{"id":3363,"date":"2018-12-11T13:51:43","date_gmt":"2018-12-11T18:51:43","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-rational-equations\/"},"modified":"2018-12-11T13:51:43","modified_gmt":"2018-12-11T18:51:43","slug":"solve-rational-equations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-rational-equations\/","title":{"raw":"Solve Rational Equations","rendered":"Solve Rational Equations"},"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 equations<\/li><li>Use rational functions<\/li><li>Solve a rational equation for a specific variable<\/li><\/ul><\/div><div data-type=\"note\" class=\"be-prepared\"><p id=\"fs-id1167831928853\">Before you get started, take this readiness quiz.<\/p><ol type=\"1\"><li>Solve: \\(\\frac{1}{6}x+\\frac{1}{2}=\\frac{1}{3}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167833239741\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve: \\({n}^{2}-5n-36=0.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836625705\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve the formula \\(5x+2y=10\\) for \\(y.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/b03538a1-8a7b-4158-a68b-e0e8a24c9fd4#fs-id1167835229496\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><p id=\"fs-id1167832052437\">After defining the terms \u2018expression\u2019 and \u2018equation\u2019 earlier, we have used them throughout this book. We have <em data-effect=\"italics\">simplified<\/em> many kinds of <em data-effect=\"italics\">expressions<\/em> and <em data-effect=\"italics\">solved<\/em> many kinds of <em data-effect=\"italics\">equations<\/em>. We have simplified many rational expressions so far in this chapter. Now we will <em data-effect=\"italics\">solve<\/em> a <span data-type=\"term\">rational equation<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167835640377\"><div data-type=\"title\">Rational Equation<\/div><p>A <strong data-effect=\"bold\">rational equation<\/strong> is an equation that contains a rational expression.<\/p><\/div><p id=\"fs-id1167835369643\">You must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign.<\/p><div data-type=\"equation\" id=\"fs-id1167827958702\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\hfill \\text{Rational Expression}\\hfill &amp; &amp; &amp; \\hfill \\text{Rational Equation}\\hfill \\\\ \\hfill \\begin{array}{c}\\hfill \\frac{1}{8}x+\\frac{1}{2}\\hfill \\\\ \\hfill \\frac{y+6}{{y}^{2}-36}\\hfill \\\\ \\hfill \\frac{1}{n-3}+\\frac{1}{n+4}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\begin{array}{ccc}\\hfill \\frac{1}{8}x+\\frac{1}{2}&amp; =\\hfill &amp; \\frac{1}{4}\\hfill \\\\ \\hfill \\frac{y+6}{{y}^{2}-36}&amp; =\\hfill &amp; y+1\\hfill \\\\ \\hfill \\frac{1}{n-3}+\\frac{1}{n+4}&amp; =\\hfill &amp; \\frac{15}{{n}^{2}+n-12}\\hfill \\end{array}\\hfill \\end{array}\\)<\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832053699\"><h3 data-type=\"title\">Solve Rational Equations<\/h3><p id=\"fs-id1167835563859\">We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to \u201cclear\u201d the fractions.<\/p><p>We will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then, we will have an equation that does not contain rational expressions and thus is much easier for us to solve. But because the original equation may have a variable in a denominator, we must be careful that we don\u2019t end up with a solution that would make a denominator equal to zero.<\/p><p id=\"fs-id1167835360295\">So before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.<\/p><p id=\"fs-id1167831912071\">An algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an <span data-type=\"term\">extraneous solution to a rational equation<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167835320224\"><div data-type=\"title\">Extraneous Solution to a Rational Equation<\/div><p id=\"fs-id1167831239457\">An <strong data-effect=\"bold\">extraneous solution to a rational equation<\/strong> is an algebraic solution that would cause any of the expressions in the original equation to be undefined.<\/p><\/div><p id=\"fs-id1167835319662\">We note any possible extraneous solutions, <em data-effect=\"italics\">c<\/em>, by writing \\(x\\ne c\\) next to the equation.<\/p><div data-type=\"example\" id=\"fs-id1167832212123\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a Rational Equation<\/div><div data-type=\"exercise\" id=\"fs-id1167835373684\"><div data-type=\"problem\"><p id=\"fs-id1167834516061\">Solve: \\(\\frac{1}{x}+\\frac{1}{3}=\\frac{5}{6}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834222446\"><span data-type=\"media\" id=\"fs-id1167831882177\" data-alt=\"Step 1 is to find any value of the variable that makes the denominator of the zero. Remember that if x is equal to 0, then 1 divided by x is undefined. So the equation becomes the sum of 1 divided by x and one-third is equal to five-sixths, where x is not equal to 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find any value of the variable that makes the denominator of the zero. Remember that if x is equal to 0, then 1 divided by x is undefined. So the equation becomes the sum of 1 divided by x and one-third is equal to five-sixths, where x is not equal to 0.\"><\/span><span data-type=\"media\" id=\"fs-id1167835373808\" data-alt=\"Step 2 is to find the least common denominator of all the fractions in the problem, 1 divided by x, one-third, and five-sixths. The least common denominator is 6 x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find the least common denominator of all the fractions in the problem, 1 divided by x, one-third, and five-sixths. The least common denominator is 6 x.\"><\/span><span data-type=\"media\" data-alt=\"Step 3 is to clear the fractions in the equation by multiplying each side by the least common denominator. The result is 6 x times the sum of 1 divided by x and one-third is equal to 6 x times five-sixths. Simplify using the distributive property. The result is 6 x times the quantity1 divided by x plus 6 x times one-third is equal to 6 x times five-sixths, which simplifies to 6 plus 2 x is equal to 5 x. This simplifies to 6 is equal to 3 x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to clear the fractions in the equation by multiplying each side by the least common denominator. The result is 6 x times the sum of 1 divided by x and one-third is equal to 6 x times five-sixths. Simplify using the distributive property. The result is 6 x times the quantity1 divided by x plus 6 x times one-third is equal to 6 x times five-sixths, which simplifies to 6 plus 2 x is equal to 5 x. This simplifies to 6 is equal to 3 x.\"><\/span><span data-type=\"media\" id=\"fs-id1167834060144\" data-alt=\"Step 4 is to solve the equation that results. The result is 2 is equal to x\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to solve the equation that results. The result is 2 is equal to x\"><\/span><span data-type=\"media\" id=\"fs-id1167834133754\" data-alt=\"Step 5 is to check the solution. Remember that any solutions that makes the original expression undefined must be discarded. The solution is not 0. Substitute x is equal to 2 into the original equation, 1 divided by x plus one-third is equal to five-sixths. Is one-half plus one-third is equal to five-sixths a true equation? Is three-sixths plus two-sixth is equal to five-sixths a true equation? Three-sixths plus two-sixth is equal to five-sixths. Five-sixth is equal to five-sixth is a true equation. So, the solution is x is equal to 2\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to check the solution. Remember that any solutions that makes the original expression undefined must be discarded. The solution is not 0. Substitute x is equal to 2 into the original equation, 1 divided by x plus one-third is equal to five-sixths. Is one-half plus one-third is equal to five-sixths a true equation? Is three-sixths plus two-sixth is equal to five-sixths a true equation? Three-sixths plus two-sixth is equal to five-sixths. Five-sixth is equal to five-sixth is a true equation. So, the solution is x is equal to 2\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835419818\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834280170\"><div data-type=\"problem\" id=\"fs-id1167835416569\"><p id=\"fs-id1167835367835\">Solve: \\(\\frac{1}{y}+\\frac{2}{3}=\\frac{1}{5}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831086720\"><p id=\"fs-id1167826813938\">\\(y=-\\frac{7}{15}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835283352\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830704153\"><div data-type=\"problem\" id=\"fs-id1167834387408\"><p>Solve: \\(\\frac{2}{3}+\\frac{1}{5}=\\frac{1}{x}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835369741\"><p id=\"fs-id1167835595617\">\\(x=\\frac{13}{15}\\)<\/p><\/div><\/div><\/div><p>The steps of this method are shown.<\/p><div data-type=\"note\" id=\"fs-id1167835378137\" class=\"howto\"><div data-type=\"title\">Solve equations with rational expressions.<\/div><ol type=\"1\" class=\"stepwise\"><li>Note any value of the variable that would make any denominator zero.<\/li><li>Find the least common denominator of <em data-effect=\"italics\">all<\/em> denominators in the equation.<\/li><li>Clear the fractions by multiplying both sides of the equation by the LCD.<\/li><li>Solve the resulting equation.<\/li><li>Check: <ul data-bullet-style=\"bullet\"><li>If any values found in Step 1 are algebraic solutions, discard them.<\/li><li>Check any remaining solutions in the original equation.<\/li><\/ul><\/li><\/ol><\/div><p id=\"fs-id1167835528067\">We always start by noting the values that would cause any denominators to be zero.<\/p><div data-type=\"example\" id=\"fs-id1167831228721\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a Rational Equation using the Zero Product Property<\/div><div data-type=\"exercise\" id=\"fs-id1167826987271\"><div data-type=\"problem\" id=\"fs-id1167835510388\"><p id=\"fs-id1167835513316\">Solve: \\(1-\\frac{5}{y}=-\\frac{6}{{y}^{2}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834536690\"><table class=\"unnumbered unstyled can-break\" summary=\"1 minus the quantity 5 divided by y is equal to the negative of the quantity 6 divided by y squared. Identify any value of y that would make any denominator 0. Add y is not equal to 0 to the equation. The least common denominator of the fractions in the equation, the quantity 5 divided by y and the negative of the quantity 6 divided by y squared is y squared. Clear the fractions in the equation by multiplying each side by y squared. y squared times the quantity 1 minus the quantity 5 divided by y is equal to y squared times the negative of the quantity 6 divided by y squared. Distribute y squared. The result is y squared times 1 minus y squared times the quantity 5 divided by y is equal to the negative of the quantity 6 divided by y squared. Multiply on both sides of the equation. The result is y squared minus 5 y is equal to negative 6. To solve this equation, first write it in standard form. Its standard form is y squared minus 5 y plus 6 is equal to 0. Factor the left side of the equation. The result is the quantity y minus 2 times the quantity y minus 3 is equal to zero. Using the Zero Product Property. the equation becomes y minus 2 is equal to 0 or y minus 3 is equal to 0. Solving each equation results in the solutions, y is equal to 2 or y is equal to 3. Check the solutions by substituting them into the original equation, 1 minus the quantity 5 divided by y is equal to the negative of 6 divided by y squared. Note that 0 was not an algebraic solution. Is the equation 1 minus the quantity 5 divided by 2 is equal to the quantity 6 divided by 2 squared true? When each side is simplified, the result is negative 3 divided by 2 is equal negative 3 divided by 2, which is a true equation. Is the equation 1 minus the quantity 5 divided by 3 is equal to the quantity 6 divided by 3 squared true? When each side is simplified, the result is negative 2 divided by 3 is equal to negative 2 divided by 3, which is a true equation. The solution is y is equal to 2 or y is equal to 3.\" data-label=\"\"><tbody><tr><td><\/td><td 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_04_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Note any value of the variable that would make<div data-type=\"newline\"><br><\/div>any denominator zero.<\/td><td 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_04_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Find the least common denominator of all denominators in<div data-type=\"newline\"><br><\/div>the equation. The LCD is <em data-effect=\"italics\">y<\/em><sup>2<\/sup>.<\/td><td><\/td><\/tr><tr><td data-align=\"left\">Clear the fractions by multiplying both sides of<div data-type=\"newline\"><br><\/div>the equation by the LCD.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834299943\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Distribute.<\/td><td 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_04_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Multiply.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834049006\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve the resulting equation. First<div data-type=\"newline\"><br><\/div>write the quadratic equation in standard form.<\/td><td 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_04_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Factor.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832060135\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Use the Zero Product Property.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834120431\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831910249\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Check.<div data-type=\"newline\"><br><\/div>We did not get 0 as an algebraic solution.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835370931\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><td><\/td><\/tr><tr><td><\/td><td data-align=\"left\">The solution is \\(y=2,\\) \\(y=3.\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835317276\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835417937\"><div data-type=\"problem\" id=\"fs-id1167835334526\"><p id=\"fs-id1167835364449\">Solve: \\(1-\\frac{2}{x}=\\frac{15}{{x}^{2}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835350474\"><p id=\"fs-id1167835234636\">\\(x=-3,x=5\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831958129\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834239193\"><div data-type=\"problem\" id=\"fs-id1167831116689\"><p id=\"fs-id1167835254163\">Solve: \\(1-\\frac{4}{y}=\\frac{12}{{y}^{2}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835529923\"><p id=\"fs-id1167835353178\">\\(y=-2,y=6\\)<\/p><\/div><\/div><\/div><p>In the next example, the last denominators is a difference of squares. Remember to factor it first to find the LCD.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834061864\"><div data-type=\"problem\" id=\"fs-id1167831846925\"><p id=\"fs-id1167828435100\">Solve: \\(\\frac{2}{x+2}+\\frac{4}{x-2}=\\frac{x-1}{{x}^{2}-4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832036216\"><table id=\"fs-id1167835596565\" class=\"unnumbered unstyled can-break\" summary=\"2 divided by the quantity x plus 2 plus 4 divided by the quantity x minus 2 is equal to the quantity x minus 1 divided by the quantity x squared minus 4. Notice that x is equal to negative 2 and x is equal to 2 would make denominators zero. Factor the denominator on the right side of the equation. The result is 2 divided by the quantity x plus 2 plus 4 divided by the quantity x minus 2 is equal to the quantity x minus 1 divided by the quantity x plus 2 times the quantity x minus 2, x is not equal to negative and x is not equal to 2. The least common denominator of all the denominators in the equation is the quantity x plus 2 times the quantity x minus 2. Clear the fractions by multiplying both sides of the equation by the least common denominator, the quantity x plus 2 times the quantity x minus 2. Distribute the least common denominator to each expression in the equation. The result is the quantity x plus 2 times the quantity x minus 2 times 2 divided by the quantity x plus 2 plus the quantity x plus 2 times the quantity x minus 2 times 4 divided by the quantity x minus 2 is equal to the quantity x plus 2 times the quantity x minus 2 times the quantity x minus 1 divided by the quantity x squared minus 4. Remove the common factors from the equation. The result is 2 times the quantity x minus 2 plus 4 times the quantity x plus 2 is equal to x minus 1. Distribute the constants on the left side of the equation. The result is 2 x minus 4 plus 4 x plus 8 is equal to x minus 1. Solve the equation by simplifying each side of the equation. The result is 6 x plus 4 is equal to x minus 1. Isolating the variable term, the result is 5 x is equal to negative 5. Solving for x, the solution is x is equal to negative 1. Check the solution. Remember x cannot be equal to 2 and x cannot be equal to negative 2. Substitute x is equal to negative into the original equation, 2 divided by the quantity x plus 2 plus 4 divided by the quantity x minus 2 is equal to the quantity x minus 1 divided by the quantity x squared minus 4. Is 2 divided by the quantity negative 1 plus 2 plus 4 divided by the quantity negative 1 minus 2 is equal to the quantity negative 1 minus 1 all divided by the square of negative 1 minus 4? When each side of the equation is simplified, the result is 2 divided by 3 is equal to 2 divided by 3, which is true. So, the solution is x is equal to negative 1.\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835215786\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Note any value of the variable<div data-type=\"newline\"><br><\/div>that would make any denominator<div data-type=\"newline\"><br><\/div>zero.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831036916\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Find the least common<div data-type=\"newline\"><br><\/div>denominator of all denominators<div data-type=\"newline\"><br><\/div>in the equation.<div data-type=\"newline\"><br><\/div>The LCD is \\(\\left(x+2\\right)\\left(x-2\\right).\\)<\/td><td><\/td><\/tr><tr><td data-align=\"left\">Clear the fractions by multiplying<div data-type=\"newline\"><br><\/div>both sides of the equation by the<div data-type=\"newline\"><br><\/div>LCD.<\/td><td 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_04_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Distribute.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835381726\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Remove common factors.<\/td><td 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_04_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td 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_04_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Distribute.<\/td><td 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_04_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve.<\/td><td 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_04_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835545413\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167831883495\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div>We did not get 2 or \u22122 as algebraic solutions.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167828377151\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><td><\/td><\/tr><tr><td><\/td><td data-align=\"left\">The solution is \\(x=-1.\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834505929\"><div data-type=\"problem\"><p id=\"fs-id1167834395459\">Solve: \\(\\frac{2}{x+1}+\\frac{1}{x-1}=\\frac{1}{{x}^{2}-1}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831066244\"><p>\\(x=\\frac{2}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835489220\"><div data-type=\"problem\" id=\"fs-id1167826997270\"><p id=\"fs-id1167832214483\">Solve: \\(\\frac{5}{y+3}+\\frac{2}{y-3}=\\frac{5}{{y}^{2}-9}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835351509\"><p>\\(y=2\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835304069\">In the next example, the first denominator is a <span data-type=\"term\" class=\"no-emphasis\">trinomial<\/span>. Remember to factor it first to find the LCD.<\/p><div data-type=\"example\" id=\"fs-id1167834053796\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834556934\"><div data-type=\"problem\"><p id=\"fs-id1167834195564\">Solve: \\(\\frac{m+11}{{m}^{2}-5m+4}=\\frac{5}{m-4}-\\frac{3}{m-1}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835310614\"><table class=\"unnumbered unstyled\" summary=\"Solve the quantity m plus 11 all divided by the quantity m squared minus 5 m plus 4 is equal to 5 divided by the quantity m minus 4 minus 3 divided by the quantity m minus 1. Factor the quadratic denominator. The result is the quantity m plus 11 all divided by the quantity m minus 4 times the quantity m minus 1 is equal to 5 divided by the quantity m minus 4 minus 3 divided by the quantity m minus 1. Notice that m is equal to 4 and m is equal to 1 would make a denominator 0. The least common denominator of all the denominators is the quantity m minus 4 times the quantity m minus 1. Clear the fractions in the equation by multiplying each side of the equation by the least common denominator. The result is the quantity m minus 4 times the quantity m minus 1 times the quantity m plus 11 all divided by the quantity m minus 4 times the quantity m minus 1 is equal to the quantity m minus 4 times the difference between 5 divided by the quantity m minus 4 and 3 divided by m minus 1. Distribute the quantity m minus 4 times the quantity m minus 1 to each term in the equation. The result is the quantity m minus 4 times the quantity m minus 1 times the quantity m plus 11 all divided by the quantity m minus 4 is equal to the quantity m minus 4 times the quantity m minus 1 times 5 divided by the quantity m minus 4 minus the quantity m minus 4 times the quantity m minus 1 times 3 divided by the quantity m minus 1. Removing the common factors, the result is m plus 11 is equal to 5 times the quantity x minus 1 minus 3 times the quantity m minus 4. Simplify the equation, the result is m plus 11 is equal to 5 m minus 5 minus 3 m plus 12. Solving the equation, the result is 4 is equal to m. Check the solution. Recall that a solution m is equal to 4would make a denominator 0, which means the solution is extraneous. There is no solution, which can be confirmed by substituting m is equal to 4 into the original equation, the quantity m plus 11 all divided by the quantity m squared minus 5 m plus 4 is equal to 5 divided by the quantity m minus 4 minus 3 divided by the quantity m minus 1. Simplifying each side of the equation, the result is the quantity 4 plus 11 all divided by the quantity 4 squared minus 5 times 4 plus 4 is equal to 5 divided by 0 minus 3 divided by the quantity 4 minus 1. 5 divided by 0 is undefined. The substitution confirmed that m is equal to 4 is extraneous, and that there is no solution.\" data-label=\"\"><tbody><tr><td><\/td><td 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_04_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Note any value of the variable that<div data-type=\"newline\"><br><\/div>would make any denominator zero.<div data-type=\"newline\"><br><\/div>Use the factored form of the quadratic<div data-type=\"newline\"><br><\/div>denominator.<\/td><td 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_04_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Find the least common denominator<div data-type=\"newline\"><br><\/div>of all denominators in the equation.<div data-type=\"newline\"><br><\/div>The LCD is \\(\\left(m-4\\right)\\left(m-1\\right).\\)<\/td><td><\/td><\/tr><tr><td data-align=\"left\">Clear the fractions by<div data-type=\"newline\"><br><\/div>multiplying both sides of the<div data-type=\"newline\"><br><\/div>equation by the LCD.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835614716\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Distribute.<\/td><td 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_04_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Remove common factors.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835288140\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832059589\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve the resulting equation.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832043823\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835361972\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Check.<div data-type=\"newline\"><br><\/div>The only algebraic solution<div data-type=\"newline\"><br><\/div>was 4, but we said that 4 would make<div data-type=\"newline\"><br><\/div>a denominator equal to zero. The algebraic solution is an<div data-type=\"newline\"><br><\/div>extraneous solution.<\/td><td><\/td><\/tr><tr><td><\/td><td data-align=\"center\">There is no solution to this equation.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167828420283\"><p id=\"fs-id1167828420312\">Solve: \\(\\frac{x+13}{{x}^{2}-7x+10}=\\frac{6}{x-5}-\\frac{4}{x-2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835341884\"><p id=\"fs-id1167835239505\">There is no solution.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834134765\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834191403\"><div data-type=\"problem\" id=\"fs-id1167834289524\"><p id=\"fs-id1167832153360\">Solve: \\(\\frac{y-6}{{y}^{2}+3y-4}=\\frac{2}{y+4}+\\frac{7}{y-1}.\\)<\/p><\/div><div data-type=\"solution\"><p>There is no solution.<\/p><\/div><\/div><\/div><p id=\"fs-id1167835329756\">The equation we solved in the previous example had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. In the next example we get two algebraic solutions. Here one or both could be extraneous solutions.<\/p><div data-type=\"example\" id=\"fs-id1167831821768\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834224774\"><div data-type=\"problem\"><p>Solve: \\(\\frac{y}{y+6}=\\frac{72}{{y}^{2}-36}+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835512341\"><table class=\"unnumbered unstyled\" summary=\"Solve y divided by the quantity y plus 6 is equal to the quantity 72 divided by the difference between y squared and 36 plus 4. Factor all of the denominators. The result is y divided by the quantity y plus 6 is equal to 72 divided by the sum of the quantity y minus 6 times y plus 6 and 4. This shows y is not equal to 6 and y is not equal to negative 6. They make a denominator 0. The least common denominator of all of the denominators is the quantity y minus 6 times the quantity y plus 6. Clear the fractions by multiplying each side by the least common denominator. The result is the quantity y minus 6 times y is equal to 72 plus the quantity y minus 6 times the quantity y plus 6 times 4. Simplifying the equation, the result is y times the quantity y minus 6 is equal to 72 plus 4 times the quantity y squared minus 36. Now the equation can be solved. Write the equation so that 0 is on one side. The result is 0 is equal to 3 y squared plus 6 y minus 72. After factoring, the equation becomes 0 is equal to 3 times the quantity y plus 6 times the quantity y minus 4. The results are y is equal to negative 6 or y is equal to 4. Remember that y is equal to negative 6 is an extraneous solution. Check y is equal to 4 by substituting it into the original equation. Is 4 divided by the sum of 4 and 6 is equal to 72 divided by the difference between 4 squared and 36 plus 4 an true equation? After simplifying, the result is four-tenths is equal to four-tenths, which is a true equation. So, the solution is y is equal to 4.\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828436496\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Factor all the denominators,<div data-type=\"newline\"><br><\/div>so we can note any value of<div data-type=\"newline\"><br><\/div>the variable that would make<div data-type=\"newline\"><br><\/div>any denominator zero.<\/td><td 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_04_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Find the least common denominator.<div data-type=\"newline\"><br><\/div>The LCD is \\(\\left(y-6\\right)\\left(y+6\\right).\\)<\/td><td><\/td><\/tr><tr><td data-align=\"left\">Clear the fractions.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228808\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831880791\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830951924\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve the resulting equation.<\/td><td 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_04_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835237890\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835236556\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834372182\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Check.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><td><\/td><\/tr><tr><td><\/td><td data-align=\"center\">The solution is \\(y=4.\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835191293\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834161557\"><div data-type=\"problem\" id=\"fs-id1167834111780\"><p>Solve: \\(\\frac{x}{x+4}=\\frac{32}{{x}^{2}-16}+5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835515151\"><p>\\(x=3\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835416535\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835311069\"><div data-type=\"problem\" id=\"fs-id1167835329368\"><p id=\"fs-id1167831086631\">Solve: \\(\\frac{y}{y+8}=\\frac{128}{{y}^{2}-64}+9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835532579\"><p id=\"fs-id1167831106924\">\\(y=7\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165927749652\">In some cases, all the algebraic solutions are extraneous.<\/p><div data-type=\"example\" id=\"fs-id1167835379244\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834377067\"><div data-type=\"problem\" id=\"fs-id1167835163926\"><p id=\"fs-id1167835513958\">Solve: \\(\\frac{x}{2x-2}-\\frac{2}{3x+3}=\\frac{5{x}^{2}-2x+9}{12{x}^{2}-12}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835622176\"><table id=\"fs-id1167835267213\" class=\"unnumbered unstyled\" summary=\"Solve the difference between x divided by the quantity 2 x minus 2 and 2 divided by the quantity 3x plus 3 is equal to the quantity 5 x squared minus 2 x plus 9 all over the quantity 12 x squared minus 12. Factor all of the denominators in the equation to identify extraneous solutions and the least common denominator. The result is the difference between x divided by the product of 2 and x minus 1 and 2 divided by the product of 3 and x plus 1 is equal to the quantity 5 x squared minus 2 x plus 9 all divided by the product of 12 x minus 1 and x plus 1. Notice that x is not equal to and x is not equal to negative 1, and that the least common denominator 12 times the quantity x minus 1 times the quantity x plus 1. Multiply each side of the equation by the least common denominator to clear the fractions. The result tis 6 times the quantity x plus 1 times x minus 4 times the quantity x minus 1 times 2 is equal to 5 x squared minus 2 x plus 9. Simplify the equation by writing the constant factors first. The result is 6 x times the quantity x plus 1 minus 4 times 2 times the quantity x minus 1 is equal to 5 x squared minus 2 x plus 9. Distribute the factors to further simplify. The result is 6 x squared plus 6x minus 8 x plus 8 is equal to 5 x squared minus 2 x plus 9. Begin solving the equation by writing it with 0 on one side. The result is x squared minus 1 is equal to 0. After factoring, the equation becomes the quantity x minus 1 times the quantity x plus 1 is equal to 0. The results are x is equal to 1 or x is equal to negative 1. Recall that x is not equal to 1 or x is not equal to negative 1. Those values are extraneous, so there is no solution.\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834376186\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">We will start by factoring all<div data-type=\"newline\"><br><\/div>denominators, to make it easier<div data-type=\"newline\"><br><\/div>to identify extraneous solutions and the LCD.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835229823\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Note any value of the variable<div data-type=\"newline\"><br><\/div>that would make any denominator zero.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835356455\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Find the least common<div data-type=\"newline\"><br><\/div>denominator.<div data-type=\"newline\"><br><\/div>The LCD is \\(12\\left(x-1\\right)\\left(x+1\\right).\\)<\/td><td><\/td><\/tr><tr><td data-align=\"left\">Clear the fractions.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835233322\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835336257\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832066186\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve the resulting equation.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835381243\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835366586\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834190380\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">Check.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\u2003\\(x=1\\) and \\(x=-1\\) are extraneous solutions.<\/td><\/tr><tr><td><\/td><td data-align=\"center\">The equation has no solution.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835214096\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835358566\"><p id=\"fs-id1167832057089\">Solve: \\(\\frac{y}{5y-10}-\\frac{5}{3y+6}=\\frac{2{y}^{2}-19y+54}{15{y}^{2}-60}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835420450\"><p id=\"fs-id1167835352167\">There is no solution.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835364530\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835479971\"><div data-type=\"problem\" id=\"fs-id1167830770198\"><p id=\"fs-id1167835338212\">Solve: \\(\\frac{z}{2z+8}-\\frac{3}{4z-8}=\\frac{3{z}^{2}-16z-16}{8{z}^{2}+2z-64}.\\)<\/p><\/div><div data-type=\"solution\"><p>There is no solution.<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167834177880\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835515393\"><div data-type=\"problem\" id=\"fs-id1167835304792\"><p id=\"fs-id1167830702633\">Solve: \\(\\frac{4}{3{x}^{2}-10x+3}+\\frac{3}{3{x}^{2}+2x-1}=\\frac{2}{{x}^{2}-2x-3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834539321\"><table id=\"fs-id1167827943907\" class=\"unnumbered unstyled\" summary=\"Solve 4 divided by the quantity 3 x squared minus 10 x plus 3 plus 3 divided by the quantity 3 x squared plus 2 x minus 1 is equal to 2 divided by the quantity x squared minus 2 x minus 3. Factor all of the denominators in the equation. The result is 4 divided by the quantity 3 x minus 1 times the quantity x minus 3 plus 3 divided by the quantity 3 x minus 1 times the quantity x plus 1 is equal to 2 divided by the quantity x minus 3 times the quantity x plus 1. Notice that x is not equal to negative 1, x is not equal to one-third, and x is not equal to 3. Those values make a denominator 0. The least common denominator is the quantity 3 x minus 1 times the quantity x plus 1 times the quantity x minus 3. Clear the fractions by multiplying each side of the equation by the least common denominator. The quantity 3 x minus 1 times the quantity x plus 1 times the quantity x minus 3 times the quantity of divided by the quantity 3 x minus 1 times the quantity x minus 3 plus 3 divided by the quantity 3 x minus 1 times the quantity x plus 1 is equal to the quantity 3 x minus 1 times the quantity x plus 1 times the quantity x minus 3 times 2 divided by the quantity x minus 3 times the quantity x plus 1. Simplify by removing common factors. The result is 4 times the quantity x plus 1 plus 3 times the quantity x minus 3 is equal to 2 times the quantity 3 x minus 1. Simplify the equation by distributing. The result is 4 x plus 4 plus 3 x minus 9 is equal to 6 x minus 2. further simplifying, the result is 7 x minus 5 is equal to 6 x minus 2. Solve the equation. The result is x is equal to 3. The only algebraic solution was but we said that would make a denominator equal to zero. The algebraic solution is an extraneous solution. There is no solution to this equation.\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835370982\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Factor all the denominators, so we can note any value of the variable that would make any denominator<div data-type=\"newline\"><br><\/div>zero.<\/td><td 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_04_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div>\\(x\\ne \\text{\u2212}1,x\\ne \\frac{1}{3},x\\ne 3\\)<\/td><\/tr><tr><td data-align=\"left\">Find the least common denominator. The LCD is \\(\\left(3x-1\\right)\\left(x+1\\right)\\left(x-3\\right).\\)<\/td><td><\/td><\/tr><tr><td data-align=\"left\">Clear the fractions.<\/td><td><\/td><\/tr><tr><td colspan=\"2\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167826781751\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834189271\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Distribute.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830702476\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834382438\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td 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_04_007g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">The only algebraic solution was \\(x=3,\\) but we said that \\(x=3\\) would make a denominator equal to zero. The algebraic solution is an extraneous solution.<\/td><td><\/td><\/tr><tr><td><\/td><td data-align=\"center\">There is no solution to this equation.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835166517\"><div data-type=\"problem\" id=\"fs-id1167826779764\"><p id=\"fs-id1167835244637\">Solve: \\(\\frac{15}{{x}^{2}+x-6}-\\frac{3}{x-2}=\\frac{2}{x+3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830921037\"><p id=\"fs-id1167830924303\">There is no solution.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835622929\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835615834\"><p id=\"fs-id1167834431535\">Solve: \\(\\frac{5}{{x}^{2}+2x-3}-\\frac{3}{{x}^{2}+x-2}=\\frac{1}{{x}^{2}+5x+6}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835241523\"><p>There is no solution.<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167830866178\"><h3 data-type=\"title\">Use Rational Functions<\/h3><p id=\"fs-id1167832057008\">Working with functions that are defined by rational expressions often lead to rational equations. Again, we use the same techniques to solve them.<\/p><div data-type=\"example\" id=\"fs-id1167835187467\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835515194\"><div data-type=\"problem\" id=\"fs-id1167835596552\"><p id=\"fs-id1167835237617\">For rational function, \\(f\\left(x\\right)=\\frac{2x-6}{{x}^{2}-8x+15},\\) <span class=\"token\">\u24d0<\/span> find the domain of the function, <span class=\"token\">\u24d1<\/span> solve \\(f\\left(x\\right)=1,\\) and <span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835358480\"><p id=\"fs-id1167835400376\"><span class=\"token\">\u24d0<\/span> The domain of a rational function is all real numbers except those that make the rational expression undefined. So to find them, we will set the denominator equal to zero and solve.<\/p><p id=\"fs-id1167831919514\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill {x}^{2}-8x+15=0\\\\ \\text{Factor the trinomial.}\\hfill &amp; &amp; &amp; \\hfill \\left(x-3\\right)\\left(x-5\\right)=0\\\\ \\text{Use the Zero Product Property.}\\hfill &amp; &amp; &amp; \\hfill x-3=0\\phantom{\\rule{1em}{0ex}}x-5=0\\\\ \\text{Solve.}\\hfill &amp; &amp; &amp; \\hfill x=3\\phantom{\\rule{1em}{0ex}}x=5\\\\ &amp; &amp; &amp; \\hfill \\text{The domain is all real numbers except}\\phantom{\\rule{0.2em}{0ex}}x\\ne 3,x\\ne 5\\text{.}\\end{array}\\).<\/p><p id=\"fs-id1167835363306\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167834539060\" class=\"unnumbered unstyled\" summary=\"The function f is equal to 1. Substitute the quantity 2 x minus 6 divided by the quantity x squared minus 8 x plus 15 for f. Factoring the denominator, the result is the quantity 2 x minus 6 all divided by the x squared minus 8 x plus 15 is equal to 1. Factoring the denominator, the result is the quantity 2 x minus 6 divided by the quantity x minus 3 times the quantity x minus 5 is equal to 1. Multiply each side of the equation by the least common denominator, the quantity x minus 3 times the quantity x minus 5. Simplifying, the result is 2 x minus 6 is equal to x squared minus 8 x plus 15. Solve the equation so that terms with variables are one side. The result is 0 is equal to x squared minus 10 x plus 21. Factor the right side of the equation. The result is 0 is equal to the quantity x minus 7 times the quantity x minus 3. Using the Zero Product Property, the result is x minus 7 is equal to 0 or x minus 3 is equal to 0. Solve each equation. The results is x is equal to 7 or x is equal to 3.\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830700608\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Substitute in the rational expression.<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834430970\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Factor the denominator.<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835589718\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Multiply both sides by the LCD,<div data-type=\"newline\"><br><\/div>\\(\\left(x-3\\right)\\left(x-5\\right).\\)<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835345154\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835306830\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835281480\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Factor.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228009\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Use the Zero Product Property.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830959814\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826819802\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167831836352\"><span class=\"token\">\u24d2<\/span> The value of the function is 1 when \\(x=7,x=3.\\) So the points on the graph of this function when \\(f\\left(x\\right)=1,\\) will be \\(\\left(7,1\\right),\\left(3,1\\right).\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831923032\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834234019\"><div data-type=\"problem\" id=\"fs-id1167834234021\"><p id=\"fs-id1167831923565\">For rational function, \\(f\\left(x\\right)=\\frac{8-x}{{x}^{2}-7x+12},\\) <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> solve \\(f\\left(x\\right)=3\\) <span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835364523\"><p id=\"fs-id1167835280319\"><span class=\"token\">\u24d0<\/span> The domain is all real numbers except \\(x\\ne 3\\) and \\(x\\ne 4.\\) <span class=\"token\">\u24d1<\/span> \\(x=2,x=\\frac{14}{3}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> \\(\\left(2,3\\right),\\left(\\frac{14}{3},3\\right)\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835420479\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834060719\"><div data-type=\"problem\" id=\"fs-id1167835361379\"><p id=\"fs-id1167835361381\">For rational function, \\(f\\left(x\\right)=\\frac{x-1}{{x}^{2}-6x+5},\\) <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> solve \\(f\\left(x\\right)=4\\) <span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835215922\"><p id=\"fs-id1167834120573\"><span class=\"token\">\u24d0<\/span> The domain is all real numbers except \\(x\\ne 1\\) and \\(x\\ne 5.\\) <span class=\"token\">\u24d1<\/span> \\(x=\\frac{21}{4}\\) <span class=\"token\">\u24d2<\/span> \\(\\left(\\frac{21}{4},4\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835348471\"><h3 data-type=\"title\">Solve a Rational Equation for a Specific Variable<\/h3><p id=\"fs-id1167826997467\">When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.<\/p><p id=\"fs-id1167835232465\">When we developed the point-slope formula from our slope formula, we cleared the fractions by multiplying by the LCD.<\/p><p id=\"fs-id1167835303521\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill m&amp; =\\hfill &amp; \\frac{y-{y}_{1}}{x-{x}_{1}}\\hfill \\\\ \\text{Multiply both sides of the equation by}\\phantom{\\rule{0.2em}{0ex}}x-{x}_{1}.\\hfill &amp; &amp; &amp; \\hfill m\\left(x-{x}_{1}\\right)&amp; =\\hfill &amp; \\left(\\frac{y-{y}_{1}}{x-{x}_{1}}\\right)\\left(x-{x}_{1}\\right)\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill m\\left(x-{x}_{1}\\right)&amp; =\\hfill &amp; y-{y}_{1}\\hfill \\\\ \\text{Rewrite the equation with the}\\phantom{\\rule{0.2em}{0ex}}y\\phantom{\\rule{0.2em}{0ex}}\\text{terms on the left.}\\hfill &amp; &amp; &amp; \\hfill y-{y}_{1}&amp; =\\hfill &amp; m\\left(x-{x}_{1}\\right)\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167826801694\">In the next example, we will use the same technique with the formula for slope that we used to get the point-slope form of an equation of a line through the point \\(\\left(2,3\\right).\\) We will add one more step to solve for <em data-effect=\"italics\">y<\/em>.<\/p><div data-type=\"example\" id=\"fs-id1167834194630\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167835379570\">Solve:\\(m=\\frac{y-2}{x-3}\\) for \\(y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831086839\"><table class=\"unnumbered unstyled\" summary=\"Solve the quantity y minus 2 divided by the quantity x minus 3 for y. The solution cannot be x is equal to 3 because it will make a denominator 0. Clear the fractions on both sides of the equation by multiplying each one by the lowest common denominator, x minus 3. The result is the quantity x minus 3 times m is equal to the quantity x minus 3 times the quantity y minus 2 all divided by the quantity x minus 3. Simplifying on each side, the result is x m minus 3 m is equal to y minus 2. Isolating the term with y, the result is x m minus 3 m plus 2 is equal to y.\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835307680\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Note any value of the variable that would<div data-type=\"newline\"><br><\/div>make any denominator zero.<\/td><td 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_04_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Clear the fractions by multiplying both sides of<div data-type=\"newline\"><br><\/div>the equation by the LCD, \\(x-3.\\)<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345481\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828421394\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Isolate the term with <em data-effect=\"italics\">y<\/em>.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834535951\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834280129\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834280132\"><div data-type=\"problem\" id=\"fs-id1167835357901\"><p id=\"fs-id1167835357904\">Solve: \\(m=\\frac{y-5}{x-4}\\)for \\(y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834134460\"><p>\\(y=mx-4m+5\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835358529\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834132867\"><div data-type=\"problem\" id=\"fs-id1167834132869\"><p id=\"fs-id1167828421267\">Solve: \\(m=\\frac{y-1}{x+5}\\) for \\(y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835376441\"><p id=\"fs-id1167835376443\">\\(y=mx+5m+1\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167831116847\">Remember to multiply both sides by the LCD in the next example.<\/p><div data-type=\"example\" id=\"fs-id1167831116850\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834433514\"><div data-type=\"problem\" id=\"fs-id1167834433516\"><p id=\"fs-id1167835368438\">Solve: \\(\\frac{1}{c}+\\frac{1}{m}=1\\) for <em data-effect=\"italics\">c<\/em>.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835582748\"><table id=\"fs-id1167835582750\" class=\"unnumbered unstyled\" summary=\"Solve the quantity 1 divided by c plus the quantity 1 divided by m is equal to 1 for c. Notice that the values that would make any denominator 0 are c is equal to 0 and m is equal to 0. The least common denominator of the denominators of the fractions is c m. Clear the fractions in the equation by multiplying each side by c m. Distribute c m to each term. The result is c m times the quantity 1 divided by c plus c m times the quantity 1 divided by m is equal to c m times 1. When simplified, the equation becomes m plus c is equal to c m. Collect the terns with c on the right side of the equation. The result is m is equal to c m minus c. When the right side is factored, the result is m is equal to c times the quantity m minus 1. Divide each side of the equation by the quantity m minus 1 to isolate c. When the common factors are removed, the result is m divided by the quantity m minus 1 is equal to c. Notice that m is not equal to 1.\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835319356\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Note any value of the variable that would make<div data-type=\"newline\"><br><\/div>any denominator zero.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834306898\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Clear the fractions by multiplying both sides of<div data-type=\"newline\"><br><\/div>the equations by the LCD, <em data-effect=\"italics\">cm<\/em>.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834431316\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Distribute.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826977623\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831922161\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Collect the terms with <em data-effect=\"italics\">c<\/em> to the right.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834505759\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Factor the expression on the right.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835280666\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">To isolate <em data-effect=\"italics\">c<\/em>, divide both sides by \\(m-1.\\)<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834376913\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify by removing common factors.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835257626\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">Notice that even though we excluded \\(c=0,m=0\\) from the original equation, we must also now state that \\(m\\ne 1.\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834196626\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835328642\"><div data-type=\"problem\" id=\"fs-id1167835328644\"><p id=\"fs-id1167835240600\">Solve: \\(\\frac{1}{a}+\\frac{1}{b}=c\\) for <em data-effect=\"italics\">a<\/em>.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835300418\"><p id=\"fs-id1167831871932\">\\(a=\\frac{b}{cb-1}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835342182\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835342186\"><div data-type=\"problem\" id=\"fs-id1167835359612\"><p id=\"fs-id1167835359614\">Solve: \\(\\frac{2}{x}+\\frac{1}{3}=\\frac{1}{y}\\) for <em data-effect=\"italics\">y<\/em>.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835310191\"><p id=\"fs-id1167835310193\">\\(y=\\frac{3x}{x+6}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826780819\" class=\"media-2\"><p id=\"fs-id1167835374004\">Access this online resource for additional instruction and practice with equations with rational expressions.<\/p><ul id=\"fs-id1167835377566\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37EqRatExp\">Equations with Rational Expressions<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167826783875\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167834190024\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">How to solve equations with rational expressions.<\/strong><ol id=\"fs-id1167835322041\" type=\"1\" class=\"stepwise\"><li>Note any value of the variable that would make any denominator zero.<\/li><li>Find the least common denominator of all denominators in the equation.<\/li><li>Clear the fractions by multiplying both sides of the equation by the LCD.<\/li><li>Solve the resulting equation.<\/li><li>Check: <ul id=\"fs-id1167831890544\" data-bullet-style=\"bullet\"><li>If any values found in Step 1 are algebraic solutions, discard them.<\/li><li>Check any remaining solutions in the original equation.<\/li><\/ul><\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831949120\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834429105\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167830897851\"><strong data-effect=\"bold\">Solve Rational Equations<\/strong><\/p><p id=\"fs-id1167835596642\">In the following exercises, solve each rational equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167835596645\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835309564\"><p id=\"fs-id1167835309566\">\\(\\frac{1}{a}+\\frac{2}{5}=\\frac{1}{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835309673\"><p id=\"fs-id1167835381730\">\\(a=10\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826781539\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826781541\"><p id=\"fs-id1167826781543\">\\(\\frac{6}{3}-\\frac{2}{d}=\\frac{4}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830704645\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830704647\"><p id=\"fs-id1167830704649\">\\(\\frac{4}{5}+\\frac{1}{4}=\\frac{2}{v}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835527876\"><p id=\"fs-id1167835423110\">\\(v=\\frac{40}{21}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835310329\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835310331\"><p id=\"fs-id1167834279098\">\\(\\frac{3}{8}+\\frac{2}{y}=\\frac{1}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835319658\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831106962\"><p id=\"fs-id1167831106964\">\\(1-\\frac{2}{m}=\\frac{8}{{m}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835358672\"><p>\\(m=-2,m=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832060052\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832060054\"><p id=\"fs-id1167832060056\">\\(1+\\frac{4}{n}=\\frac{21}{{n}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835365115\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835365117\"><p id=\"fs-id1167832128656\">\\(1+\\frac{9}{p}=\\frac{-20}{{p}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830961108\"><p id=\"fs-id1167830961110\">\\(p=-5,p=-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834423050\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835368406\"><p id=\"fs-id1167835368408\">\\(1-\\frac{7}{q}=\\frac{-6}{{q}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835304732\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835304734\"><p id=\"fs-id1167835304736\">\\(\\frac{5}{3v-2}=\\frac{7}{4v}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834505336\"><p id=\"fs-id1167834505338\">\\(v=14\\)<\/p><\/div><\/div><div data-type=\"exercise\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834063671\"><p id=\"fs-id1167834063673\">\\(\\frac{8}{2w+1}=\\frac{3}{w}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831040520\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831040523\"><p id=\"fs-id1167831040525\">\\(\\frac{3}{x+4}+\\frac{7}{x-4}=\\frac{8}{{x}^{2}-16}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835348754\"><p id=\"fs-id1167835334884\">\\(x=-\\frac{4}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835229217\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835229219\"><p id=\"fs-id1167835229221\">\\(\\frac{5}{y-9}+\\frac{1}{y+9}=\\frac{18}{{y}^{2}-81}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834190044\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834190046\"><p id=\"fs-id1167834190048\">\\(\\frac{8}{z-10}-\\frac{7}{z+10}=\\frac{5}{{z}^{2}-100}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835479340\"><p id=\"fs-id1167831835428\">\\(z=-145\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835479358\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835479360\"><p id=\"fs-id1167835479362\">\\(\\frac{9}{a+11}-\\frac{6}{a-11}=\\frac{6}{{a}^{2}-121}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834111880\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834111883\"><p id=\"fs-id1167834111885\">\\(\\frac{-10}{q-2}-\\frac{7}{q+4}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826994126\"><p id=\"fs-id1167826994129\">\\(q=-18,q=-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834079305\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834146932\"><p id=\"fs-id1167834146934\">\\(\\frac{2}{s+7}-\\frac{3}{s-3}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831881752\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831881755\"><p id=\"fs-id1167830959746\">\\(\\frac{v-10}{{v}^{2}-5v+4}=\\frac{3}{v-1}-\\frac{6}{v-4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835365624\"><p id=\"fs-id1167831881738\">\\(\\text{no&nbsp;solution}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827966802\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167827966804\"><p id=\"fs-id1167835237656\">\\(\\frac{w+8}{{w}^{2}-11w+28}=\\frac{5}{w-7}+\\frac{2}{w-4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830704522\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834552530\"><p id=\"fs-id1167834552532\">\\(\\frac{x-10}{{x}^{2}+8x+12}=\\frac{3}{x+2}+\\frac{4}{x+6}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835304338\"><p id=\"fs-id1167835304340\">\\(\\text{no&nbsp;solution}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831880161\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835371420\"><p id=\"fs-id1167835371422\">\\(\\frac{y-5}{{y}^{2}-4y-5}=\\frac{1}{y+1}+\\frac{1}{y-5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835234777\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835234779\"><p id=\"fs-id1167835234781\">\\(\\frac{b+3}{3b}+\\frac{b}{24}=\\frac{1}{b}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835380533\"><p id=\"fs-id1167835380536\">\\(b=-8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830837155\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830837157\"><p id=\"fs-id1167835340875\">\\(\\frac{c+3}{12c}+\\frac{c}{36}=\\frac{1}{4c}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835230880\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835230882\"><p id=\"fs-id1167834064438\">\\(\\frac{d}{d+3}=\\frac{18}{{d}^{2}-9}+4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830704484\"><p id=\"fs-id1167830704486\">\\(d=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834555265\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834555267\"><p id=\"fs-id1167834124400\">\\(\\frac{m}{m+5}=\\frac{50}{{m}^{2}-25}+6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835339503\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835339505\"><p id=\"fs-id1167835339507\">\\(\\frac{n}{n+2}-3=\\frac{8}{{n}^{2}-4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834526292\"><p id=\"fs-id1167834526295\">\\(m=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835380884\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167828447056\"><p id=\"fs-id1167828447059\">\\(\\frac{p}{p+7}-8=\\frac{98}{{p}^{2}-49}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831887701\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834448880\"><p id=\"fs-id1167834448882\">\\(\\frac{q}{3q-9}-\\frac{3}{4q+12}=\\frac{7{q}^{2}+6q+63}{24{q}^{2}-216}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835218128\"><p id=\"fs-id1167835218130\">\\(\\text{no&nbsp;solution}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835348954\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831887557\"><p id=\"fs-id1167831887559\">\\(\\frac{r}{3r-15}-\\frac{1}{4r+20}=\\frac{3{r}^{2}+17r+40}{12{r}^{2}-300}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835254283\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167835512058\">\\(\\frac{s}{2s+6}-\\frac{2}{5s+5}=\\frac{5{s}^{2}-3s-7}{10{s}^{2}+40s+30}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834430234\"><p id=\"fs-id1167834430236\">\\(s=\\frac{5}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835511057\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831117195\"><p id=\"fs-id1167831117197\">\\(\\frac{t}{6t-12}-\\frac{5}{2t+10}=\\frac{{t}^{2}-23t+70}{12{t}^{2}+36t-120}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835419937\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835419939\"><p id=\"fs-id1167834063967\">\\(\\frac{2}{{x}^{2}+2x-8}-\\frac{1}{{x}^{2}+9x+20}=\\frac{4}{{x}^{2}+3x-10}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835364061\"><p id=\"fs-id1167835364063\">\\(x=-\\frac{4}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835514064\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835514066\"><p id=\"fs-id1167835338292\">\\(\\frac{5}{{x}^{2}+4x+3}+\\frac{2}{{x}^{2}+x-6}=\\frac{3}{{x}^{2}-x-2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826978010\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826978012\"><p id=\"fs-id1167830925343\">\\(\\frac{3}{{x}^{2}-5x-6}+\\frac{3}{{x}^{2}-7x+6}=\\frac{6}{{x}^{2}-1}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834537641\"><p>no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826978331\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826978333\"><p>\\(\\frac{2}{{x}^{2}+2x-3}+\\frac{3}{{x}^{2}+4x+3}=\\frac{6}{{x}^{2}-1}\\)<\/p><\/div><\/div><p id=\"fs-id1167826778769\"><strong data-effect=\"bold\">Solve Rational Equations that Involve Functions<\/strong><\/p><div data-type=\"exercise\" id=\"fs-id1167835356003\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835356005\"><p id=\"fs-id1167835356007\">For rational function, \\(f\\left(x\\right)=\\frac{x-2}{{x}^{2}+6x+8},\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> find the domain of the function<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> solve \\(f\\left(x\\right)=5\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/div><div data-type=\"solution\" id=\"fs-id1167835349026\"><p id=\"fs-id1167835301328\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> The domain is all real numbers except \\(x\\ne \\text{\u2212}2\\) and \\(x\\ne \\text{\u2212}4.\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(x=-3,x=-\\frac{14}{5}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> \\(\\left(-3,5\\right),\\left(-\\frac{14}{5},5\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835319215\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835319217\"><p id=\"fs-id1167831239531\">For rational function, \\(f\\left(x\\right)=\\frac{x+1}{{x}^{2}-2x-3},\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> find the domain of the function<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> solve \\(f\\left(x\\right)=1\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835287710\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835287712\"><p id=\"fs-id1167835287714\">For rational function, \\(f\\left(x\\right)=\\frac{2-x}{{x}^{2}-7x+10},\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> find the domain of the function<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> solve \\(f\\left(x\\right)=2\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/div><div data-type=\"solution\" id=\"fs-id1167834430848\"><p id=\"fs-id1167834535542\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> The domain is all real numbers except \\(x\\ne 2\\) and \\(x\\ne 5.\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(x=\\frac{9}{2},\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> \\(\\left(\\frac{9}{2},2\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834066254\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834066256\"><p id=\"fs-id1167835354887\">For rational function, \\(f\\left(x\\right)=\\frac{5-x}{{x}^{2}+5x+6},\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> find the domain of the function<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> solve \\(f\\left(x\\right)=3\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> the points on the graph at this function value.<\/div><\/div><p id=\"fs-id1167835336553\"><strong data-effect=\"bold\">Solve a Rational Equation for a Specific Variable<\/strong><\/p><p id=\"fs-id1167830693613\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167830693617\"><div data-type=\"problem\" id=\"fs-id1167832053198\"><p id=\"fs-id1167832053200\">\\(\\frac{C}{r}=2\\pi \\) for \\(r.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834124437\"><p>\\(r=\\frac{C}{2\\pi }\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835173771\"><div data-type=\"problem\" id=\"fs-id1167834244190\"><p id=\"fs-id1167834244192\">\\(\\frac{I}{r}=P\\) for \\(r.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835237516\"><div data-type=\"problem\" id=\"fs-id1167835375434\"><p id=\"fs-id1167835375436\">\\(\\frac{v+3}{w-1}=\\frac{1}{2}\\) for \\(w.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831913245\"><p id=\"fs-id1167831913247\">\\(w=2v+7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831884542\"><div data-type=\"problem\" id=\"fs-id1167831884544\"><p id=\"fs-id1167834539247\">\\(\\frac{x+5}{2-y}=\\frac{4}{3}\\) for \\(y.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834130086\"><div data-type=\"problem\" id=\"fs-id1167835365713\"><p id=\"fs-id1167835365715\">\\(a=\\frac{b+3}{c-2}\\) for \\(c.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835319924\"><p id=\"fs-id1167835368012\">\\(c=\\frac{b+3+2a}{a}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834423786\"><div data-type=\"problem\" id=\"fs-id1167834423788\"><p id=\"fs-id1167834099281\">\\(m=\\frac{n}{2-n}\\) for \\(n.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826782306\"><div data-type=\"problem\" id=\"fs-id1167826782308\"><p id=\"fs-id1167830704311\">\\(\\frac{1}{p}+\\frac{2}{q}=4\\) for \\(p.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835331700\"><p id=\"fs-id1167831106942\">\\(p=\\frac{q}{4q-2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831823292\"><div data-type=\"problem\" id=\"fs-id1167835254155\"><p id=\"fs-id1167835254157\">\\(\\frac{3}{s}+\\frac{1}{t}=2\\) for \\(s.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835380032\"><div data-type=\"problem\" id=\"fs-id1167832041363\"><p id=\"fs-id1167832041365\">\\(\\frac{2}{v}+\\frac{1}{5}=\\frac{3}{w}\\) for \\(w.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834473638\"><p id=\"fs-id1167834473640\">\\(w=\\frac{15v}{10+v}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835347802\"><div data-type=\"problem\" id=\"fs-id1167830838410\"><p id=\"fs-id1167830838412\">\\(\\frac{6}{x}+\\frac{2}{3}=\\frac{1}{y}\\) for \\(y.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834539307\"><div data-type=\"problem\"><p id=\"fs-id1167834539311\">\\(\\frac{m+3}{n-2}=\\frac{4}{5}\\) for \\(n.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826977638\"><p id=\"fs-id1167826977640\">\\(n=\\frac{5m+23}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831912314\"><div data-type=\"problem\" id=\"fs-id1167831912316\"><p id=\"fs-id1167834191152\">\\(r=\\frac{s}{3-t}\\) for \\(t.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831846917\"><div data-type=\"problem\" id=\"fs-id1167831846919\"><p id=\"fs-id1167835519767\">\\(\\frac{E}{c}={m}^{2}\\) for \\(c.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832152860\"><p id=\"fs-id1167832152862\">\\(c=\\frac{E}{{m}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834184720\"><div data-type=\"problem\" id=\"fs-id1167834184722\"><p id=\"fs-id1167834146999\">\\(\\frac{R}{T}=W\\) for \\(T.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835321864\"><div data-type=\"problem\" id=\"fs-id1167835321866\"><p id=\"fs-id1167834587420\">\\(\\frac{3}{x}-\\frac{5}{y}=\\frac{1}{4}\\) for \\(y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835319640\"><p id=\"fs-id1167835319643\">\\(y=\\frac{20x}{12-x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832057452\"><div data-type=\"problem\" id=\"fs-id1167832057454\"><p id=\"fs-id1167835258331\">\\(c=\\frac{2}{a}\\phantom{\\rule{0.2em}{0ex}}+\\frac{b}{5}\\) for \\(a.\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834157025\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167831921246\"><div data-type=\"problem\" id=\"fs-id1167831921248\"><p id=\"fs-id1167832060427\">Your class mate is having trouble in this section. Write down the steps you would use to explain how to solve a rational equation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831883616\"><p id=\"fs-id1167831883618\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834066032\"><div data-type=\"problem\" id=\"fs-id1167834066034\"><p id=\"fs-id1167834066036\">Alek thinks the equation \\(\\frac{y}{y+6}=\\frac{72}{{y}^{2}-36}+4\\) has two solutions, \\(y=-6\\) and \\(y=4.\\) Explain why Alek is wrong.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832053662\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167835330833\"><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-id1167834061744\" data-alt=\"This table has four columns and four 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 equations. In row 3, the I can was solve rational equations involving functions. In row 4, the I can was solve rational equations for a specific variable.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and four 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 equations. In row 3, the I can was solve rational equations involving functions. In row 4, the I can was solve rational equations for a specific variable.\"><\/span><p id=\"fs-id1167834094796\"><span class=\"token\">\u24d1<\/span> On a scale of \\(1-10,\\) how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167832068260\"><dt>extraneous solution to a rational equation<\/dt><dd id=\"fs-id1167832068263\">An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.<\/dd><\/dl><dl id=\"fs-id1167835342941\"><dt>rational equation<\/dt><dd id=\"fs-id1167835342944\">A rational equation is an equation 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 equations<\/li>\n<li>Use rational functions<\/li>\n<li>Solve a rational equation for a specific variable<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" class=\"be-prepared\">\n<p id=\"fs-id1167831928853\">Before you get started, take this readiness quiz.<\/p>\n<ol type=\"1\">\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9af382a4950145a19a4bc79a5bfa669f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"90\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167833239741\" 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-88a559b7a0b7dac74f9908cc7a4e93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#110;&#45;&#51;&#54;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"136\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836625705\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79b727dd3be137db9ceda850e7b18c09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#43;&#50;&#121;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -4px;\" \/> 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;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/b03538a1-8a7b-4158-a68b-e0e8a24c9fd4#fs-id1167835229496\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167832052437\">After defining the terms \u2018expression\u2019 and \u2018equation\u2019 earlier, we have used them throughout this book. We have <em data-effect=\"italics\">simplified<\/em> many kinds of <em data-effect=\"italics\">expressions<\/em> and <em data-effect=\"italics\">solved<\/em> many kinds of <em data-effect=\"italics\">equations<\/em>. We have simplified many rational expressions so far in this chapter. Now we will <em data-effect=\"italics\">solve<\/em> a <span data-type=\"term\">rational equation<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835640377\">\n<div data-type=\"title\">Rational Equation<\/div>\n<p>A <strong data-effect=\"bold\">rational equation<\/strong> is an equation that contains a rational expression.<\/p>\n<\/div>\n<p id=\"fs-id1167835369643\">You must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167827958702\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c165157fe36afb7587a96773f8fde244_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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#97;&#116;&#105;&#111;&#110;&#97;&#108;&#32;&#69;&#120;&#112;&#114;&#101;&#115;&#115;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#97;&#116;&#105;&#111;&#110;&#97;&#108;&#32;&#69;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#54;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#43;&#52;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#54;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#121;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#43;&#52;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#110;&#45;&#49;&#50;&#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;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"91\" width=\"401\" style=\"vertical-align: -43px;\" \/><\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832053699\">\n<h3 data-type=\"title\">Solve Rational Equations<\/h3>\n<p id=\"fs-id1167835563859\">We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to \u201cclear\u201d the fractions.<\/p>\n<p>We will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then, we will have an equation that does not contain rational expressions and thus is much easier for us to solve. But because the original equation may have a variable in a denominator, we must be careful that we don\u2019t end up with a solution that would make a denominator equal to zero.<\/p>\n<p id=\"fs-id1167835360295\">So before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.<\/p>\n<p id=\"fs-id1167831912071\">An algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an <span data-type=\"term\">extraneous solution to a rational equation<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835320224\">\n<div data-type=\"title\">Extraneous Solution to a Rational Equation<\/div>\n<p id=\"fs-id1167831239457\">An <strong data-effect=\"bold\">extraneous solution to a rational equation<\/strong> is an algebraic solution that would cause any of the expressions in the original equation to be undefined.<\/p>\n<\/div>\n<p id=\"fs-id1167835319662\">We note any possible extraneous solutions, <em data-effect=\"italics\">c<\/em>, by writing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9ea43d66987857075e4093e0f56e7a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/> next to the equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832212123\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Rational Equation<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835373684\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834516061\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-463bcdbb9618c1dbd746c6d5f2110173_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;&#49;&#125;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"80\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834222446\"><span data-type=\"media\" id=\"fs-id1167831882177\" data-alt=\"Step 1 is to find any value of the variable that makes the denominator of the zero. Remember that if x is equal to 0, then 1 divided by x is undefined. So the equation becomes the sum of 1 divided by x and one-third is equal to five-sixths, where x is not equal to 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find any value of the variable that makes the denominator of the zero. Remember that if x is equal to 0, then 1 divided by x is undefined. So the equation becomes the sum of 1 divided by x and one-third is equal to five-sixths, where x is not equal to 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835373808\" data-alt=\"Step 2 is to find the least common denominator of all the fractions in the problem, 1 divided by x, one-third, and five-sixths. The least common denominator is 6 x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to find the least common denominator of all the fractions in the problem, 1 divided by x, one-third, and five-sixths. The least common denominator is 6 x.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 3 is to clear the fractions in the equation by multiplying each side by the least common denominator. The result is 6 x times the sum of 1 divided by x and one-third is equal to 6 x times five-sixths. Simplify using the distributive property. The result is 6 x times the quantity1 divided by x plus 6 x times one-third is equal to 6 x times five-sixths, which simplifies to 6 plus 2 x is equal to 5 x. This simplifies to 6 is equal to 3 x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to clear the fractions in the equation by multiplying each side by the least common denominator. The result is 6 x times the sum of 1 divided by x and one-third is equal to 6 x times five-sixths. Simplify using the distributive property. The result is 6 x times the quantity1 divided by x plus 6 x times one-third is equal to 6 x times five-sixths, which simplifies to 6 plus 2 x is equal to 5 x. This simplifies to 6 is equal to 3 x.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167834060144\" data-alt=\"Step 4 is to solve the equation that results. The result is 2 is equal to x\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to solve the equation that results. The result is 2 is equal to x\" \/><\/span><span data-type=\"media\" id=\"fs-id1167834133754\" data-alt=\"Step 5 is to check the solution. Remember that any solutions that makes the original expression undefined must be discarded. The solution is not 0. Substitute x is equal to 2 into the original equation, 1 divided by x plus one-third is equal to five-sixths. Is one-half plus one-third is equal to five-sixths a true equation? Is three-sixths plus two-sixth is equal to five-sixths a true equation? Three-sixths plus two-sixth is equal to five-sixths. Five-sixth is equal to five-sixth is a true equation. So, the solution is x is equal to 2\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to check the solution. Remember that any solutions that makes the original expression undefined must be discarded. The solution is not 0. Substitute x is equal to 2 into the original equation, 1 divided by x plus one-third is equal to five-sixths. Is one-half plus one-third is equal to five-sixths a true equation? Is three-sixths plus two-sixth is equal to five-sixths a true equation? Three-sixths plus two-sixth is equal to five-sixths. Five-sixth is equal to five-sixth is a true equation. So, the solution is x is equal to 2\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835419818\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834280170\">\n<div data-type=\"problem\" id=\"fs-id1167835416569\">\n<p id=\"fs-id1167835367835\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7156fba739120d9441d73c94c5b92214_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"80\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831086720\">\n<p id=\"fs-id1167826813938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54e30e49790e871b370040d53a5e2006_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"63\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835283352\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830704153\">\n<div data-type=\"problem\" id=\"fs-id1167834387408\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acb556b2262e82dd7099803d9a6718a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"80\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835369741\">\n<p id=\"fs-id1167835595617\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96a604f73aa16e1a21548273f782fc37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"50\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The steps of this method are shown.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835378137\" class=\"howto\">\n<div data-type=\"title\">Solve equations with rational expressions.<\/div>\n<ol type=\"1\" class=\"stepwise\">\n<li>Note any value of the variable that would make any denominator zero.<\/li>\n<li>Find the least common denominator of <em data-effect=\"italics\">all<\/em> denominators in the equation.<\/li>\n<li>Clear the fractions by multiplying both sides of the equation by the LCD.<\/li>\n<li>Solve the resulting equation.<\/li>\n<li>Check:\n<ul data-bullet-style=\"bullet\">\n<li>If any values found in Step 1 are algebraic solutions, discard them.<\/li>\n<li>Check any remaining solutions in the original equation.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167835528067\">We always start by noting the values that would cause any denominators to be zero.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831228721\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Rational Equation using the Zero Product Property<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826987271\">\n<div data-type=\"problem\" id=\"fs-id1167835510388\">\n<p id=\"fs-id1167835513316\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6896094adbc58d073454df02a12319d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#121;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"100\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834536690\">\n<table class=\"unnumbered unstyled can-break\" summary=\"1 minus the quantity 5 divided by y is equal to the negative of the quantity 6 divided by y squared. Identify any value of y that would make any denominator 0. Add y is not equal to 0 to the equation. The least common denominator of the fractions in the equation, the quantity 5 divided by y and the negative of the quantity 6 divided by y squared is y squared. Clear the fractions in the equation by multiplying each side by y squared. y squared times the quantity 1 minus the quantity 5 divided by y is equal to y squared times the negative of the quantity 6 divided by y squared. Distribute y squared. The result is y squared times 1 minus y squared times the quantity 5 divided by y is equal to the negative of the quantity 6 divided by y squared. Multiply on both sides of the equation. The result is y squared minus 5 y is equal to negative 6. To solve this equation, first write it in standard form. Its standard form is y squared minus 5 y plus 6 is equal to 0. Factor the left side of the equation. The result is the quantity y minus 2 times the quantity y minus 3 is equal to zero. Using the Zero Product Property. the equation becomes y minus 2 is equal to 0 or y minus 3 is equal to 0. Solving each equation results in the solutions, y is equal to 2 or y is equal to 3. Check the solutions by substituting them into the original equation, 1 minus the quantity 5 divided by y is equal to the negative of 6 divided by y squared. Note that 0 was not an algebraic solution. Is the equation 1 minus the quantity 5 divided by 2 is equal to the quantity 6 divided by 2 squared true? When each side is simplified, the result is negative 3 divided by 2 is equal negative 3 divided by 2, which is a true equation. Is the equation 1 minus the quantity 5 divided by 3 is equal to the quantity 6 divided by 3 squared true? When each side is simplified, the result is negative 2 divided by 3 is equal to negative 2 divided by 3, which is a true equation. The solution is y is equal to 2 or y is equal to 3.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td 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_04_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Note any value of the variable that would make<\/p>\n<div data-type=\"newline\"><\/div>\n<p>any denominator zero.<\/td>\n<td 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_04_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Find the least common denominator of all denominators in<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the equation. The LCD is <em data-effect=\"italics\">y<\/em><sup>2<\/sup>.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Clear the fractions by multiplying both sides of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the equation by the LCD.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834299943\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Distribute.<\/td>\n<td 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_04_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Multiply.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834049006\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve the resulting equation. First<\/p>\n<div data-type=\"newline\"><\/div>\n<p>write the quadratic equation in standard form.<\/td>\n<td 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_04_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Factor.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832060135\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Use the Zero Product Property.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834120431\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831910249\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Check.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>We did not get 0 as an algebraic solution.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835370931\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-488a2426705328554c03c03bb86b08ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f0eb02c9be9d3f22890308095c7a3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835317276\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835417937\">\n<div data-type=\"problem\" id=\"fs-id1167835334526\">\n<p id=\"fs-id1167835364449\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21a1b668b0a2001457d08d7355ef3ac2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"87\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835350474\">\n<p id=\"fs-id1167835234636\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-036f1e4a90ab5f3e58302b12c8180e7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#44;&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831958129\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834239193\">\n<div data-type=\"problem\" id=\"fs-id1167831116689\">\n<p id=\"fs-id1167835254163\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92b736a9f1e30803020ea1badd1f6de1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#121;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"86\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835529923\">\n<p id=\"fs-id1167835353178\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d32f00bfb2b0d4ddb67467eb3591f5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#44;&#121;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, the last denominators is a difference of squares. Remember to factor it first to find the LCD.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834061864\">\n<div data-type=\"problem\" id=\"fs-id1167831846925\">\n<p id=\"fs-id1167828435100\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cce88f8d576e6b6c0cec1d53bc481528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#43;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#45;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"142\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832036216\">\n<table id=\"fs-id1167835596565\" class=\"unnumbered unstyled can-break\" summary=\"2 divided by the quantity x plus 2 plus 4 divided by the quantity x minus 2 is equal to the quantity x minus 1 divided by the quantity x squared minus 4. Notice that x is equal to negative 2 and x is equal to 2 would make denominators zero. Factor the denominator on the right side of the equation. The result is 2 divided by the quantity x plus 2 plus 4 divided by the quantity x minus 2 is equal to the quantity x minus 1 divided by the quantity x plus 2 times the quantity x minus 2, x is not equal to negative and x is not equal to 2. The least common denominator of all the denominators in the equation is the quantity x plus 2 times the quantity x minus 2. Clear the fractions by multiplying both sides of the equation by the least common denominator, the quantity x plus 2 times the quantity x minus 2. Distribute the least common denominator to each expression in the equation. The result is the quantity x plus 2 times the quantity x minus 2 times 2 divided by the quantity x plus 2 plus the quantity x plus 2 times the quantity x minus 2 times 4 divided by the quantity x minus 2 is equal to the quantity x plus 2 times the quantity x minus 2 times the quantity x minus 1 divided by the quantity x squared minus 4. Remove the common factors from the equation. The result is 2 times the quantity x minus 2 plus 4 times the quantity x plus 2 is equal to x minus 1. Distribute the constants on the left side of the equation. The result is 2 x minus 4 plus 4 x plus 8 is equal to x minus 1. Solve the equation by simplifying each side of the equation. The result is 6 x plus 4 is equal to x minus 1. Isolating the variable term, the result is 5 x is equal to negative 5. Solving for x, the solution is x is equal to negative 1. Check the solution. Remember x cannot be equal to 2 and x cannot be equal to negative 2. Substitute x is equal to negative into the original equation, 2 divided by the quantity x plus 2 plus 4 divided by the quantity x minus 2 is equal to the quantity x minus 1 divided by the quantity x squared minus 4. Is 2 divided by the quantity negative 1 plus 2 plus 4 divided by the quantity negative 1 minus 2 is equal to the quantity negative 1 minus 1 all divided by the square of negative 1 minus 4? When each side of the equation is simplified, the result is 2 divided by 3 is equal to 2 divided by 3, which is true. So, the solution is x is equal to negative 1.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835215786\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Note any value of the variable<\/p>\n<div data-type=\"newline\"><\/div>\n<p>that would make any denominator<\/p>\n<div data-type=\"newline\"><\/div>\n<p>zero.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831036916\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Find the least common<\/p>\n<div data-type=\"newline\"><\/div>\n<p>denominator of all denominators<\/p>\n<div data-type=\"newline\"><\/div>\n<p>in the equation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The LCD is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26c47c2d8bb3c28cc1747c72a5a924d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Clear the fractions by multiplying<\/p>\n<div data-type=\"newline\"><\/div>\n<p>both sides of the equation by the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>LCD.<\/td>\n<td 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_04_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Distribute.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835381726\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Remove common factors.<\/td>\n<td 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_04_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td 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_04_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Distribute.<\/td>\n<td 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_04_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve.<\/td>\n<td 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_04_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835545413\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831883495\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p>We did not get 2 or \u22122 as algebraic solutions.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167828377151\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6de4e73609a66312d9714a253f9ae3a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/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-id1167834505929\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834395459\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b996c7c3ae0e18c5b664ea4124909728_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#43;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#45;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"142\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831066244\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd606513e0b1ba31937cdc32d1e43458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835489220\">\n<div data-type=\"problem\" id=\"fs-id1167826997270\">\n<p id=\"fs-id1167832214483\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-286e94fb8d215abe872e917ea8dde0b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#121;&#43;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#121;&#45;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"140\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835351509\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-552d8ed773e160e229551b39aff39445_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835304069\">In the next example, the first denominator is a <span data-type=\"term\" class=\"no-emphasis\">trinomial<\/span>. Remember to factor it first to find the LCD.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834053796\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834556934\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834195564\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f41087306712bd959f5bcd86b170de1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#43;&#49;&#49;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#109;&#43;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#109;&#45;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#109;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"185\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835310614\">\n<table class=\"unnumbered unstyled\" summary=\"Solve the quantity m plus 11 all divided by the quantity m squared minus 5 m plus 4 is equal to 5 divided by the quantity m minus 4 minus 3 divided by the quantity m minus 1. Factor the quadratic denominator. The result is the quantity m plus 11 all divided by the quantity m minus 4 times the quantity m minus 1 is equal to 5 divided by the quantity m minus 4 minus 3 divided by the quantity m minus 1. Notice that m is equal to 4 and m is equal to 1 would make a denominator 0. The least common denominator of all the denominators is the quantity m minus 4 times the quantity m minus 1. Clear the fractions in the equation by multiplying each side of the equation by the least common denominator. The result is the quantity m minus 4 times the quantity m minus 1 times the quantity m plus 11 all divided by the quantity m minus 4 times the quantity m minus 1 is equal to the quantity m minus 4 times the difference between 5 divided by the quantity m minus 4 and 3 divided by m minus 1. Distribute the quantity m minus 4 times the quantity m minus 1 to each term in the equation. The result is the quantity m minus 4 times the quantity m minus 1 times the quantity m plus 11 all divided by the quantity m minus 4 is equal to the quantity m minus 4 times the quantity m minus 1 times 5 divided by the quantity m minus 4 minus the quantity m minus 4 times the quantity m minus 1 times 3 divided by the quantity m minus 1. Removing the common factors, the result is m plus 11 is equal to 5 times the quantity x minus 1 minus 3 times the quantity m minus 4. Simplify the equation, the result is m plus 11 is equal to 5 m minus 5 minus 3 m plus 12. Solving the equation, the result is 4 is equal to m. Check the solution. Recall that a solution m is equal to 4would make a denominator 0, which means the solution is extraneous. There is no solution, which can be confirmed by substituting m is equal to 4 into the original equation, the quantity m plus 11 all divided by the quantity m squared minus 5 m plus 4 is equal to 5 divided by the quantity m minus 4 minus 3 divided by the quantity m minus 1. Simplifying each side of the equation, the result is the quantity 4 plus 11 all divided by the quantity 4 squared minus 5 times 4 plus 4 is equal to 5 divided by 0 minus 3 divided by the quantity 4 minus 1. 5 divided by 0 is undefined. The substitution confirmed that m is equal to 4 is extraneous, and that there is no solution.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td 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_04_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Note any value of the variable that<\/p>\n<div data-type=\"newline\"><\/div>\n<p>would make any denominator zero.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Use the factored form of the quadratic<\/p>\n<div data-type=\"newline\"><\/div>\n<p>denominator.<\/td>\n<td 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_04_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Find the least common denominator<\/p>\n<div data-type=\"newline\"><\/div>\n<p>of all denominators in the equation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The LCD is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14a2901f15cbc1b2a0cc319823f22e37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Clear the fractions by<\/p>\n<div data-type=\"newline\"><\/div>\n<p>multiplying both sides of the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equation by the LCD.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835614716\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Distribute.<\/td>\n<td 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_04_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Remove common factors.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835288140\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832059589\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve the resulting equation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832043823\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835361972\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_004i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Check.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The only algebraic solution<\/p>\n<div data-type=\"newline\"><\/div>\n<p>was 4, but we said that 4 would make<\/p>\n<div data-type=\"newline\"><\/div>\n<p>a denominator equal to zero. The algebraic solution is an<\/p>\n<div data-type=\"newline\"><\/div>\n<p>extraneous solution.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\">There is no solution to this equation.<\/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\">\n<div data-type=\"problem\" id=\"fs-id1167828420283\">\n<p id=\"fs-id1167828420312\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b4e5941bab71ff7b6120418cc7e4e99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#49;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#120;&#45;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#45;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"174\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835341884\">\n<p id=\"fs-id1167835239505\">There is no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834134765\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834191403\">\n<div data-type=\"problem\" id=\"fs-id1167834289524\">\n<p id=\"fs-id1167832153360\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0bfaf7370c0e813318c0a9368845156_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#54;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#121;&#45;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#121;&#43;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#121;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"165\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p>There is no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835329756\">The equation we solved in the previous example had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. In the next example we get two algebraic solutions. Here one or both could be extraneous solutions.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831821768\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834224774\">\n<div data-type=\"problem\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b42de2935dd228faa4ff17806d6c3ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#121;&#43;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#50;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"127\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835512341\">\n<table class=\"unnumbered unstyled\" summary=\"Solve y divided by the quantity y plus 6 is equal to the quantity 72 divided by the difference between y squared and 36 plus 4. Factor all of the denominators. The result is y divided by the quantity y plus 6 is equal to 72 divided by the sum of the quantity y minus 6 times y plus 6 and 4. This shows y is not equal to 6 and y is not equal to negative 6. They make a denominator 0. The least common denominator of all of the denominators is the quantity y minus 6 times the quantity y plus 6. Clear the fractions by multiplying each side by the least common denominator. The result is the quantity y minus 6 times y is equal to 72 plus the quantity y minus 6 times the quantity y plus 6 times 4. Simplifying the equation, the result is y times the quantity y minus 6 is equal to 72 plus 4 times the quantity y squared minus 36. Now the equation can be solved. Write the equation so that 0 is on one side. The result is 0 is equal to 3 y squared plus 6 y minus 72. After factoring, the equation becomes 0 is equal to 3 times the quantity y plus 6 times the quantity y minus 4. The results are y is equal to negative 6 or y is equal to 4. Remember that y is equal to negative 6 is an extraneous solution. Check y is equal to 4 by substituting it into the original equation. Is 4 divided by the sum of 4 and 6 is equal to 72 divided by the difference between 4 squared and 36 plus 4 an true equation? After simplifying, the result is four-tenths is equal to four-tenths, which is a true equation. So, the solution is y is equal to 4.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828436496\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Factor all the denominators,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>so we can note any value of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the variable that would make<\/p>\n<div data-type=\"newline\"><\/div>\n<p>any denominator zero.<\/td>\n<td 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_04_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Find the least common denominator.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The LCD is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-644685a5e232a723fd1a041bc22a98ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Clear the fractions.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228808\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831880791\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830951924\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve the resulting equation.<\/td>\n<td 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_04_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835237890\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><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_04_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835236556\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834372182\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_005k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Check.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><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_04_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-270d2ebaae94f65bccc2c78eddebf555_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835191293\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834161557\">\n<div data-type=\"problem\" id=\"fs-id1167834111780\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bb6ef06816a3917837f26158d3d20a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#120;&#43;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#43;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"128\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835515151\">\n<p><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;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835416535\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835311069\">\n<div data-type=\"problem\" id=\"fs-id1167835329368\">\n<p id=\"fs-id1167831086631\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edfbb493dd3d21d4c43e289d0e5128e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#121;&#43;&#56;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#56;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;&#125;&#43;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"127\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835532579\">\n<p id=\"fs-id1167831106924\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5deadc78bd3ef1af0f597eca7a46a78d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165927749652\">In some cases, all the algebraic solutions are extraneous.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835379244\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834377067\">\n<div data-type=\"problem\" id=\"fs-id1167835163926\">\n<p id=\"fs-id1167835513958\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-364f7344f0788d228c910c70b52b4027_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#120;&#45;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#120;&#43;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#43;&#57;&#125;&#123;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"188\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835622176\">\n<table id=\"fs-id1167835267213\" class=\"unnumbered unstyled\" summary=\"Solve the difference between x divided by the quantity 2 x minus 2 and 2 divided by the quantity 3x plus 3 is equal to the quantity 5 x squared minus 2 x plus 9 all over the quantity 12 x squared minus 12. Factor all of the denominators in the equation to identify extraneous solutions and the least common denominator. The result is the difference between x divided by the product of 2 and x minus 1 and 2 divided by the product of 3 and x plus 1 is equal to the quantity 5 x squared minus 2 x plus 9 all divided by the product of 12 x minus 1 and x plus 1. Notice that x is not equal to and x is not equal to negative 1, and that the least common denominator 12 times the quantity x minus 1 times the quantity x plus 1. Multiply each side of the equation by the least common denominator to clear the fractions. The result tis 6 times the quantity x plus 1 times x minus 4 times the quantity x minus 1 times 2 is equal to 5 x squared minus 2 x plus 9. Simplify the equation by writing the constant factors first. The result is 6 x times the quantity x plus 1 minus 4 times 2 times the quantity x minus 1 is equal to 5 x squared minus 2 x plus 9. Distribute the factors to further simplify. The result is 6 x squared plus 6x minus 8 x plus 8 is equal to 5 x squared minus 2 x plus 9. Begin solving the equation by writing it with 0 on one side. The result is x squared minus 1 is equal to 0. After factoring, the equation becomes the quantity x minus 1 times the quantity x plus 1 is equal to 0. The results are x is equal to 1 or x is equal to negative 1. Recall that x is not equal to 1 or x is not equal to negative 1. Those values are extraneous, so there is no solution.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834376186\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We will start by factoring all<\/p>\n<div data-type=\"newline\"><\/div>\n<p>denominators, to make it easier<\/p>\n<div data-type=\"newline\"><\/div>\n<p>to identify extraneous solutions and the LCD.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835229823\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Note any value of the variable<\/p>\n<div data-type=\"newline\"><\/div>\n<p>that would make any denominator zero.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835356455\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Find the least common<\/p>\n<div data-type=\"newline\"><\/div>\n<p>denominator.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The LCD is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90a59f9b3c92f89d124158ac468d67bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Clear the fractions.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835233322\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835336257\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832066186\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve the resulting equation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835381243\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835366586\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><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_04_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834190380\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_006j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">Check.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>\u2003<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;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad143a0d979362a51b48a48c9ca9f59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/> are extraneous solutions.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\">The equation has no solution.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835214096\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835358566\">\n<p id=\"fs-id1167832057089\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4664335dcc109954cb5c55b9a3e98e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#53;&#121;&#45;&#49;&#48;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#121;&#43;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#57;&#121;&#43;&#53;&#52;&#125;&#123;&#49;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"206\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835420450\">\n<p id=\"fs-id1167835352167\">There is no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835364530\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835479971\">\n<div data-type=\"problem\" id=\"fs-id1167830770198\">\n<p id=\"fs-id1167835338212\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bfdfa8e3a8b684cae72c894b794f273_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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"198\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p>There is no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834177880\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835515393\">\n<div data-type=\"problem\" id=\"fs-id1167835304792\">\n<p id=\"fs-id1167830702633\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772c03515b7e181e29c02c9912249de8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"252\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834539321\">\n<table id=\"fs-id1167827943907\" class=\"unnumbered unstyled\" summary=\"Solve 4 divided by the quantity 3 x squared minus 10 x plus 3 plus 3 divided by the quantity 3 x squared plus 2 x minus 1 is equal to 2 divided by the quantity x squared minus 2 x minus 3. Factor all of the denominators in the equation. The result is 4 divided by the quantity 3 x minus 1 times the quantity x minus 3 plus 3 divided by the quantity 3 x minus 1 times the quantity x plus 1 is equal to 2 divided by the quantity x minus 3 times the quantity x plus 1. Notice that x is not equal to negative 1, x is not equal to one-third, and x is not equal to 3. Those values make a denominator 0. The least common denominator is the quantity 3 x minus 1 times the quantity x plus 1 times the quantity x minus 3. Clear the fractions by multiplying each side of the equation by the least common denominator. The quantity 3 x minus 1 times the quantity x plus 1 times the quantity x minus 3 times the quantity of divided by the quantity 3 x minus 1 times the quantity x minus 3 plus 3 divided by the quantity 3 x minus 1 times the quantity x plus 1 is equal to the quantity 3 x minus 1 times the quantity x plus 1 times the quantity x minus 3 times 2 divided by the quantity x minus 3 times the quantity x plus 1. Simplify by removing common factors. The result is 4 times the quantity x plus 1 plus 3 times the quantity x minus 3 is equal to 2 times the quantity 3 x minus 1. Simplify the equation by distributing. The result is 4 x plus 4 plus 3 x minus 9 is equal to 6 x minus 2. further simplifying, the result is 7 x minus 5 is equal to 6 x minus 2. Solve the equation. The result is x is equal to 3. The only algebraic solution was but we said that would make a denominator equal to zero. The algebraic solution is an extraneous solution. There is no solution to this equation.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835370982\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Factor all the denominators, so we can note any value of the variable that would make any denominator<\/p>\n<div data-type=\"newline\"><\/div>\n<p>zero.<\/td>\n<td 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_04_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1062995fdf56acfade717db0c0d6834d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#44;&#120;&#92;&#110;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#120;&#92;&#110;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"146\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Find the least common denominator. The LCD is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc2340d8bdf9546d5b9651e4b162d6de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Clear the fractions.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"right\"><span data-type=\"media\" id=\"fs-id1167826781751\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834189271\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Distribute.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830702476\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834382438\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_007g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The only algebraic solution was <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;\" \/> but we said that <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;\" \/> would make a denominator equal to zero. The algebraic solution is an extraneous solution.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\">There is no solution to this equation.<\/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-id1167835166517\">\n<div data-type=\"problem\" id=\"fs-id1167826779764\">\n<p id=\"fs-id1167835244637\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d27046e2de044b93e88b5f15b7e2e3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#45;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#43;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"160\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830921037\">\n<p id=\"fs-id1167830924303\">There is no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835622929\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835615834\">\n<p id=\"fs-id1167834431535\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25553b8557c90bbd3c75bdfe8914dbc2_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;&#43;&#50;&#120;&#45;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#54;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"224\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835241523\">\n<p>There is no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167830866178\">\n<h3 data-type=\"title\">Use Rational Functions<\/h3>\n<p id=\"fs-id1167832057008\">Working with functions that are defined by rational expressions often lead to rational equations. Again, we use the same techniques to solve them.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835187467\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835515194\">\n<div data-type=\"problem\" id=\"fs-id1167835596552\">\n<p id=\"fs-id1167835237617\">For rational function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5edb3effccabf4b026ed12c23badd9c_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;&#120;&#45;&#54;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#53;&#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-0827ec0f399ef51d9fba7afd09c7c7a6_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;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> and <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-id1167835358480\">\n<p id=\"fs-id1167835400376\"><span class=\"token\">\u24d0<\/span> The domain of a rational function is all real numbers except those that make the rational expression undefined. So to find them, we will set the denominator equal to zero and solve.<\/p>\n<p id=\"fs-id1167831919514\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6ebfc4631b883052846f69acc8841b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#53;&#61;&#48;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#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;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#90;&#101;&#114;&#111;&#32;&#80;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#51;&#61;&#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;&#53;&#61;&#48;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#61;&#51;&#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;&#53;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#100;&#111;&#109;&#97;&#105;&#110;&#32;&#105;&#115;&#32;&#97;&#108;&#108;&#32;&#114;&#101;&#97;&#108;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#101;&#120;&#99;&#101;&#112;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#110;&#101;&#32;&#51;&#44;&#120;&#92;&#110;&#101;&#32;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"694\" style=\"vertical-align: -48px;\" \/>.<\/p>\n<p id=\"fs-id1167835363306\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834539060\" class=\"unnumbered unstyled\" summary=\"The function f is equal to 1. Substitute the quantity 2 x minus 6 divided by the quantity x squared minus 8 x plus 15 for f. Factoring the denominator, the result is the quantity 2 x minus 6 all divided by the x squared minus 8 x plus 15 is equal to 1. Factoring the denominator, the result is the quantity 2 x minus 6 divided by the quantity x minus 3 times the quantity x minus 5 is equal to 1. Multiply each side of the equation by the least common denominator, the quantity x minus 3 times the quantity x minus 5. Simplifying, the result is 2 x minus 6 is equal to x squared minus 8 x plus 15. Solve the equation so that terms with variables are one side. The result is 0 is equal to x squared minus 10 x plus 21. Factor the right side of the equation. The result is 0 is equal to the quantity x minus 7 times the quantity x minus 3. Using the Zero Product Property, the result is x minus 7 is equal to 0 or x minus 3 is equal to 0. Solve each equation. The results is x is equal to 7 or x is equal to 3.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830700608\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Substitute in the rational expression.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834430970\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Factor the denominator.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835589718\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Multiply both sides by the LCD,<\/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-0004931ec635cf5f72836be048958d8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835345154\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835306830\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835281480\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Factor.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228009\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Use the Zero Product Property.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830959814\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826819802\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_008i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167831836352\"><span class=\"token\">\u24d2<\/span> The value of the function is 1 when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6269ac1586d7341b90ab9423c3daad18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#55;&#44;&#120;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"97\" style=\"vertical-align: -4px;\" \/> So the points on the graph of this function when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0827ec0f399ef51d9fba7afd09c7c7a6_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;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> will be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc2546e0b0e4a1da47e891e931706305_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831923032\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834234019\">\n<div data-type=\"problem\" id=\"fs-id1167834234021\">\n<p id=\"fs-id1167831923565\">For rational function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ddac62bebf314fbc4e85fc8dda9fdb9_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;&#56;&#45;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#49;&#50;&#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-39245420f2f66fc09003b33dbfd1d858_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;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" 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-id1167835364523\">\n<p id=\"fs-id1167835280319\"><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-d9acec13ad728ef3614e055d39d0f65c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"43\" 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-1510cf64e4145ec5565b1af55d077e42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"100\" style=\"vertical-align: -6px;\" \/><\/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-c1fa0a98de292d7df5c99bfc91fdab6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#51;&#125;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"98\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835420479\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834060719\">\n<div data-type=\"problem\" id=\"fs-id1167835361379\">\n<p id=\"fs-id1167835361381\">For rational function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-493fd2af27da28a514ae2279ba0d4c9c_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;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#53;&#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-84b4b7bce144a709e7eb9d0ff3c1feae_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;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" 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-id1167835215922\">\n<p id=\"fs-id1167834120573\"><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-07652aaac10652c3ffcd03e780128f56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#49;\" 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-acd72a8868181d6019bd755d4c41f281_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#53;&#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-23458f301aad6bd48668ead987cec31f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc7b421058b714a9a7b6cc3c1b4018ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#125;&#123;&#52;&#125;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"47\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835348471\">\n<h3 data-type=\"title\">Solve a Rational Equation for a Specific Variable<\/h3>\n<p id=\"fs-id1167826997467\">When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.<\/p>\n<p id=\"fs-id1167835232465\">When we developed the point-slope formula from our slope formula, we cleared the fractions by multiplying by the LCD.<\/p>\n<p id=\"fs-id1167835303521\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4d9365a572d38d4e38b561d98daff65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#98;&#111;&#116;&#104;&#32;&#115;&#105;&#100;&#101;&#115;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#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;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#119;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#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;&#101;&#114;&#109;&#115;&#32;&#111;&#110;&#32;&#116;&#104;&#101;&#32;&#108;&#101;&#102;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#45;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"685\" style=\"vertical-align: -43px;\" \/><\/p>\n<p id=\"fs-id1167826801694\">In the next example, we will use the same technique with the formula for slope that we used to get the point-slope form of an equation of a line through the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5594171aa6d641e4c22675b30f97faad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> We will add one more step to solve for <em data-effect=\"italics\">y<\/em>.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834194630\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835379570\">Solve:<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1604379d2fe6dce8eb1470bf1dbb8a6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#50;&#125;&#123;&#120;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"67\" style=\"vertical-align: -6px;\" \/> 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 data-type=\"solution\" id=\"fs-id1167831086839\">\n<table class=\"unnumbered unstyled\" summary=\"Solve the quantity y minus 2 divided by the quantity x minus 3 for y. The solution cannot be x is equal to 3 because it will make a denominator 0. Clear the fractions on both sides of the equation by multiplying each one by the lowest common denominator, x minus 3. The result is the quantity x minus 3 times m is equal to the quantity x minus 3 times the quantity y minus 2 all divided by the quantity x minus 3. Simplifying on each side, the result is x m minus 3 m is equal to y minus 2. Isolating the term with y, the result is x m minus 3 m plus 2 is equal to y.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835307680\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Note any value of the variable that would<\/p>\n<div data-type=\"newline\"><\/div>\n<p>make any denominator zero.<\/td>\n<td 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_04_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Clear the fractions by multiplying both sides of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the equation by the LCD, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cee246bedf0a5d020cd46bff5510a79e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"45\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345481\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167828421394\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Isolate the term with <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834535951\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834280129\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834280132\">\n<div data-type=\"problem\" id=\"fs-id1167835357901\">\n<p id=\"fs-id1167835357904\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ba40fa1f6414ebf45339bdc54ea4caa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#53;&#125;&#123;&#120;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"67\" style=\"vertical-align: -6px;\" \/>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 data-type=\"solution\" id=\"fs-id1167834134460\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f846eac38d3b10d9e1f5b02abd95604b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#109;&#120;&#45;&#52;&#109;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835358529\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834132867\">\n<div data-type=\"problem\" id=\"fs-id1167834132869\">\n<p id=\"fs-id1167828421267\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49eb376be34558567f61c3f85d4d7cde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#49;&#125;&#123;&#120;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"67\" style=\"vertical-align: -8px;\" \/> 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 data-type=\"solution\" id=\"fs-id1167835376441\">\n<p id=\"fs-id1167835376443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ea9c9b3fc433e3d894351022c1d0367_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#109;&#120;&#43;&#53;&#109;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831116847\">Remember to multiply both sides by the LCD in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831116850\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834433514\">\n<div data-type=\"problem\" id=\"fs-id1167834433516\">\n<p id=\"fs-id1167835368438\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1fb656d192e82b5961b8786f7ee0cc70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#109;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"78\" style=\"vertical-align: -6px;\" \/> for <em data-effect=\"italics\">c<\/em>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835582748\">\n<table id=\"fs-id1167835582750\" class=\"unnumbered unstyled\" summary=\"Solve the quantity 1 divided by c plus the quantity 1 divided by m is equal to 1 for c. Notice that the values that would make any denominator 0 are c is equal to 0 and m is equal to 0. The least common denominator of the denominators of the fractions is c m. Clear the fractions in the equation by multiplying each side by c m. Distribute c m to each term. The result is c m times the quantity 1 divided by c plus c m times the quantity 1 divided by m is equal to c m times 1. When simplified, the equation becomes m plus c is equal to c m. Collect the terns with c on the right side of the equation. The result is m is equal to c m minus c. When the right side is factored, the result is m is equal to c times the quantity m minus 1. Divide each side of the equation by the quantity m minus 1 to isolate c. When the common factors are removed, the result is m divided by the quantity m minus 1 is equal to c. Notice that m is not equal to 1.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835319356\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Note any value of the variable that would make<\/p>\n<div data-type=\"newline\"><\/div>\n<p>any denominator zero.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834306898\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Clear the fractions by multiplying both sides of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the equations by the LCD, <em data-effect=\"italics\">cm<\/em>.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834431316\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Distribute.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826977623\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831922161\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Collect the terms with <em data-effect=\"italics\">c<\/em> to the right.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834505759\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Factor the expression on the right.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835280666\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">To isolate <em data-effect=\"italics\">c<\/em>, divide both sides by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab1cd2f6c2f0505d5a1a2e9eda38fa74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834376913\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify by removing common factors.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835257626\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_010i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">Notice that even though we excluded <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c468089cacb3e79b5e4a295a43d87fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#48;&#44;&#109;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\" \/> from the original equation, we must also now state that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ad3d7e4136fbde101477abf223c9ebb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#110;&#101;&#32;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834196626\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835328642\">\n<div data-type=\"problem\" id=\"fs-id1167835328644\">\n<p id=\"fs-id1167835240600\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76186261e108b04d92ec22b0a6b49e1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#125;&#61;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"73\" style=\"vertical-align: -6px;\" \/> for <em data-effect=\"italics\">a<\/em>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835300418\">\n<p id=\"fs-id1167831871932\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4518bed56f9d9d095b9cd6e2a5a81109_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#98;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"65\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835342182\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835342186\">\n<div data-type=\"problem\" id=\"fs-id1167835359612\">\n<p id=\"fs-id1167835359614\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32ebe147a26451ae3124b9377ab3810f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"76\" style=\"vertical-align: -9px;\" \/> for <em data-effect=\"italics\">y<\/em>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835310191\">\n<p id=\"fs-id1167835310193\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16e2f12abb265953b3c93f0aea295a27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#120;&#43;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"61\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826780819\" class=\"media-2\">\n<p id=\"fs-id1167835374004\">Access this online resource for additional instruction and practice with equations with rational expressions.<\/p>\n<ul id=\"fs-id1167835377566\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37EqRatExp\">Equations with Rational Expressions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167826783875\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167834190024\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to solve equations with rational expressions.<\/strong>\n<ol id=\"fs-id1167835322041\" type=\"1\" class=\"stepwise\">\n<li>Note any value of the variable that would make any denominator zero.<\/li>\n<li>Find the least common denominator of all denominators in the equation.<\/li>\n<li>Clear the fractions by multiplying both sides of the equation by the LCD.<\/li>\n<li>Solve the resulting equation.<\/li>\n<li>Check:\n<ul id=\"fs-id1167831890544\" data-bullet-style=\"bullet\">\n<li>If any values found in Step 1 are algebraic solutions, discard them.<\/li>\n<li>Check any remaining solutions in the original equation.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831949120\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834429105\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167830897851\"><strong data-effect=\"bold\">Solve Rational Equations<\/strong><\/p>\n<p id=\"fs-id1167835596642\">In the following exercises, solve each rational equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835596645\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835309564\">\n<p id=\"fs-id1167835309566\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a7fe87ba8edad19473e2510bcb869fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"74\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835309673\">\n<p id=\"fs-id1167835381730\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9996f50c8ca415aa204376da45f502e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826781539\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826781541\">\n<p id=\"fs-id1167826781543\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62bd32d8d6c3b6b488ea593a7d89b73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"74\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830704645\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830704647\">\n<p id=\"fs-id1167830704649\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1ff067e1b13305e08ccf89f3f0ee9ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835527876\">\n<p id=\"fs-id1167835423110\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-918aa03e1fb232273a605ff0ccced4ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#48;&#125;&#123;&#50;&#49;&#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-id1167835310329\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835310331\">\n<p id=\"fs-id1167834279098\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11c66fa8a337dd5353c186007565532a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#121;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"74\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835319658\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831106962\">\n<p id=\"fs-id1167831106964\"><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 data-type=\"solution\" id=\"fs-id1167835358672\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dddc9fb9a502c38a4298f3a9a5715322_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#50;&#44;&#109;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832060052\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832060054\">\n<p id=\"fs-id1167832060056\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43ec7706cc33fcc094869b2d25e46c02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"83\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835365115\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835365117\">\n<p id=\"fs-id1167832128656\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32178d373ae7b281bf65438b200a418c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#112;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#48;&#125;&#123;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"91\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830961108\">\n<p id=\"fs-id1167830961110\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa0774ede736047aa7cbe8839b09aa2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#53;&#44;&#112;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834423050\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835368406\">\n<p id=\"fs-id1167835368408\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9d061c53f16ced3548b137650048a6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#113;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"84\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835304732\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835304734\">\n<p id=\"fs-id1167835304736\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ceeea39fbc6cb63f1287779d5f3b32d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#118;&#45;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"74\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834505336\">\n<p id=\"fs-id1167834505338\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ca649ac5ca50bb40262b8be9961a419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834063671\">\n<p id=\"fs-id1167834063673\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a579fab5b58f71da4862e66d9555588_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#50;&#119;&#43;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"73\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831040520\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831040523\">\n<p id=\"fs-id1167831040525\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c274a4af71d23c721704308922e4329f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#120;&#45;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"143\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835348754\">\n<p id=\"fs-id1167835334884\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c23c8728b5d487d70c3c91d603846b1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835229217\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835229219\">\n<p id=\"fs-id1167835229221\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-beed7eea86a2876f7f4850fc39eeb330_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#121;&#45;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#43;&#57;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"142\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834190044\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834190046\">\n<p id=\"fs-id1167834190048\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e07d36569e1e4780c45e9fa6d4f9932a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#122;&#45;&#49;&#48;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#122;&#43;&#49;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"162\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835479340\">\n<p id=\"fs-id1167831835428\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b09a55fbcf45c41a89198b9b429d9a4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#45;&#49;&#52;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"73\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835479358\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835479360\">\n<p id=\"fs-id1167835479362\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98b8179bc7c5c7c4f68142be7f593b95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#97;&#43;&#49;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#97;&#45;&#49;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"163\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834111880\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834111883\">\n<p id=\"fs-id1167834111885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-960edef91e76403b20e443ab495e142b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#48;&#125;&#123;&#113;&#45;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#113;&#43;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"108\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826994126\">\n<p id=\"fs-id1167826994129\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09a2fe95aa1186191c8267d1e6e939f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#45;&#49;&#56;&#44;&#113;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834079305\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834146932\">\n<p id=\"fs-id1167834146934\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61f68647a4e8d1a7e4d5efddc7cef16f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#115;&#43;&#55;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#115;&#45;&#51;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"107\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831881752\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831881755\">\n<p id=\"fs-id1167830959746\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cc35b2595643cf695673991a0dc1f0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#45;&#49;&#48;&#125;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#118;&#43;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#118;&#45;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#118;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"159\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835365624\">\n<p id=\"fs-id1167831881738\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26364633689fe59cdc4c5a06cbc0396f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"87\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827966802\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167827966804\">\n<p id=\"fs-id1167835237656\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32acf77e3600d785114201dd038f48de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#119;&#43;&#56;&#125;&#123;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#119;&#43;&#50;&#56;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#119;&#45;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#119;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"185\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830704522\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834552530\">\n<p id=\"fs-id1167834552532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1829c46419eba3c82c98b56416007f55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#49;&#48;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#49;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#43;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"169\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835304338\">\n<p id=\"fs-id1167835304340\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26364633689fe59cdc4c5a06cbc0396f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"87\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831880161\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835371420\">\n<p id=\"fs-id1167835371422\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3e3c1ad4da70cd782521d9b6ae1a1bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#53;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#121;&#45;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#43;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"160\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835234777\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835234779\">\n<p id=\"fs-id1167835234781\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48ccbede8a7e1ecf886bf17d1ce653cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#43;&#51;&#125;&#123;&#51;&#98;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835380533\">\n<p id=\"fs-id1167835380536\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54d3fc67ae103fb9fbced469462556f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830837155\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830837157\">\n<p id=\"fs-id1167835340875\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8026c9b46992434ccfd793e9a2231fb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#43;&#51;&#125;&#123;&#49;&#50;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#51;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"103\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835230880\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835230882\">\n<p id=\"fs-id1167834064438\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b35341fbaba37aac5166c789598ccf41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#100;&#125;&#123;&#100;&#43;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#125;&#123;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"116\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830704484\">\n<p id=\"fs-id1167830704486\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90899c4debe310841a2f67893781f68c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834555265\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834555267\">\n<p id=\"fs-id1167834124400\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df7b9ee9149725cb0af285c2dace18c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#109;&#43;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#48;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"133\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835339503\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835339505\">\n<p id=\"fs-id1167835339507\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f02b763a136b878b57bebf6b9fe3c1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#110;&#43;&#50;&#125;&#45;&#51;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"117\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834526292\">\n<p id=\"fs-id1167834526295\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d94dded0dc9c883f82d566d62e2d4b42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#49;\" 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-id1167835380884\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828447056\">\n<p id=\"fs-id1167828447059\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-700ea52cb1ff49a50fd7044392fe171f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#112;&#43;&#55;&#125;&#45;&#56;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#56;&#125;&#123;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"123\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831887701\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834448880\">\n<p id=\"fs-id1167834448882\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e32cbcb72e1b15063bd7042d53d89b68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#51;&#113;&#45;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#113;&#43;&#49;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#113;&#43;&#54;&#51;&#125;&#123;&#50;&#52;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"191\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835218128\">\n<p id=\"fs-id1167835218130\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26364633689fe59cdc4c5a06cbc0396f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"87\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835348954\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831887557\">\n<p id=\"fs-id1167831887559\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1c3a89df79a5821ea11b1f9e58ebd33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#125;&#123;&#51;&#114;&#45;&#49;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#114;&#43;&#50;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#114;&#43;&#52;&#48;&#125;&#123;&#49;&#50;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"205\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835254283\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835512058\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b3ab0354f64eea5f813c8f54bd6d80b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#115;&#125;&#123;&#50;&#115;&#43;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#115;&#43;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#115;&#45;&#55;&#125;&#123;&#49;&#48;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#48;&#115;&#43;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"198\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834430234\">\n<p id=\"fs-id1167834430236\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b7dc2e5482a13ccca6412f0495bed8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835511057\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831117195\">\n<p id=\"fs-id1167831117197\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-608243d27f9c18c6f2c2c649158dde03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#54;&#116;&#45;&#49;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#116;&#43;&#49;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#51;&#116;&#43;&#55;&#48;&#125;&#123;&#49;&#50;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#54;&#116;&#45;&#49;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"213\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835419937\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835419939\">\n<p id=\"fs-id1167834063967\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a75ba0e6f4f106a34a5fdfe17338485_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#43;&#50;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"240\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835364061\">\n<p id=\"fs-id1167835364063\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b9321a73d1108395bc0a2c0a4e734f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835514064\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835514066\">\n<p id=\"fs-id1167835338292\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d02e16db008cf071962c2e5e6283bf1d_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;&#43;&#52;&#120;&#43;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"212\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826978010\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826978012\">\n<p id=\"fs-id1167830925343\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2880f4c8c99dc13559ffdbf00385a38e_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;&#45;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"201\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537641\">\n<p>no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826978331\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826978333\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0575dfedc7d18423ec095b750d2cd342_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"201\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826778769\"><strong data-effect=\"bold\">Solve Rational Equations that Involve Functions<\/strong><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835356003\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835356005\">\n<p id=\"fs-id1167835356007\">For rational function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d130ebaa7378316041d4e6e3eadef4e1_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;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#56;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"126\" style=\"vertical-align: -9px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> find the domain of the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89e7d21e9b6530d6e6731a14e8a0a291_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;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835349026\">\n<p id=\"fs-id1167835301328\">\n<div data-type=\"newline\"><\/div>\n<p><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-778c54a0c02107e335f913dc6731c2a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#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-83e08a23d1a261fff4bb1802cc49253c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1ec1fc1c01fc4ff51266da54e225961_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"128\" style=\"vertical-align: -6px;\" \/><\/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-0e9cd5a64b17c6ffd953d0fc422cb06b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#53;&#125;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835319215\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835319217\">\n<p id=\"fs-id1167831239531\">For rational function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ce70502b04f0e34de160b01055f72c7_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;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"126\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> find the domain of the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><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;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835287710\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835287712\">\n<p id=\"fs-id1167835287714\">For rational function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04a57b49793399c627d1cf20f642629d_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;&#45;&#55;&#120;&#43;&#49;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"133\" style=\"vertical-align: -9px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> find the domain of the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><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;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> find the points on the graph at this function value.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834430848\">\n<p id=\"fs-id1167834535542\">\n<div data-type=\"newline\"><\/div>\n<p><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-acd72a8868181d6019bd755d4c41f281_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#110;&#101;&#32;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4372881d9dd6d94aa055739ac51ae52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"49\" style=\"vertical-align: -6px;\" \/><\/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-9b664e9810fab6a66ad2c726358b173b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834066254\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834066256\">\n<p id=\"fs-id1167835354887\">For rational function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f755906c38740809a582e1b29c5cbf20_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;&#53;&#45;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#54;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"126\" style=\"vertical-align: -9px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> find the domain of the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39245420f2f66fc09003b33dbfd1d858_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;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> the points on the graph at this function value.<\/div>\n<\/div>\n<p id=\"fs-id1167835336553\"><strong data-effect=\"bold\">Solve a Rational Equation for a Specific Variable<\/strong><\/p>\n<p id=\"fs-id1167830693613\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830693617\">\n<div data-type=\"problem\" id=\"fs-id1167832053198\">\n<p id=\"fs-id1167832053200\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cef0301b0b2df5c665b93c93b1835fc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#67;&#125;&#123;&#114;&#125;&#61;&#50;&#92;&#112;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa03a29f511592c1a1ecc8b306b0cf0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834124437\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87d9c30fd61188b167d8a041c23126d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#67;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#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-id1167835173771\">\n<div data-type=\"problem\" id=\"fs-id1167834244190\">\n<p id=\"fs-id1167834244192\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd4b90e2425650e43d2561e87df6f413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#73;&#125;&#123;&#114;&#125;&#61;&#80;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa03a29f511592c1a1ecc8b306b0cf0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835237516\">\n<div data-type=\"problem\" id=\"fs-id1167835375434\">\n<p id=\"fs-id1167835375436\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62f6374ac7e87696dbc874ad4dea0f88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#43;&#51;&#125;&#123;&#119;&#45;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -7px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c885ae2387fc01f58967c91d7b5a6b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831913245\">\n<p id=\"fs-id1167831913247\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3c6df9a6ad299fd350e7e38cc7a8fa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#50;&#118;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"86\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831884542\">\n<div data-type=\"problem\" id=\"fs-id1167831884544\">\n<p id=\"fs-id1167834539247\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f9b4ac46c6d1b1eb968fd335ee2a140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#53;&#125;&#123;&#50;&#45;&#121;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"60\" 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-id1167834130086\">\n<div data-type=\"problem\" id=\"fs-id1167835365713\">\n<p id=\"fs-id1167835365715\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e117a57ef687e5ffd6f4683adad8809_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#43;&#51;&#125;&#123;&#99;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9642cdcdda4b3ba04e90b50a66ecb92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835319924\">\n<p id=\"fs-id1167835368012\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6333b1947f6fae932e31444c65ee9bf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#43;&#51;&#43;&#50;&#97;&#125;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834423786\">\n<div data-type=\"problem\" id=\"fs-id1167834423788\">\n<p id=\"fs-id1167834099281\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c31591f5ed7f5286495f47673e4e89e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#45;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ecab3f5df4767e23a2660ceb88ceabd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826782306\">\n<div data-type=\"problem\" id=\"fs-id1167826782308\">\n<p id=\"fs-id1167830704311\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00f4165c3dcb1898ef617427afeeff7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#112;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#113;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"75\" style=\"vertical-align: -9px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa6ef6ec04c2dccd40c7f3e3be899df7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835331700\">\n<p id=\"fs-id1167831106942\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-790fab69906d72eb344d6ebc204c08db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#52;&#113;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"67\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831823292\">\n<div data-type=\"problem\" id=\"fs-id1167835254155\">\n<p id=\"fs-id1167835254157\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2467cf87a4ed8bfa0aed0c582233e27a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#115;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#116;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"73\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23a7daa116b8874af1538c91f8d239de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835380032\">\n<div data-type=\"problem\" id=\"fs-id1167832041363\">\n<p id=\"fs-id1167832041365\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7064ffed5165adcc64dc39d178e57102_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#118;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"78\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c885ae2387fc01f58967c91d7b5a6b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834473638\">\n<p id=\"fs-id1167834473640\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-804d8c386d8ac9888954cd85d11f4e32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#118;&#125;&#123;&#49;&#48;&#43;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"71\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835347802\">\n<div data-type=\"problem\" id=\"fs-id1167830838410\">\n<p id=\"fs-id1167830838412\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4203d27d5b3af8faf5cbc2bf1bb12036_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"76\" 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-id1167834539307\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834539311\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7268d38ffde884d631db44695fc1ac94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#43;&#51;&#125;&#123;&#110;&#45;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ecab3f5df4767e23a2660ceb88ceabd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826977638\">\n<p id=\"fs-id1167826977640\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4356a43535ce2e4a5ff8f23c356e9ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#109;&#43;&#50;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"80\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831912314\">\n<div data-type=\"problem\" id=\"fs-id1167831912316\">\n<p id=\"fs-id1167834191152\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef2e0e31ff32a8f2f7dd8c1fd12f5697_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#115;&#125;&#123;&#51;&#45;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4998f61a094184afa02f41dd4ab518c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831846917\">\n<div data-type=\"problem\" id=\"fs-id1167831846919\">\n<p id=\"fs-id1167835519767\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-deb34591b373a412391e793ebe76d94c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#69;&#125;&#123;&#99;&#125;&#61;&#123;&#109;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9642cdcdda4b3ba04e90b50a66ecb92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832152860\">\n<p id=\"fs-id1167832152862\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15e6c3fbf1cc7e5a73d5d9f93779778f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#69;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834184720\">\n<div data-type=\"problem\" id=\"fs-id1167834184722\">\n<p id=\"fs-id1167834146999\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5219566e40571a954870de6faa167657_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#82;&#125;&#123;&#84;&#125;&#61;&#87;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acf5fe4fa9802c795bc63e65b548daf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835321864\">\n<div data-type=\"problem\" id=\"fs-id1167835321866\">\n<p id=\"fs-id1167834587420\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd0bd7201865e4eed8f335907eb26372_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#121;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"75\" 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 data-type=\"solution\" id=\"fs-id1167835319640\">\n<p id=\"fs-id1167835319643\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d43c3c43f2b0ef299a52f0e19d678b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#120;&#125;&#123;&#49;&#50;&#45;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"68\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832057452\">\n<div data-type=\"problem\" id=\"fs-id1167832057454\">\n<p id=\"fs-id1167835258331\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0dab87578c98ce36897ccd66210e5b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#97;&#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;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"77\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ebc59bdf10d3d739bfa532b65c85287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834157025\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167831921246\">\n<div data-type=\"problem\" id=\"fs-id1167831921248\">\n<p id=\"fs-id1167832060427\">Your class mate is having trouble in this section. Write down the steps you would use to explain how to solve a rational equation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831883616\">\n<p id=\"fs-id1167831883618\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834066032\">\n<div data-type=\"problem\" id=\"fs-id1167834066034\">\n<p id=\"fs-id1167834066036\">Alek thinks the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a11032da1c46a578386c23f789663d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#121;&#43;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#50;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"123\" style=\"vertical-align: -10px;\" \/> has two solutions, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9ec985c3bc0aa35a95da9940e4def8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-270d2ebaae94f65bccc2c78eddebf555_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> Explain why Alek is wrong.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832053662\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167835330833\"><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-id1167834061744\" data-alt=\"This table has four columns and four 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 equations. In row 3, the I can was solve rational equations involving functions. In row 4, the I can was solve rational equations for a specific variable.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_04_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and four 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 equations. In row 3, the I can was solve rational equations involving functions. In row 4, the I can was solve rational equations for a specific variable.\" \/><\/span><\/p>\n<p id=\"fs-id1167834094796\"><span class=\"token\">\u24d1<\/span> On a scale of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3e3c048222c38a09c036de10daa1700_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#49;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/> how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167832068260\">\n<dt>extraneous solution to a rational equation<\/dt>\n<dd id=\"fs-id1167832068263\">An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167835342941\">\n<dt>rational equation<\/dt>\n<dd id=\"fs-id1167835342944\">A rational equation is an equation that contains a rational expression.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3363","chapter","type-chapter","status-publish","hentry"],"part":3130,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3363","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\/3363\/revisions"}],"predecessor-version":[{"id":15248,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3363\/revisions\/15248"}],"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\/3363\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3363"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3363"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3363"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3363"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}