a4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#45;&#55;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"38\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826780637\">\n<div data-type=\"problem\" id=\"fs-id1167826780639\">\n<p id=\"fs-id1167826780641\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45035eb83fb034cc584bcf150ef7a7a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#56;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"36\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826780667\">\n<p id=\"fs-id1167826780669\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0828f2f523b251cb7799759efc9e3921_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834473956\">\n<div data-type=\"problem\" id=\"fs-id1167834473958\">\n<p id=\"fs-id1167834473960\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-996b148b12ee8d9fb4c7ce0e92eba674_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"58\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835361743\"><strong data-effect=\"bold\">Simplify Rational Expressions<\/strong><\/p>\n<p id=\"fs-id1167835361748\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835361752\">\n<div data-type=\"problem\" id=\"fs-id1167835361754\">\n<p id=\"fs-id1167835361756\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec2db507888c7522e7972e76b0f45a02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830705625\">\n<p id=\"fs-id1167830705627\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-469181e98c8ec4b1bce8f57ea4ba4e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830705638\">\n<div data-type=\"problem\" id=\"fs-id1167830705641\">\n<p id=\"fs-id1167830705643\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebdb6f02d823939a64f4e04fa470583d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#49;&#56;&#109;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"42\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831923697\">\n<div data-type=\"problem\" id=\"fs-id1167831923700\">\n<p id=\"fs-id1167831923702\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82d89cf9317e943e31473fea6ef052f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#49;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"65\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834219693\">\n<p id=\"fs-id1167834219695\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27626e0396da7c692e719e8b76887711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#51;&#125;&#123;&#120;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"26\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835390012\">\n<div data-type=\"problem\" id=\"fs-id1167835390014\">\n<p id=\"fs-id1167835390016\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf4087b311bd5e9b728ef9e9c7176903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#118;&#45;&#51;&#53;&#125;&#123;&#50;&#53;&#45;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"39\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834473291\"><strong data-effect=\"bold\">Multiply Rational Expressions<\/strong><\/p>\n<p id=\"fs-id1167834473296\">In the following exercises, multiply.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834473299\">\n<div data-type=\"problem\" id=\"fs-id1167834473301\">\n<p id=\"fs-id1167834473304\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a1b6043ace8c7a81a7e402a96eb201c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835281051\">\n<p id=\"fs-id1167835281053\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c27e6c52f9cb0bf70516c13f69448e9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835281065\">\n<div data-type=\"problem\" id=\"fs-id1167835281067\">\n<p id=\"fs-id1167835281069\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80df86a4fa81dc5b02ee93b288db9393_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#56;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#52;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"60\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831881468\">\n<div data-type=\"problem\" id=\"fs-id1167831086784\">\n<p id=\"fs-id1167831086786\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea5053e59e20d0f88b2650ab5f7a68b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#50;&#120;&#45;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#56;&#120;&#43;&#51;&#50;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#50;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"137\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835415767\">\n<p id=\"fs-id1167835415769\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9070cdd5ed4b2d6866fa21823e978136_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#120;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"26\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835415784\">\n<div data-type=\"problem\" id=\"fs-id1167835415786\">\n<p id=\"fs-id1167835415788\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f92f2321a7e0d576daaaabe4f986c132_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#45;&#49;&#48;&#125;&#123;&#57;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#121;&#43;&#57;&#125;&#123;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#57;&#121;&#45;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"152\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835389850\"><strong data-effect=\"bold\">Divide Rational Expressions<\/strong><\/p>\n<p id=\"fs-id1167835389856\">In the following exercises, divide.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835389859\">\n<div data-type=\"problem\" id=\"fs-id1167835389861\">\n<p id=\"fs-id1167835389863\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0df77d9a6512789ff4210975d99ed3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#49;&#50;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;&#123;&#51;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"108\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831995222\">\n<p id=\"fs-id1167831995224\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-445628792aed92c394f6ea78e2a519e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"73\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835530095\">\n<div data-type=\"problem\" id=\"fs-id1167835530097\">\n<p id=\"fs-id1167835530099\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9ad51a11495ab5f30e70aa8ea68a089_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#123;&#52;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#125;&#123;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#43;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"113\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834517331\">\n<div data-type=\"problem\" id=\"fs-id1167834517333\">\n<p id=\"fs-id1167834517336\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-786c638e741b074dc31711e7da0e3452_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#43;&#119;&#125;&#123;&#119;&#45;&#57;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#49;&#45;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#45;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834282574\">\n<p id=\"fs-id1167834282576\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51d716501a5ee6f0440efb16a226acd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#49;&#43;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"35\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832026070\">\n<div data-type=\"problem\" id=\"fs-id1167832026072\">\n<p id=\"fs-id1167832026074\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32dbaac00384352b1873bec3f9667e70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#121;&#45;&#54;&#51;&#125;&#123;&#52;&#121;&#43;&#51;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"171\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834554815\">\n<div data-type=\"problem\" id=\"fs-id1167834554817\">\n<p id=\"fs-id1167834554819\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efa91a33eb86dc8b73d41e4cbc9a762f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;&#125;&#123;&#51;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#54;&#99;&#43;&#49;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#99;&#45;&#51;&#50;&#125;&#123;&#49;&#53;&#99;&#43;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"70\" style=\"vertical-align: -18px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828235423\">\n<p id=\"fs-id1167830700855\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-adc9f8a0ae7b9334f1fc951fba2b7913_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#99;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"24\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830700872\">\n<div data-type=\"problem\" id=\"fs-id1167830700875\">\n<p id=\"fs-id1167830700877\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f0290537c09ab1bf66835283790468e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#97;&#125;&#123;&#97;&#45;&#52;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#45;&#50;&#52;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#97;&#43;&#49;&#48;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#97;&#125;&#123;&#97;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"170\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834282615\"><strong data-effect=\"bold\">Multiply and Divide Rational Functions<\/strong><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834282621\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834282623\">\n<p id=\"fs-id1167834282625\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a19d698605e6f0a4acb05b75c0c8025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf25b3823a2fbdc0403fe09617b84061_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"122\" style=\"vertical-align: -7px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ae8319342d0f542ea74ef960c5a05c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"121\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834141892\">\n<p id=\"fs-id1167834141894\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d719481b525ae46eff322dbce0d4e2ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831911673\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831911675\">\n<p id=\"fs-id1167831911677\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0b221e9e07eaf1d1995ed2df80b9d32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"93\" style=\"vertical-align: -9px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4265c6343c713bf6364e60eab2a1ca7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#120;&#45;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"103\" style=\"vertical-align: -7px;\" \/> 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-2f003d6f4272c96a3a3eb3971dc4c9ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#52;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#52;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835609846\">\n<h4 data-type=\"title\"><a href=\"\/contents\/6c5aa31f-890a-445a-9a9c-44a7941355f7\" class=\"target-chapter\">Add and Subtract Rational Expressions<\/a><\/h4>\n<p id=\"fs-id1167835609854\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions with a Common Denominator<\/strong><\/p>\n<p id=\"fs-id1167835609860\">In the following exercises, perform the indicated operations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835609863\">\n<div data-type=\"problem\" id=\"fs-id1167835609865\">\n<p id=\"fs-id1167835609867\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da5adda75eb5281bcb047408129d6c35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"53\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834533452\">\n<p id=\"fs-id1167834533454\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834533462\">\n<div data-type=\"problem\" id=\"fs-id1167834533464\">\n<p id=\"fs-id1167834533466\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c48c98916f07a1a1c98cc39d888459ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#97;&#45;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#97;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"89\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831884309\">\n<div data-type=\"problem\" id=\"fs-id1167831884311\">\n<p id=\"fs-id1167831892787\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87d39ce7ddba3b3c4c13e2e04cf08282_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#121;&#125;&#123;&#121;&#43;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#125;&#123;&#121;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"97\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167831892830\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04318527779f27842ef75812bf820d23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835509750\">\n<div data-type=\"problem\" id=\"fs-id1167835509752\">\n<p id=\"fs-id1167835509754\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e3c4c610e2bb6b6a751e6ad68353d00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831919688\">\n<div data-type=\"problem\" id=\"fs-id1167831919690\">\n<p id=\"fs-id1167831919692\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ffc239d82f4998bfb9d0516243f7c06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#120;&#45;&#55;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#43;&#50;&#56;&#125;&#123;&#120;&#45;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"91\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835359033\">\n<p id=\"fs-id1167835359036\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cd392abb5a6fd3a660e3b5912589956_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831883771\">\n<div data-type=\"problem\" id=\"fs-id1167831883773\">\n<p id=\"fs-id1167831883775\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1683deae6a31d758581b75057517e72a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#121;&#43;&#49;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#49;&#125;&#123;&#121;&#43;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"89\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834111847\">\n<div data-type=\"problem\" id=\"fs-id1167834111849\">\n<p id=\"fs-id1167834111851\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01cb964741ca4830097669e4157b4c1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#113;&#43;&#51;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#113;&#43;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#113;&#45;&#54;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#113;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"137\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832074864\">\n<p id=\"fs-id1167832074866\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-546bcaec7d7fb6ee9f91fdae44735a59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#45;&#51;&#125;&#123;&#113;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"25\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832074888\">\n<div data-type=\"problem\" id=\"fs-id1167832074890\">\n<p id=\"fs-id1167832074892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0837dbc2c6c2177f009c19b8935f8f97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#116;&#43;&#52;&#116;&#43;&#51;&#125;&#123;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#116;&#45;&#51;&#50;&#125;&#123;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831862571\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions Whose Denominators Are Opposites<\/strong><\/p>\n<p id=\"fs-id1167831862576\">In the following exercises, add and subtract.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834079478\">\n<p id=\"fs-id1167834079480\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c91ad69c283e354638a2d2e720b55bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#119;&#125;&#123;&#54;&#119;&#45;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#119;&#45;&#50;&#125;&#123;&#49;&#45;&#54;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"95\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834133902\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6c2c5e4cb5ac887b697f3305597b1cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#119;&#43;&#50;&#125;&#123;&#54;&#119;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"42\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834133931\">\n<div data-type=\"problem\" id=\"fs-id1167834133933\">\n<p id=\"fs-id1167834133935\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-441887a80b82b51821a409468a7ee858_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#97;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#97;&#45;&#56;&#125;&#123;&#52;&#45;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"96\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835414934\">\n<div data-type=\"problem\" id=\"fs-id1167835414936\">\n<p id=\"fs-id1167835414938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1dc32a19fe27a2fe4d13bc90c7d19df0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#98;&#45;&#49;&#53;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#98;&#45;&#49;&#125;&#123;&#52;&#57;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"154\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826857038\">\n<p id=\"fs-id1167826857040\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85a95a0249908720b807124febd2ae0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#98;&#45;&#50;&#125;&#123;&#98;&#43;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"31\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826857064\">\n<div data-type=\"problem\" id=\"fs-id1167826857066\">\n<p id=\"fs-id1167826857068\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd7494af6be586d7f45baabd38954860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#121;&#43;&#55;&#125;&#123;&#50;&#121;&#45;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#121;&#43;&#50;&#125;&#123;&#53;&#45;&#50;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"160\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834454265\"><strong data-effect=\"bold\">Find the Least Common Denominator of Rational Expressions<\/strong><\/p>\n<p id=\"fs-id1167834454270\">In the following exercises, find the LCD.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834454273\">\n<div data-type=\"problem\" id=\"fs-id1167834454276\">\n<p id=\"fs-id1167834454278\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8378fbcc4eefe31aafc5ce53534fc2c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#97;&#45;&#49;&#48;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#97;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#97;&#45;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"132\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831920685\">\n<p id=\"fs-id1167831920687\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ff843de2dc806e20e9da5596bdaa4c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"165\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835320099\">\n<div data-type=\"problem\" id=\"fs-id1167835320101\">\n<p id=\"fs-id1167835320103\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed22a5f7c59f619522f232a8b2d81ddd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#110;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#110;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"103\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831822005\">\n<div data-type=\"problem\" id=\"fs-id1167831822007\">\n<p id=\"fs-id1167835622856\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25b5b57b9f4ea761d5766725fcf151d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#112;&#45;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#109;&#125;&#123;&#51;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#51;&#112;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"152\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834376197\">\n<p id=\"fs-id1167834376200\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4ea59e59ad24fce486f9f294f6b7602_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"173\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834327388\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions with Unlike Denominators<\/strong><\/p>\n<p id=\"fs-id1167834327394\">In the following exercises, perform the indicated operations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834327397\">\n<div data-type=\"problem\" id=\"fs-id1167834327399\">\n<p id=\"fs-id1167834327401\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b688356d1042c8036b8cc343ebd4bd2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#97;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"53\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835301259\">\n<div data-type=\"problem\" id=\"fs-id1167835301261\">\n<p id=\"fs-id1167835301264\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2062e07e297dd521e86bce63b0a809f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#99;&#45;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#99;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"73\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834299889\">\n<p id=\"fs-id1167834299891\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c3ac3a346fc7d2624fbc1009e4187d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#99;&#45;&#49;&#50;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"69\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826995141\">\n<div data-type=\"problem\" id=\"fs-id1167826995143\">\n<p id=\"fs-id1167826995145\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13c2bcf4a7c4309bad07b8aa8f660104_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"116\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835215385\">\n<div data-type=\"problem\" id=\"fs-id1167835215387\">\n<p id=\"fs-id1167835215389\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70b08b6afe5c19b5593153afbf3dac0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#50;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"162\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835417204\">\n<p id=\"fs-id1167835417206\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57ad214808e3db9b29214beec01e267c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#54;&#120;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"109\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826808680\">\n<div data-type=\"problem\" id=\"fs-id1167826808682\">\n<p id=\"fs-id1167826808684\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-deefbfd7b6f2f7d95c28d1abf1b4154e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#113;&#125;&#123;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#113;&#45;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#113;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"102\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835410632\">\n<div data-type=\"problem\" id=\"fs-id1167835410635\">\n<p id=\"fs-id1167830698010\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a47af9fc92bef986c8696fd9d13932cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#121;&#125;&#123;&#121;&#43;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#50;&#125;&#123;&#121;&#43;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"76\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830698047\">\n<p id=\"fs-id1167830698050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed468e26986193a26437e9ad7a4df22e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"84\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835550842\">\n<div data-type=\"problem\" id=\"fs-id1167835550844\">\n<p id=\"fs-id1167835550846\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a622cc7119853291d242602778b2225_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#119;&#45;&#49;&#53;&#125;&#123;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#119;&#45;&#50;&#48;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#119;&#43;&#50;&#125;&#123;&#52;&#45;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"116\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834525788\">\n<div data-type=\"problem\" id=\"fs-id1167834525790\">\n<p id=\"fs-id1167834525792\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a360246730ab6bde3a374cf8e5dc35af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#109;&#43;&#51;&#125;&#123;&#109;&#43;&#50;&#125;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"68\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835302571\">\n<p id=\"fs-id1167835302573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33e4f223206c1c5a02ef83c3709e89a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#109;&#45;&#55;&#125;&#123;&#109;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"37\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835302598\">\n<div data-type=\"problem\" id=\"fs-id1167835302600\">\n<p id=\"fs-id1167835320304\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-724bae2b8de83c93e114a16eb6af2d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#110;&#43;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#110;&#45;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#45;&#57;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"136\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535865\">\n<div data-type=\"problem\" id=\"fs-id1167834535867\">\n<p id=\"fs-id1167834535869\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85eeaf2939b2f9fbe297265926f9ff94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#97;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#97;&#43;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"89\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826863901\">\n<p id=\"fs-id1167826863903\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4cfb0fa20c708c1f659b7b38a3bbbfc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#97;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826863920\">\n<div data-type=\"problem\" id=\"fs-id1167826863923\">\n<p id=\"fs-id1167826863925\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e49ed86a5683f035c3b6f5ebfe32a4e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#48;&#120;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"97\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832042426\"><strong data-effect=\"bold\">Add and Subtract Rational Functions<\/strong><\/p>\n<p id=\"fs-id1167832042431\">In the following exercises, find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c9ba76d92ef455d911f821204f4be95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d629b05700538f1d987aa1572837c00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/> are given.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834397163\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834397165\">\n<p id=\"fs-id1167834397167\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e3b4bac56e31683025bd417e423fe33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#49;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#49;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"147\" style=\"vertical-align: -9px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3645d98985b27b1ca1a0b0771edf375_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#50;&#45;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826779457\">\n<p id=\"fs-id1167826779459\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf58006c964c674dc309cb451b83c395_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#56;&#125;&#123;&#120;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"92\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830698105\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830698107\">\n<p id=\"fs-id1167830698109\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e612885836b2e8675426c1a56ab815b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#52;&#120;&#43;&#51;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#51;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"126\" style=\"vertical-align: -9px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-893810656262413f2fadd41361d7d4d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#120;&#43;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"88\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835533765\">In the following exercises, find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7de960a332c98ab624de84fc3632a5b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-984a3dd11ed3c9a1f42d61a2defb75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d629b05700538f1d987aa1572837c00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/> are given.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167827966859\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167827966861\">\n<p id=\"fs-id1167827966863\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-904c5c7f5a266af7bbb88b68cb06c7dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"115\" style=\"vertical-align: -8px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4f3ebb6ca6ba95551f2465a067dda59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#45;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"95\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831031167\">\n<p id=\"fs-id1167831031169\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f33af1796b012a1a86995b38d607b6f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#43;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"99\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831031198\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831031200\">\n<p id=\"fs-id1167831031202\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d429933e80ee2f2bec168d3102c4002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#120;&#43;&#54;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"94\" style=\"vertical-align: -8px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a66aa4e5d4056f4df50316de13ff186_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"101\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834183710\">\n<h4 data-type=\"title\"><a href=\"\/contents\/9c6e36cd-6eb0-4457-8db5-0da97afca2ac\" class=\"target-chapter\">Simplify Complex Rational Expressions<\/a><\/h4>\n<p id=\"fs-id1167834183718\"><strong data-effect=\"bold\">Simplify a Complex Rational Expression by Writing It as Division<\/strong><\/p>\n<p id=\"fs-id1167834525808\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834525811\">\n<div data-type=\"problem\" id=\"fs-id1167834525813\">\n<p id=\"fs-id1167834525815\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1cbf1ed3b0350fc80198c08ebcb595ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#120;&#125;&#123;&#120;&#43;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"33\" style=\"vertical-align: -18px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831825653\">\n<p id=\"fs-id1167831825655\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f00f61a5ef3b641c8e97f88c17e0ea2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#50;&#125;&#123;&#50;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"26\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831825675\">\n<div data-type=\"problem\" id=\"fs-id1167831825678\">\n<p id=\"fs-id1167831825680\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac15f9d056e8cd352bea9dc1ae1f93fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831115347\">\n<div data-type=\"problem\" id=\"fs-id1167831115349\">\n<p id=\"fs-id1167831115351\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61546a2e9313ed558cba75141abd1b20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#120;&#43;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#43;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#45;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"63\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826864294\">\n<p id=\"fs-id1167826864296\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b781b27822e582a607ea131596724fc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826932489\">\n<div data-type=\"problem\" id=\"fs-id1167826932492\">\n<p id=\"fs-id1167826932494\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c48a3b704db68d10200d7e7d1a9eb386_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#109;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#110;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#109;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"40\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835378244\"><strong data-effect=\"bold\">Simplify a Complex Rational Expression by Using the LCD<\/strong><\/p>\n<p id=\"fs-id1167835378250\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835378253\">\n<div data-type=\"problem\" id=\"fs-id1167835368010\">\n<p id=\"fs-id1167835368013\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f1396f664b16d1589fb891397858fa7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"37\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835368048\">\n<p id=\"fs-id1167835368050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9deb028bb2822163ee3aa88f47c67bfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831881490\">\n<div data-type=\"problem\" id=\"fs-id1167831881493\">\n<p id=\"fs-id1167831881495\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d587f6ed8ceffb36c42d5934bb2c18b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"38\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835180539\">\n<div data-type=\"problem\" id=\"fs-id1167835180542\">\n<p id=\"fs-id1167835180544\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e65cc80083063e199cb4ba357ae88db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#122;&#43;&#55;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#122;&#43;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#122;&#45;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"74\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831116404\">\n<p id=\"fs-id1167831116407\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-666133e74bce93188c5a24de8e201a5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#122;&#45;&#53;&#125;&#123;&#50;&#51;&#122;&#43;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"46\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831116431\">\n<div data-type=\"problem\" id=\"fs-id1167831116433\">\n<p id=\"fs-id1167831116435\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ad8255c9933187370dc8a89ced1946c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#121;&#45;&#51;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#121;&#45;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#43;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"62\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834130696\">\n<h4 data-type=\"title\"><a href=\"\/contents\/114b6c20-ac2e-4d26-8ede-f6f4a0bce191\" class=\"target-chapter\">7.4 Solve Rational Equations<\/a><\/h4>\n<p id=\"fs-id1167831887997\"><strong data-effect=\"bold\">Solve Rational Equations<\/strong><\/p>\n<p id=\"fs-id1167831888003\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831888006\">\n<div data-type=\"problem\" id=\"fs-id1167831888008\">\n<p id=\"fs-id1167831888010\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3faddacf410e71c3da82fb21649727d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831888036\">\n<p id=\"fs-id1167831888038\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c1a68497798f908bb216e7be7efa051_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#55;&#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-id1167831103003\">\n<div data-type=\"problem\" id=\"fs-id1167831103005\">\n<p id=\"fs-id1167831103007\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f208a6a52ad101669fa41587ab536492_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#109;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"90\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831922363\">\n<div data-type=\"problem\" id=\"fs-id1167831922365\">\n<p id=\"fs-id1167831922367\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f4dd7bae5828e4a2e5fa8e434c90ee5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#45;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#43;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"131\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835531491\">\n<p id=\"fs-id1167835531493\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31245fe5defdd234d8523ded31e7db29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835531509\">\n<div data-type=\"problem\" id=\"fs-id1167835531511\">\n<p id=\"fs-id1167835531513\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bc133e913124122c6bcf3473ec9b189_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#113;&#43;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#113;&#45;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"107\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826940578\">\n<div data-type=\"problem\" id=\"fs-id1167826940580\">\n<p id=\"fs-id1167826940582\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d35305ac4d8d7f320d2634fe9a5e4c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#45;&#49;&#53;&#125;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#118;&#43;&#49;&#56;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#118;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#118;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"166\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835621206\">\n<p id=\"fs-id1167835621208\">no solution <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835621214\">\n<div data-type=\"problem\" id=\"fs-id1167835621216\">\n<p id=\"fs-id1167835621218\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d0e03c9a64c4a2f36ec72cfc268e977_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#122;&#125;&#123;&#49;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#122;&#43;&#51;&#125;&#123;&#51;&#122;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"98\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835414568\"><strong data-effect=\"bold\">Solve Rational Equations that Involve Functions<\/strong><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835414573\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834429412\">\n<p id=\"fs-id1167834429414\">For rational function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3aba66c70c4be510cabeee2424a44a4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#56;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"126\" style=\"vertical-align: -9px;\" \/> <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d446cc08ae856e689f64d9c128c6986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835621277\">\n<p id=\"fs-id1167835621279\"><span class=\"token\">\u24d0<\/span> The domain is all real numbers except <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-452dfa3e30d0e3e80e17898832f59cfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7112ae50e9efd1b5a002a64d57f978bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" 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-50d0235abc5f48751d150bd26dc0dbe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#44;&#120;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"93\" 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-618bec1d74df8f2783cf0f6fdc4be51f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832153353\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832153356\">\n<p id=\"fs-id1167832153358\">For rational function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4c0b6b5f7979920e607db7c8c9bcee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#45;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#49;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"133\" style=\"vertical-align: -9px;\" \/> <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a863a7bff6459540d0d241a71243de8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834539223\"><strong data-effect=\"bold\">Solve a Rational Equation for a Specific Variable<\/strong><\/p>\n<p id=\"fs-id1167834539228\">In the following exercises, solve for the indicated variable.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834539232\">\n<div data-type=\"problem\" id=\"fs-id1167834539234\">\n<p id=\"fs-id1167834539236\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9fef1b66de034dbc460c9b8d4df02ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#86;&#125;&#123;&#108;&#125;&#61;&#104;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31529be6267191e2fa79454af14e828f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830702695\">\n<p id=\"fs-id1167830702697\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6cabc085fb2e4358a566edc85d2f06c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#86;&#125;&#123;&#104;&#119;&#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-id1167830702716\">\n<div data-type=\"problem\" id=\"fs-id1167830702719\">\n<p id=\"fs-id1167830702721\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29dd87ee6391f282c7e82c65860dce13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#121;&#125;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"74\" style=\"vertical-align: -9px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62f853fa6f372493298c507883a9f490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834191397\">\n<div data-type=\"problem\" id=\"fs-id1167834191399\">\n<p id=\"fs-id1167834191401\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a6ec16fb6a9f583a7e1dbd2b17552bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#53;&#125;&#123;&#122;&#45;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"61\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f61c5f97aacff00c045bf87511685b00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834309683\">\n<p id=\"fs-id1167834309685\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac152ccf65d90d50b1a9710a35eafb32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#53;&#43;&#55;&#120;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"86\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835353171\">\n<div data-type=\"problem\" id=\"fs-id1167835353173\">\n<p id=\"fs-id1167835353175\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ae67b53989a88e23d236b34f8ffe82e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#107;&#125;&#123;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"51\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d6b5f653e06cb8340667948d4fa1c05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835353217\">\n<h4 data-type=\"title\"><a href=\"\/contents\/b694470e-ede5-454b-8052-34b9bc92ae11\" class=\"target-chapter\">Solve Applications with Rational Equations<\/a><\/h4>\n<p id=\"fs-id1167834239074\"><strong data-effect=\"bold\">Solve Proportions<\/strong><\/p>\n<p id=\"fs-id1167834239080\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834239083\">\n<div data-type=\"problem\" id=\"fs-id1167834239085\">\n<p id=\"fs-id1167834239087\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce018f0644c696137fce473e8ee33039_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834239105\">\n<p id=\"fs-id1167834239107\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20b253d669aa678ca53fc72f135aa5e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831909917\">\n<div data-type=\"problem\" id=\"fs-id1167831909920\">\n<p id=\"fs-id1167831909922\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36c09ce2e5e7766fd7e5e1ac681d0b1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#121;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"42\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831909954\">\n<div data-type=\"problem\" id=\"fs-id1167831909956\">\n<p id=\"fs-id1167831909958\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7d49e4d1ac55d7586cacb5b6472f6cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#115;&#125;&#123;&#115;&#43;&#50;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"65\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830698152\">\n<p id=\"fs-id1167830698154\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b59223d2c162a0fdd163af8597f0845e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830698162\">\n<div data-type=\"problem\" id=\"fs-id1167830698164\">\n<p id=\"fs-id1167830698167\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cbfe4bd0b31354b9532b6c7f472766d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#45;&#51;&#125;&#123;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#43;&#50;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835621389\"><strong data-effect=\"bold\">Solve Using Proportions<\/strong><\/p>\n<p id=\"fs-id1167835621395\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835621398\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835621400\">\n<p id=\"fs-id1167835621402\">Rachael had a 21-ounce strawberry shake that has 739 calories. How many calories are there in a 32-ounce shake?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835621408\">\n<p id=\"fs-id1167835621410\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89d5f96b1734ee2d90f64df976c45b56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#54;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"34\" style=\"vertical-align: -1px;\" \/> calories<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835621419\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835621421\">\n<p id=\"fs-id1167835621424\">Leo went to Mexico over Christmas break and changed ?525 dollars into Mexican pesos. At that time, the exchange rate had ?1 US is equal to 16.25 Mexican pesos. How many Mexican pesos did he get for his trip?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832215360\"><strong data-effect=\"bold\">Solve Similar Figure Applications<\/strong><\/p>\n<p id=\"fs-id1171790307463\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832215366\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832215368\">\n<p id=\"fs-id1167832215371\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91acd2d3a8297e2d0caa8e3d4bfae3ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> is similar to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32a93b0a8112ce101de969b1d2e44893_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#88;&#89;&#90;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/> The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167834192033\" data-alt=\"The first figure is triangle A B C with side A B 8 units long, side B C 7 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 2 and two-thirds units long, side Y Z x units long, and side X Z 3 units long.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B 8 units long, side B C 7 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 2 and two-thirds units long, side Y Z x units long, and side X Z 3 units long.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834192003\">\n<p id=\"fs-id1167834192005\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b574dbd58b3bbc79d5bf5740ee9d4723_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#57;&#59;&#120;&#61;&#50;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"100\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834192043\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826803599\">\n<p id=\"fs-id1167826803602\">On a map of Europe, Paris, Rome, and Vienna form a triangle whose sides are shown in the figure below. If the actual distance from Rome to Vienna is 700 miles, find the distance from<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> Paris to Rome<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Paris to Vienna<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167826803633\" data-alt=\"The figure is a triangle formed by Paris, Vienna, and Rome. The distance between Paris and Vienna is 7.7 centimeters. The distance between Vienna and Rome is 7 centimeters. The distance between Rome and Paris is 8.9 centimeters.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by Paris, Vienna, and Rome. The distance between Paris and Vienna is 7.7 centimeters. The distance between Vienna and Rome is 7 centimeters. The distance between Rome and Paris is 8.9 centimeters.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826803645\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826803647\">\n<p id=\"fs-id1167826803649\">Francesca is 5.75 feet tall. Late one afternoon, her shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835317998\">\n<p id=\"fs-id1167835318000\">23 feet<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835318006\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835318008\">\n<p id=\"fs-id1167835318010\">The height of a lighthouse in Pensacola, Florida is 150 feet. Standing next to the statue, 5.5-foot-tall Natasha cast a 1.1-foot shadow. How long would the shadow of the lighthouse be?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835318023\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p>\n<p id=\"fs-id1167835318029\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835318032\">\n<div data-type=\"problem\" id=\"fs-id1167835318034\">\n<p id=\"fs-id1167835318036\">When making the 5-hour drive home from visiting her parents, Lolo ran into bad weather. She was able to drive 176 miles while the weather was good, but then driving 10 mph slower, went 81 miles when it turned bad. How fast did she drive when the weather was bad?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835318043\">\n<p id=\"fs-id1167835318045\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a494309aca1bdc45eb593f8e8497226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835621321\">\n<div data-type=\"problem\" id=\"fs-id1167835621323\">\n<p id=\"fs-id1167835621325\">Mark is riding on a plane that can fly 490 miles with a tailwind of 20 mph in the same time that it can fly 350 miles against a tailwind of 20 mph. What is the speed of the plane?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835621343\">\n<div data-type=\"problem\" id=\"fs-id1167835621345\">\n<p id=\"fs-id1167835621347\">Josue can ride his bicycle 8 mph faster than Arjun can ride his bike. It takes Luke 3 hours longer than Josue to ride 48 miles. How fast can John ride his bike?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835621353\">\n<p id=\"fs-id1167835621355\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8dde8d4cade60a4f3ca0e779512b974c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835621364\">\n<div data-type=\"problem\" id=\"fs-id1167835621366\">\n<p id=\"fs-id1167835621369\">Curtis was training for a triathlon. He ran 8 kilometers and biked 32 kilometers in a total of 3 hours. His running speed was 8 kilometers per hour less than his biking speed. What was his running speed?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831949094\"><strong data-effect=\"bold\">Solve Work Applications<\/strong><\/p>\n<p id=\"fs-id1167831949099\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167831949104\">\n<p id=\"fs-id1167831949106\">Brandy can frame a room in 1 hour, while Jake takes 4 hours. How long could they frame a room working together?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831949112\">\n<p id=\"fs-id1167831949114\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c995a9c919f066bf6863b7f22d9cc88f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> hour<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831949127\">\n<div data-type=\"problem\" id=\"fs-id1167831949129\">\n<p id=\"fs-id1167831949131\">Prem takes 3 hours to mow the lawn while her cousin, Barb, takes 2 hours. How long will it take them working together?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832116068\">\n<div data-type=\"problem\" id=\"fs-id1167832116071\">\n<p id=\"fs-id1167832116073\">Jeffrey can paint a house in 6 days, but if he gets a helper he can do it in 4 days. How long would it take the helper to paint the house alone?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832116078\">\n<p id=\"fs-id1167832116080\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40f7428479801bf4ef27a5149e9df526_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" style=\"vertical-align: -1px;\" \/> days<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832116090\">\n<div data-type=\"problem\" id=\"fs-id1167832116092\">\n<p id=\"fs-id1167832116094\">Marta and Deb work together writing a book that takes them 90 days. If Sue worked alone it would take her 120 days. How long would it take Deb to write the book alone?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167830703231\"><strong data-effect=\"bold\">Solve Direct Variation Problems<\/strong><\/p>\n<p id=\"fs-id1167830703236\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830703239\">\n<div data-type=\"problem\" id=\"fs-id1167830703241\">\n<p id=\"fs-id1167830703243\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> varies directly as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-633f747b2de427f7a98de490b31bd0f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> and <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;\" \/> find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-544f0c217d8ac90dfa2a61195c4c02f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830809300\">\n<p id=\"fs-id1167830809303\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aca3387e59afe477960eabc2f23b3db5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830809310\">\n<div data-type=\"problem\" id=\"fs-id1167830809312\">\n<p id=\"fs-id1167830809314\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> varies inversely as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b39c88cf316710a4ae3bb2aea5385885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e211b3d19a6898a4c9192f117c1fe08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0788dd24eed136bfb85d403b3bf3ea63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830697548\">\n<div data-type=\"problem\" id=\"fs-id1167830697550\">\n<p id=\"fs-id1167830697552\">Vanessa is traveling to see her fianc\u00e9. The distance, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bca0eeb2425a54167e5e6044c55c320b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: -4px;\" \/> varies directly with the speed, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fec560ccea89a9f8adbc60a211badaed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/> she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835414678\">\n<p id=\"fs-id1167835414680\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1028393fcf0d55343fa3edfc3a8180ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835414690\">\n<div data-type=\"problem\" id=\"fs-id1167835414692\">\n<p id=\"fs-id1167835414694\">If the cost of a pizza varies directly with its diameter, and if an 8\u201d diameter pizza costs ?12, how much would a 6\u201d diameter pizza cost?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835517931\">\n<div data-type=\"problem\" id=\"fs-id1167835517933\">\n<p id=\"fs-id1167835517935\">The distance to stop a car varies directly with the square of its speed. It takes 200 feet to stop a car going 50 mph. How many feet would it take to stop a car going 60 mph?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835517941\">\n<p id=\"fs-id1167835517943\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a5a438892b7b8a4ac9923f442ddfbd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#56;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> feet<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835517952\"><strong data-effect=\"bold\">Solve Inverse Variation Problems<\/strong><\/p>\n<p id=\"fs-id1167835517958\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835517961\">\n<div data-type=\"problem\" id=\"fs-id1167835517963\">\n<p id=\"fs-id1167835517965\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> varies inversely with the square of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-428977d87d8c9ab2c3d60050f95d109b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5072b7479ca854c5e3cdea8ffff2c0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7fc2201c80103aaa425fc3eb5c3eac6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3070dfc42d639f884f54816708c8d1a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920975\">\n<div data-type=\"problem\" id=\"fs-id1167831920978\">\n<p id=\"fs-id1167831920980\">The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase 12 tickets for ?6, how many tickets can Madelyn afford to buy if the price increased to ?8?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835377879\">\n<p id=\"fs-id1167835377881\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-856c036bf740697388d177b1dc5ba52a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/> tickets<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835377890\">\n<div data-type=\"problem\" id=\"fs-id1167835377893\">\n<p id=\"fs-id1167835377895\">On a string instrument, the length of a string varies inversely with the frequency of its vibrations. If an 11-inch string on a violin has a frequency of 360 cycles per second, what frequency does a 12-inch string have?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835377914\">\n<h4 data-type=\"title\"><a href=\"\/contents\/a68b06f6-2833-4512-b24f-c0da889a8759\" class=\"target-chapter\">Solve Rational Inequalities<\/a><\/h4>\n<p id=\"fs-id1167834327303\"><strong data-effect=\"bold\">Solve Rational Inequalities<\/strong><\/p>\n<p id=\"fs-id1167834327308\">In the following exercises, solve each rational inequality and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834327312\">\n<div data-type=\"problem\" id=\"fs-id1167834327314\">\n<p id=\"fs-id1167834327316\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db6b777fac970b6bc7ba622115f2fa1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#51;&#125;&#123;&#120;&#43;&#52;&#125;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"60\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834327342\">\n<p id=\"fs-id1167834327344\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e98936559a31e5a971a9977b1631f707_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832117897\">\n<div data-type=\"problem\" id=\"fs-id1167832117899\">\n<p id=\"fs-id1167832117901\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b396771db4183cde13f4c1a76d4fbc6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#125;&#123;&#120;&#45;&#50;&#125;&#62;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826987657\">\n<div data-type=\"problem\" id=\"fs-id1167826987659\">\n<p id=\"fs-id1167826987662\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c50987e307e4da1923148f424b97fb18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#45;&#50;&#125;&#123;&#120;&#45;&#52;&#125;&#92;&#108;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826987689\">\n<p id=\"fs-id1167835497978\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5838c3f1d6839c64f46976edce1a44f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#54;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835497997\">\n<div data-type=\"problem\" id=\"fs-id1167835497999\">\n<p id=\"fs-id1167835498002\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6677273ef308372eac6de8cce1c61884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#49;&#50;&#125;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"99\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834525225\">\n<div data-type=\"problem\" id=\"fs-id1167834525227\">\n<p id=\"fs-id1167834525229\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebf0e34efb664554a4f08a61d1976d2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834525261\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8233b2ec55c91265661e20aa3f777b69_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;&#50;&#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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835414782\">\n<div data-type=\"problem\" id=\"fs-id1167835414784\">\n<p id=\"fs-id1167835414786\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5737538f83bdc264bc02beba0baa32f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#45;&#50;&#125;&#60;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"79\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832068272\"><strong data-effect=\"bold\">Solve an Inequality with Rational Functions<\/strong><\/p>\n<p id=\"fs-id1167832068277\">In the following exercises, solve each rational function inequality and write the solution in interval notation<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167828441724\">\n<div data-type=\"problem\" id=\"fs-id1167828441726\">\n<p id=\"fs-id1167828441729\">Given the function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c89c95b97b496f5d506c17e0449cbe26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#53;&#125;&#123;&#120;&#45;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/> find the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> that make the function greater than or equal to 0.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828441772\">\n<p id=\"fs-id1167828441774\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73fb9ccf49afc1442ad74175ed225033_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;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#53;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827957228\">\n<div data-type=\"problem\" id=\"fs-id1167827957230\">\n<p id=\"fs-id1167827957232\">Given the function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-817b40056de4430f3a4361e4a612616c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#120;&#43;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"97\" style=\"vertical-align: -8px;\" \/> find the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> that make the function less than or equal to 0.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827957618\">\n<div data-type=\"problem\" id=\"fs-id1167827957620\">\n<p id=\"fs-id1167827957622\">The function<\/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-f207163ea824f3ba792325d5b20f73ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#53;&#48;&#120;&#43;&#49;&#48;&#48;&#44;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"184\" style=\"vertical-align: -4px;\" \/> represents the cost to produce <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> number of items. Find <span class=\"token\">\u24d0<\/span> the average cost function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc1adfaa08ed832cb0751363955bdd21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> how many items should be produced so that the average cost is less than ?160.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834463827\">\n<p id=\"fs-id1167834463829\">\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-8900652674fe2c504c25f12ab68325ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#48;&#120;&#43;&#49;&#48;&#48;&#48;&#48;&#48;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"142\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> More than 10,000 items must be produced to keep the average cost below <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d99b8076d9e41ecc46df524253df1c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#49;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: -1px;\" \/> per item.<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835354781\">\n<div data-type=\"problem\" id=\"fs-id1167835354783\">\n<p id=\"fs-id1167835354785\">Tillman is starting his own business by selling tacos at the beach. Accounting for the cost of his food truck and ingredients for the tacos, the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b67c704239e4188214856fcff8fb339_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#43;&#54;&#44;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/> represents the cost for Tillman to produce <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> tacos. Find <span class=\"token\">\u24d0<\/span> the average cost function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc1adfaa08ed832cb0751363955bdd21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/> for Tillman\u2019s Tacos <span class=\"token\">\u24d1<\/span> how many tacos should Tillman produce so that the average cost is less than ?4.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167835575399\">\n<h3 data-type=\"title\">Practice Test<\/h3>\n<p id=\"fs-id1167835575406\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835575409\">\n<div data-type=\"problem\" id=\"fs-id1167835575411\">\n<p id=\"fs-id1167835575413\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07f560d9d770469ef219d0019a8f8658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#98;&#125;&#123;&#49;&#50;&#97;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"34\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835575441\">\n<p id=\"fs-id1167835575443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81896b935d868e3c96d7d850dde107a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#51;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"13\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826783185\">\n<div data-type=\"problem\" id=\"fs-id1167826783187\">\n<p id=\"fs-id1167826783189\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-585ff146c9871587e64ad20b0319b865_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#120;&#45;&#49;&#56;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826783235\">In the following exercises, perform the indicated operation and simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831030461\">\n<div data-type=\"problem\" id=\"fs-id1167831030463\">\n<p id=\"fs-id1167831030466\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5bb2e9689d071351d74a4ce13b12c76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#125;&#123;&#120;&#43;&#50;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#54;&#125;&#123;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831030514\">\n<p id=\"fs-id1167831030516\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6704f60ef0e02f63a251b709654a6f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#51;&#125;&#123;&#51;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"26\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834402831\">\n<div data-type=\"problem\" id=\"fs-id1167834402834\">\n<p id=\"fs-id1167834402836\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ff5db99c94b975b23e402219461c891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#121;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"92\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832215290\">\n<div data-type=\"problem\" id=\"fs-id1167832215292\">\n<p id=\"fs-id1167832215294\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52320491729cfd9048e4e936325ed7db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#43;&#50;&#48;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#45;&#55;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"162\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830699035\">\n<p id=\"fs-id1167830923399\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-805cf0bcb22d145a673929a5430ae93d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#51;&#125;&#123;&#120;&#43;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"26\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830923422\">\n<div data-type=\"problem\" id=\"fs-id1167830923424\">\n<p id=\"fs-id1167830923426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c83e173585b40fd7c3e3e629fe9d1c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#97;&#125;&#123;&#51;&#97;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#97;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#97;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"114\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826849514\">\n<div data-type=\"problem\" id=\"fs-id1167826849517\">\n<p id=\"fs-id1167826849519\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e07f8843f96e972e043d99688fb7963_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#110;&#45;&#49;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#110;&#45;&#49;&#125;&#123;&#49;&#45;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"150\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826828378\">\n<p id=\"fs-id1167826828380\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4694708cce18b58002d0499bd76aa1f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#110;&#45;&#50;&#125;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"33\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834562469\">\n<div data-type=\"problem\" id=\"fs-id1167834562471\">\n<p id=\"fs-id1167834562473\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b034c20a49f95f40a350854427b96ee4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#120;&#45;&#55;&#125;&#123;&#56;&#120;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#49;&#125;&#123;&#51;&#45;&#56;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"169\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831822896\">\n<div data-type=\"problem\" id=\"fs-id1167831822899\">\n<p id=\"fs-id1167831822901\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b54928569c7caf126d59500548674ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#109;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#109;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"37\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832059767\">\n<p id=\"fs-id1167832059769\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6895be3ba1383d2cfe4df5d6e939ee68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#45;&#109;&#125;&#123;&#109;&#43;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"32\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832059792\">In the following exercises, solve each equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832059795\">\n<div data-type=\"problem\" id=\"fs-id1167832059797\">\n<p id=\"fs-id1167832059799\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a0f5561092e147896897c73212488e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830705543\">\n<div data-type=\"problem\" id=\"fs-id1167830705545\">\n<p id=\"fs-id1167830705547\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38e4cb722816e286dde096824e2d30f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#122;&#45;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#122;&#43;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"141\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835496307\">\n<p id=\"fs-id1167835496309\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-245b65492d7318ac0623a96a41ec8879_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835496325\">\n<div data-type=\"problem\" id=\"fs-id1167835496327\">\n<p id=\"fs-id1167835496329\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91d2004530b1a1f88577891effda174d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#122;&#125;&#123;&#50;&#122;&#43;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#122;&#45;&#56;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#122;&#45;&#49;&#54;&#125;&#123;&#56;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#122;&#45;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"193\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826819292\">In the following exercises, solve each rational inequality and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826819296\">\n<div data-type=\"problem\" id=\"fs-id1167826819298\">\n<p id=\"fs-id1167826819300\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f51e6790d60c019db42299ca5028f3a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#120;&#125;&#123;&#120;&#45;&#54;&#125;&#92;&#108;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826819323\">\n<p id=\"fs-id1167826819325\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bd174d0b13ab9acb98df1da06a7568b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#51;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830700720\">\n<div data-type=\"problem\" id=\"fs-id1167830700722\">\n<p id=\"fs-id1167830700725\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36a16d020a221b9d17056ec79d6b054a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#43;&#51;&#125;&#123;&#120;&#45;&#54;&#125;&#62;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835531841\">\n<div data-type=\"problem\" id=\"fs-id1167835531843\">\n<p id=\"fs-id1167835531845\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa8bcdc076268fb9a1d253711896801a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835531876\">\n<p id=\"fs-id1167835531878\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da147a3981da20329d97947a2cb60ae1_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;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#54;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"170\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834494827\">In the following exercises, find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f264abc4b41d7f930b4d3d73b6915acb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> given <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cf2960a734fbb77b7ee136645c35dac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"128\" style=\"vertical-align: -8px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9aefec705aaea7c5356c1f11df030b1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#53;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830961400\">\n<div data-type=\"problem\" id=\"fs-id1167830961402\">\n<p id=\"fs-id1167830961404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7de960a332c98ab624de84fc3632a5b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834346852\">\n<div data-type=\"problem\" id=\"fs-id1167834346854\">\n<p id=\"fs-id1167834346856\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a19d698605e6f0a4acb05b75c0c8025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834346898\">\n<p id=\"fs-id1167834120083\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afa23ba2fb64e195fa3a3f107adc0432_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"139\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834120131\">\n<div data-type=\"problem\" id=\"fs-id1167834120134\">\n<p id=\"fs-id1167834120136\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13c4f4106f31127d3b3dd55dae014d98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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-id1167834346478\">\n<div data-type=\"problem\" id=\"fs-id1167834346480\">\n<p id=\"fs-id1167834346483\">Given the function,<\/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-24062bafa4895ed1da7970c22df9eca1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;&#53;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"136\" style=\"vertical-align: -9px;\" \/> find the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> that make the function less than or equal to 0.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835414618\">\n<p id=\"fs-id1167835414620\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-745ab7d13c2bd8ff8df4c39424bf735b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"35\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835414640\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835414643\">\n<div data-type=\"problem\" id=\"fs-id1167835414645\">\n<p id=\"fs-id1167835414647\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> varies directly with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, and <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;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17acc4d2c8e05f20df3331ef13fe72d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/> find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-415ffe3279e92b9a407bb2060515d03e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834123576\">\n<div data-type=\"problem\" id=\"fs-id1167834123578\">\n<p id=\"fs-id1167834123580\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> varies inversely with the square of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58f6088d510af3792bc3bf5981c30e7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0788dd24eed136bfb85d403b3bf3ea63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832212105\">\n<p id=\"fs-id1167832212107\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0ef2d7f8274fb8137ec050ca3af7248_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#49;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"49\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834224702\">\n<div data-type=\"problem\" id=\"fs-id1167834224704\">\n<p id=\"fs-id1167834224706\">Matheus can ride his bike for 30 miles with the wind in the same amount of time that he can go 21 miles against the wind. If the wind\u2019s speed is 6 mph, what is Matheus\u2019 speed on his bike?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834224726\">\n<div data-type=\"problem\" id=\"fs-id1167834224728\">\n<p id=\"fs-id1167834224730\">Oliver can split a truckload of logs in 8 hours, but working with his dad they can get it done in 3 hours. How long would it take Oliver\u2019s dad working alone to split the logs?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834224737\">\n<p id=\"fs-id1167834224739\">Oliver\u2019s dad would take <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98909f69f0cbe2aca9c088c014144c8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"18\" style=\"vertical-align: -6px;\" \/> hours to split the logs himself.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834224755\">\n<div data-type=\"problem\" id=\"fs-id1167834224757\">\n<p id=\"fs-id1167830964041\">The volume of a gas in a container varies inversely with the pressure on the gas. If a container of nitrogen has a volume of 29.5 liters with 2000 psi, what is the volume if the tank has a 14.7 psi rating? Round to the nearest whole number.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830964060\">\n<div data-type=\"problem\" id=\"fs-id1167830964062\">\n<p id=\"fs-id1167830964064\">The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio. The diagram gives the map distances between these cities in inches.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167830964071\" data-alt=\"The figure is a triangle formed by Cincinnati, Dayton, and Columbus. The distance between Cincinnati and Dayton is 2.4 inches. The distance between Dayton and Columbus is 3.2 inches. The distance between Columbus and Cincinnati is 5.3 inches.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_06_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by Cincinnati, Dayton, and Columbus. The distance between Cincinnati and Dayton is 2.4 inches. The distance between Dayton and Columbus is 3.2 inches. The distance between Columbus and Cincinnati is 5.3 inches.\" \/><\/span><\/p>\n<p id=\"fs-id1167830964083\">The actual distance from Dayton to Cincinnati is 48 miles. What is the actual distance between Dayton and Columbus?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830964088\">\n<p id=\"fs-id1167830964090\">The distance between Dayton and Columbus is 64 miles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167830964100\">\n<dt>critical point of a rational inequality<\/dt>\n<dd id=\"fs-id1167828420827\">The critical point of a rational inequality is a number which makes the rational expression zero or undefined.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167828420833\">\n<dt>rational inequality<\/dt>\n<dd id=\"fs-id1167828420838\">A rational inequality is an inequality that contains a rational expression.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3471","chapter","type-chapter","status-publish","hentry"],"part":3130,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3471","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3471\/revisions"}],"predecessor-version":[{"id":15250,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3471\/revisions\/15250"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/3130"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3471\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3471"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3471"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3471"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3471"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}