{"id":3228,"date":"2018-12-11T13:51:22","date_gmt":"2018-12-11T18:51:22","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/add-and-subtract-rational-expressions\/"},"modified":"2018-12-11T13:51:22","modified_gmt":"2018-12-11T18:51:22","slug":"add-and-subtract-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/add-and-subtract-rational-expressions\/","title":{"raw":"Add and Subtract Rational Expressions","rendered":"Add and Subtract Rational Expressions"},"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>Add and subtract rational expressions with a common denominator<\/li><li>Add and subtract rational expressions whose denominators are opposites<\/li><li>Find the least common denominator of rational expressions<\/li><li>Add and subtract rational expressions with unlike denominators<\/li><li>Add and subtract rational functions<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167836622513\" class=\"be-prepared\"><p id=\"fs-id1167829644630\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167836605367\" type=\"1\"><li>Add: \\(\\frac{7}{10}+\\frac{8}{15}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836553755\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Subtract: \\(\\frac{3x}{4}-\\frac{8}{9}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836518722\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Subtract: \\(6\\left(2x+1\\right)-4\\left(x-5\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829791835\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836516393\"><h3 data-type=\"title\">Add and Subtract Rational Expressions with a Common Denominator<\/h3><p id=\"fs-id1167833328016\">What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.<\/p><p id=\"fs-id1167836306628\">It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.<\/p><div data-type=\"note\" id=\"fs-id1167836628310\"><div data-type=\"title\">Rational Expression Addition and Subtraction<\/div><p id=\"fs-id1167836514001\">If <em data-effect=\"italics\">p<\/em>, <em data-effect=\"italics\">q<\/em>, and <em data-effect=\"italics\">r<\/em> are polynomials where \\(r\\ne 0,\\) then<\/p><div data-type=\"equation\" id=\"fs-id1167836312458\" class=\"unnumbered\" data-label=\"\">\\(\\frac{p}{r}+\\frac{q}{r}=\\frac{p+q}{r}\\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}\\frac{p}{r}-\\frac{q}{r}=\\frac{p-q}{r}\\)<\/div><\/div><p>To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result over the common denominator.<\/p><p id=\"fs-id1167836551229\">We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.<\/p><p>Remember, too, we do not allow values that would make the denominator zero. What value of <em data-effect=\"italics\">x<\/em> should be excluded in the next example?<\/p><div data-type=\"example\" id=\"fs-id1167836544834\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167829690445\">Add: \\(\\frac{11x+28}{x+4}+\\frac{{x}^{2}}{x+4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829908743\"><p id=\"fs-id1167836524115\">Since the denominator is \\(x+4,\\) we must exclude the value \\(x=-4.\\)<\/p><p id=\"fs-id1167836689465\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{5.8em}{0ex}}\\frac{11x+28}{x+4}+\\frac{{x}^{2}}{x+4},\\phantom{\\rule{0.8em}{0ex}}x\\ne \\text{\u2212}4\\hfill \\\\ \\begin{array}{c}\\text{The fractions have a common denominator,}\\hfill \\\\ \\text{so add the numerators and place the sum}\\hfill \\\\ \\text{over the common denominator.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{11x+28+{x}^{2}}{x+4}\\hfill \\\\ \\\\ \\\\ \\text{Write the degrees in descending order.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{{x}^{2}+11x+28}{x+4}\\hfill \\\\ \\\\ \\\\ \\text{Factor the numerator.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{\\left(x+4\\right)\\left(x+7\\right)}{x+4}\\hfill \\\\ \\\\ \\\\ \\text{Simplify by removing common factors.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{\\overline{)\\left(x+4\\right)}\\left(x+7\\right)}{\\overline{)x+4}}\\hfill \\\\ \\\\ \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}x+7\\hfill \\end{array}\\)<\/p><p>The expression simplifies to \\(x+7\\) but the original expression had a denominator of \\(x+4\\) so \\(x\\ne \\text{\u2212}4.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836519066\"><div data-type=\"problem\"><p id=\"fs-id1167836524224\">Simplify: \\(\\frac{9x+14}{x+7}+\\frac{{x}^{2}}{x+7}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836393292\"><p>\\(x+2\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836393168\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167836694802\">Simplify: \\(\\frac{{x}^{2}+8x}{x+5}+\\frac{15}{x+5}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836448476\"><p id=\"fs-id1167836286266\">\\(x+3\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836791982\">To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator. Be careful of the signs when you subtract a binomial or trinomial.<\/p><div data-type=\"example\" id=\"fs-id1167836503887\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p>Subtract: \\(\\frac{5{x}^{2}-7x+3}{{x}^{2}-3x+18}-\\frac{4{x}^{2}+x-9}{{x}^{2}-3x+18}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836615100\"><p id=\"fs-id1167829746839\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{5{x}^{2}-7x+3}{{x}^{2}-3x+18}-\\frac{4{x}^{2}+x-9}{{x}^{2}-3x+18}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Subtract the numerators and place the}\\hfill \\\\ \\text{difference over the common denominator.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{5{x}^{2}-7x+3-\\left(4{x}^{2}+x-9\\right)}{{x}^{2}-3x+18}\\hfill \\\\ \\\\ \\\\ \\text{Distribute the sign in the numerator.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}\\hfill \\\\ \\\\ \\\\ \\text{Combine like terms.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}\\hfill \\\\ \\\\ \\\\ \\text{Factor the numerator and the denominator.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{\\left(x-2\\right)\\left(x-6\\right)}{\\left(x+3\\right)\\left(x-6\\right)}\\hfill \\\\ \\\\ \\\\ \\text{Simplify by removing common factors.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{\\left(x-2\\right)\\overline{)\\left(x-6\\right)}}{\\left(x+3\\right)\\overline{)\\left(x-6\\right)}}\\hfill \\\\ \\\\ \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{\\left(x-2\\right)}{\\left(x+3\\right)}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836386277\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836293673\"><div data-type=\"problem\" id=\"fs-id1167836697800\"><p>Subtract: \\(\\frac{4{x}^{2}-11x+8}{{x}^{2}-3x+2}-\\frac{3{x}^{2}+x-3}{{x}^{2}-3x+2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836481451\"><p>\\(\\frac{x-11}{x-2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836515050\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836516618\"><div data-type=\"problem\"><p id=\"fs-id1167829695628\">Subtract: \\(\\frac{6{x}^{2}-x+20}{{x}^{2}-81}-\\frac{5{x}^{2}+11x-7}{{x}^{2}-81}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824733285\"><p id=\"fs-id1167829788807\">\\(\\frac{x-3}{x+9}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Add and Subtract Rational Expressions Whose Denominators are Opposites<\/h3><p id=\"fs-id1167829718460\">When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by \\(\\frac{-1}{-1}.\\)<\/p><p id=\"fs-id1167833051123\">Let\u2019s see how this works.<\/p><table id=\"fs-id1167836662756\" class=\"unnumbered unstyled\" summary=\"7 divided by d plus 5 divided by negative d. Multiply 5 divided by d times the fraction, negative 1 over negative 1. The result is 7 divided by d plus the quantity negative 1 times 5 all divided by the quantity negative 1times negative d. Notice that the denominators are the same. The denominators are d. 7 divided by d plus negative 5 divided by d. Simplify by adding the numerators. The result is 2 divided by d.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836319593\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the second fraction by \\(\\frac{-1}{-1}.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829746072\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The denominators are the same.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836296174\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1165927912335\">Be careful with the signs as you work with the opposites when the fractions are being subtracted.<\/p><div data-type=\"example\" id=\"fs-id1167836728843\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829692333\"><div data-type=\"problem\"><p>Subtract: \\(\\frac{{m}^{2}-6m}{{m}^{2}-1}-\\frac{3m+2}{1-{m}^{2}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836360555\"><table id=\"fs-id1167833339440\" class=\"unnumbered unstyled can-break\" summary=\"The rational expression the quantity m squared minus 6 divided by the quantity m squared minus 1 minus the rational expression the quantity 3 m plus 2 divided by the quantity 1 minus m squared. The denominators m squared minus 1 and 1 minus m squared are opposites so multiply the second rational expression by negative 1 over negative 1. The result is the quantity m squared minus 6 divided by the quantity m squared minus 1 minus negative 1 times the quantity 3 m plus 2 divided by negative 1 times the quantity 1 minus m squared. Simplify the second rational expression. The result is the rational expression, the quantity m squared minus 6 divided by the quantity m squared minus 1 minus the rational expression, the quantity negative 3 m minus 2 divided by the quantity m squared minus 1. The denominators of both rational expressions are the same, m squared minus 1, so subtract the numerators. The result is m squared minus 6 m minus the quantity negative 3 m minus 2 all divided by the quantity m squared minus 1. Combine the like terms in the numerator. The result is the quantity m squared minus 3 m plus 2 all divided by m squared minus 1. The numerator, the quantity m squared minus 3 m plus 2, factors into the quantity m minus 1 times the quantity m minus 2, and the denominator, m squared minus 1, factors into the quantity m minus 1 times the quantity m plus 1. The result is the quantity m minus 1 times the quantity m minus 2 all divided by the quantity m minus 1 times the quantity m plus 1. Simplify the expression by removing the common factor, m minus 1, from the numerator and denominator. The result is the quantity m minus 2 divided by the quantity m plus 1.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The denominators are opposites, so multiply the<div data-type=\"newline\"><br><\/div>second fraction by \\(\\frac{-1}{-1}.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836530198\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the second fraction.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836376367\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The denominators are the same. Subtract the numerators.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826170180\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1165927925224\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829809610\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the numerator and denominator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836609915\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify by removing common factors.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836550878\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836620617\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836558518\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836362962\"><div data-type=\"problem\" id=\"fs-id1167829717704\"><p id=\"fs-id1167836418052\">Subtract: \\(\\frac{{y}^{2}-5y}{{y}^{2}-4}-\\frac{6y-6}{4-{y}^{2}}.\\)<\/p><\/div><div data-type=\"solution\"><p>\\(\\frac{y+3}{y+2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836728750\"><div data-type=\"problem\" id=\"fs-id1167836699650\"><p id=\"fs-id1167836558293\">Subtract: \\(\\frac{2{n}^{2}+8n-1}{{n}^{2}-1}-\\frac{{n}^{2}-7n-1}{1-{n}^{2}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836309417\"><p id=\"fs-id1167829936836\">\\(\\frac{3n-2}{n-1}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836288812\"><h3 data-type=\"title\">Find the Least Common Denominator of Rational Expressions<\/h3><p id=\"fs-id1167836652759\">When we add or subtract rational expressions with unlike denominators, we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.<\/p><p id=\"fs-id1167836399264\">Let\u2019s look at this example: \\(\\frac{7}{12}+\\frac{5}{18}.\\) Since the denominators are not the same, the first step was to find the least common denominator (LCD).<\/p><p>To find the LCD of the fractions, we factored 12 and 18 into primes, lining up any common primes in columns. Then we \u201cbrought down\u201d one prime from each column. Finally, we multiplied the factors to find the LCD.<\/p><p id=\"fs-id1167836630479\">When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.<\/p><span data-type=\"media\" id=\"fs-id1167836299691\" data-alt=\"Seven-twelfths plus five-eighteenths. Write the prime factorizations of each denominator and line up the common factors. The denominator of the first fraction is 12. The prime factorization of 12 is 2 times 2 times 3. The denominator of the second fraction is 18. The prime factorization of 18 is 2 times 3 times 3. Bringing down a factor from each column, the lowest common denominator of 12 and 18 is 2 times 2 times 3 times 3, which is 36. Write both fractions using the lowest common denominator. To do this multiply the numerator and denominator of the first fraction by 3 and multiply the numerator and denominator of the second fraction by 2. The result is 7 times 3 all divided by 12 times 3 plus 5 times 2 all divided by 18 times 2. Simplify each fraction. 7 times 3 is 21 and 12 times 3 is 36. 5 times 2 is 10 and 18 times 2 is 36. The result is twenty-one thirty-sixths plus ten thirty-sixths.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Seven-twelfths plus five-eighteenths. Write the prime factorizations of each denominator and line up the common factors. The denominator of the first fraction is 12. The prime factorization of 12 is 2 times 2 times 3. The denominator of the second fraction is 18. The prime factorization of 18 is 2 times 3 times 3. Bringing down a factor from each column, the lowest common denominator of 12 and 18 is 2 times 2 times 3 times 3, which is 36. Write both fractions using the lowest common denominator. To do this multiply the numerator and denominator of the first fraction by 3 and multiply the numerator and denominator of the second fraction by 2. The result is 7 times 3 all divided by 12 times 3 plus 5 times 2 all divided by 18 times 2. Simplify each fraction. 7 times 3 is 21 and 12 times 3 is 36. 5 times 2 is 10 and 18 times 2 is 36. The result is twenty-one thirty-sixths plus ten thirty-sixths.\"><\/span><p id=\"fs-id1167829580440\">We do the same thing for rational expressions. However, we leave the LCD in factored form.<\/p><div data-type=\"note\" id=\"fs-id1167833239730\" class=\"howto\"><div data-type=\"title\">Find the least common denominator of rational expressions.<\/div><ol type=\"1\" class=\"stepwise\"><li>Factor each denominator completely.<\/li><li>List the factors of each denominator. Match factors vertically when possible.<\/li><li>Bring down the columns by including all factors, but do not include common factors twice.<\/li><li>Write the LCD as the product of the factors.<\/li><\/ol><\/div><p>Remember, we always exclude values that would make the denominator zero. What values of \\(x\\) should we exclude in this next example?<\/p><div data-type=\"example\" id=\"fs-id1167836331076\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833386366\"><div data-type=\"problem\"><p id=\"fs-id1167836612766\"><span class=\"token\">\u24d0<\/span> Find the LCD for the expressions \\(\\frac{8}{{x}^{2}-2x-3},\\frac{3x}{{x}^{2}+4x+3}\\) and <span class=\"token\">\u24d1<\/span> rewrite them as equivalent rational expressions with the lowest common denominator.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829694661\"><p id=\"fs-id1167836493853\"><span class=\"token\">\u24d0<\/span><\/p><table id=\"fs-id1167826130067\" class=\"unnumbered unstyled\" summary=\"Find the lowest common denominator of the rational expressions 8 divided by the quantity x squared minus 2 x minus 3 and 3 x divided by the quantity x squared plus 4 x plus 3. Line up the denominators and factor them completely. Line up the common factors. The denominator x squared minus 2 x minus 3 is equal to the quantity x plus 1 times the quantity x minus 3. The denominator x squared plus 4 x plus 3 is equal to the quantity x plus 1 times the quantity x plus 3. The quantity x plus 1 is a factor of both denominators. The quantity x minus 3 is only a factor of the denominator x squared minus 2 x minus 3, so it is in a column alone. The quantity x plus 3 is only a factor of the denominator x squared plus 4 x plus 3, so it is in a column alone. The lowest common denominator is the product of the primes from each column, the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the LCD for \\(\\frac{8}{{x}^{2}-2x-3},\\frac{3x}{{x}^{2}+4x+3}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor each denominator completely, lining up common factors.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>Bring down the columns.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the LCD as the product of the factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829594568\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836287401\"><span class=\"token\">\u24d1<\/span><\/p><table id=\"fs-id1167836692693\" class=\"unnumbered unstyled\" summary=\"Factor the denominators of the rational expressions, 8 divided by the quantity x squared minus 2 x minus 3 and 3 x divided by the quantity x squared plus 4 x plus 3. The denominator, x squared minus 2 x minus 3, factors into the quantity x plus 1 times the quantity x minus 3. The denominator, x squared plus 4 x plus 3, factors into the quantity x plus 1 times the quantity x plus 3. Write each rational expression with a factored denominator. The first rational expression is 8 divided by the quantity x plus 1 times the quantity x minus 3. The second rational expression is 3 x divided by the quantity x plus 1 times the quantity x plus 3. Multiply the numerator and denominator of each expression by the missing lowest common denominator factor. The results are 8 times the quantity x plus 3 all divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3 and the quantity 3 x times the quantity x minus 3 all divided by the quantity x plus 1 times the quantity x plus 3 times the quantity x minus 3. Simplify the numerators of the rational expressions and keep the denominators. 8 times the quantity x plus 3 is 8 x plus 24. 3 x times the quantity x minus 3 is 3 x squared minus 9. The results are 8 times the quantity x plus 3 all divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3 and the quantity 3 x squared minus 9 x all divided by the quantity x plus 1 times x plus 3 times the quantity x minus 3.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor each denominator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836532070\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply each denominator by the \u2018missing\u2019<div data-type=\"newline\"><br><\/div>LCD factor and multiply each numerator by the same factor.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836512927\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836527650\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836524675\"><div data-type=\"problem\" id=\"fs-id1167836288975\"><p id=\"fs-id1167829691761\"><span class=\"token\">\u24d0<\/span> Find the LCD for the expressions \\(\\frac{2}{{x}^{2}-x-12},\\frac{1}{{x}^{2}-16}\\) <span class=\"token\">\u24d1<\/span> rewrite them as equivalent rational expressions with the lowest common denominator.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829712961\"><p id=\"fs-id1167833021001\"><span class=\"token\">\u24d0<\/span>\\(\\left(x-4\\right)\\left(x+3\\right)\\left(x+4\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{2x+8}{\\left(x-4\\right)\\left(x+3\\right)\\left(x+4\\right)}\\),<div data-type=\"newline\"><br><\/div>\\(\\frac{x+3}{\\left(x-4\\right)\\left(x+3\\right)\\left(x+4\\right)}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836683527\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836362198\"><div data-type=\"problem\" id=\"fs-id1167836534817\"><p id=\"fs-id1167836619996\"><span class=\"token\">\u24d0<\/span> Find the LCD for the expressions \\(\\frac{3x}{{x}^{2}-3x+10},\\frac{5}{{x}^{2}+3x+2}\\) <span class=\"token\">\u24d1<\/span> rewrite them as equivalent rational expressions with the lowest common denominator.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833058297\"><p id=\"fs-id1167836321317\"><span class=\"token\">\u24d0<\/span>\\(\\left(x+2\\right)\\left(x-5\\right)\\left(x+1\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{3{x}^{2}+3x}{\\left(x+2\\right)\\left(x-5\\right)\\left(x+1\\right)}\\),<div data-type=\"newline\"><br><\/div>\\(\\frac{5x-25}{\\left(x+2\\right)\\left(x-5\\right)\\left(x+1\\right)}\\)<\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832936445\"><h3 data-type=\"title\">Add and Subtract Rational Expressions with Unlike Denominators<\/h3><p id=\"fs-id1167833378257\">Now we have all the steps we need to add or subtract rational expressions with unlike denominators.<\/p><div data-type=\"example\" id=\"fs-id1167836730276\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Add Rational Expressions with Unlike Denominators<\/div><div data-type=\"exercise\" id=\"fs-id1167836439678\"><div data-type=\"problem\" id=\"fs-id1167836578933\"><p id=\"fs-id1167833396962\">Add: \\(\\frac{3}{x-3}+\\frac{2}{x-2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829753077\"><span data-type=\"media\" id=\"fs-id1167836622324\" data-alt=\"Step 1 is to determine if the rational expressions 3 divided by the quantity x minus 3 and 2 divided by the quantity x minus 2 have a common factors. The denominators x minus 3 and x minus 2 do not have any common factors, which means the lowest common denominator of the rational expressions is the quantity x minus 3 times the quantity x minus 2. Rewrite each rational expression with the least common denominator. Multiply the numerator and denominator of 3 divided by the quantity x minus 3 by the quantity x minus 2. Multiply the numerator and denominator of 2 divided by the quantity x minus 2 by the quantity x minus 2. The result is the rational expression 3 times the quantity x minus 2 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression 2 times the quantity x minus 3 divided by the quantity x minus 2 times the quantity x minus 3. Simplify the numerators and keep the denominators factored. The numerator of the first rational expression, 3 times the quantity x minus 2, simplifies to 3 x minus 6. The numerator of the second rational expression, 2 times the quantity x minus 3, simplifies to 2 x minus 6. The result is the rational expression the quantity 3 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression, the quantity 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_006a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to determine if the rational expressions 3 divided by the quantity x minus 3 and 2 divided by the quantity x minus 2 have a common factors. The denominators x minus 3 and x minus 2 do not have any common factors, which means the lowest common denominator of the rational expressions is the quantity x minus 3 times the quantity x minus 2. Rewrite each rational expression with the least common denominator. Multiply the numerator and denominator of 3 divided by the quantity x minus 3 by the quantity x minus 2. Multiply the numerator and denominator of 2 divided by the quantity x minus 2 by the quantity x minus 2. The result is the rational expression 3 times the quantity x minus 2 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression 2 times the quantity x minus 3 divided by the quantity x minus 2 times the quantity x minus 3. Simplify the numerators and keep the denominators factored. The numerator of the first rational expression, 3 times the quantity x minus 2, simplifies to 3 x minus 6. The numerator of the second rational expression, 2 times the quantity x minus 3, simplifies to 2 x minus 6. The result is the rational expression the quantity 3 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression, the quantity 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2.\"><\/span><span data-type=\"media\" id=\"fs-id1167829746788\" data-alt=\"Step 2 is to add or subtract the rational expressions by adding the numerators, the quantity 3 x minus 6 and the quantity 2 x minus 6, and placing the sum over the denominator, the quantity x minus 3 times the quantity x minus 2. The result is the quantity 3 x minus 6 plus 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2. Simplify the numerator by combining like terms. The result is the quantity 5 x minus 12 all divided by the quantity x minus 3 times the quantity x minus 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_006b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to add or subtract the rational expressions by adding the numerators, the quantity 3 x minus 6 and the quantity 2 x minus 6, and placing the sum over the denominator, the quantity x minus 3 times the quantity x minus 2. The result is the quantity 3 x minus 6 plus 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2. Simplify the numerator by combining like terms. The result is the quantity 5 x minus 12 all divided by the quantity x minus 3 times the quantity x minus 2.\"><\/span><span data-type=\"media\" id=\"fs-id1167836728342\" data-alt=\"Step 3. Notice that 5 x minus 12 cannot be factored, so the answer is simplified.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_006c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. Notice that 5 x minus 12 cannot be factored, so the answer is simplified.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836712359\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836520469\"><div data-type=\"problem\" id=\"fs-id1167836432782\"><p id=\"fs-id1167833408057\">Add: \\(\\frac{2}{x-2}+\\frac{5}{x+3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836609856\"><p id=\"fs-id1167829614414\">\\(\\frac{7x-4}{\\left(x-2\\right)\\left(x+3\\right)}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836529491\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829931412\"><div data-type=\"problem\" id=\"fs-id1167833383016\"><p id=\"fs-id1167836477545\">Add:\\(\\frac{4}{m+3}+\\frac{3}{m+4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836628945\"><p id=\"fs-id1167836528220\">\\(\\frac{7m+25}{\\left(m+3\\right)\\left(m+4\\right)}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836630117\">The steps used to add rational expressions are summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167836493535\" class=\"howto\"><div data-type=\"title\">Add or subtract rational expressions.<\/div><ol id=\"fs-id1167836294894\" type=\"1\" class=\"stepwise\"><li>Determine if the expressions have a common denominator. <ul id=\"fs-id1167836508187\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Yes<\/strong> \u2013 go to step 2.<\/li><li><strong data-effect=\"bold\">No<\/strong> \u2013 Rewrite each rational expression with the LCD. <ul id=\"fs-id1167836609976\" data-bullet-style=\"bullet\"><li>Find the LCD.<\/li><li>Rewrite each rational expression as an equivalent rational expression with the LCD.<\/li><\/ul><\/li><\/ul><\/li><li>Add or subtract the rational expressions.<\/li><li>Simplify, if possible.<\/li><\/ol><\/div><p id=\"fs-id1167836524712\">Avoid the temptation to simplify too soon. In the example above, we must leave the first rational expression as \\(\\frac{3x-6}{\\left(x-3\\right)\\left(x-2\\right)}\\) to be able to add it to \\(\\frac{2x-6}{\\left(x-2\\right)\\left(x-3\\right)}.\\) Simplify <em data-effect=\"italics\">only<\/em> after you have combined the numerators.<\/p><div data-type=\"example\" id=\"fs-id1167836541935\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836296452\"><div data-type=\"problem\" id=\"fs-id1167836300182\"><p id=\"fs-id1167836409414\">Add: \\(\\frac{8}{{x}^{2}-2x-3}+\\frac{3x}{{x}^{2}+4x+3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829872177\"><table id=\"fs-id1167836289072\" class=\"unnumbered unstyled\" summary=\"Add the rational expressions 8 divided by the quantity x squared minus 2 x minus 3 and 3 x divided by the quantity x squared plus 4 x plus 3. Notice the denominators x squared minus 2 x minus 3 and x squared plus 4 x plus 3 do not have a common denominator. Rewrite each rational expression using the least common denominator. To find the least common denominator, factor x squared minus 2 x minus 3 and x squared plus 4 x plus 3, lining their common factors up in columns. The denominator x squared minus 2 x minus 3 is equal to the quantity x plus 1 times the quantity x minus 3. The denominator x squared plus 4 x plus 3 is equal to the quantity x plus 1 times the quantity x plus 3. The least common denominator is the product of each factor from each column, the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. Rewrite the rationale expression as an equivalent rational expression using the least common denominator, quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. The first denominator of the rational expression, 8 divided by the quantity x squared minus 2 x minus 3, can be factored into the quantity x plus 1 times the quantity x minus 3. Multiply the numerator and the denominator of the rational expression by the quantity x plus three to create the least common denominator. The equivalent rational expression is the quantity 8 times the quantity x plus one, divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. The first denominator of the rational expression, 3 x divided by the quantity x squared plus 4 x plus 3, can be factored into the quantity x plus 1 times the quantity x plus 3. Multiply the numerator and the denominator of the rational expression by the quantity x minus 3 to create the least common denominator. The equivalent rational expression is the quantity 3 x times the quantity x minus 1, divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. Simplify the numerators. The numerator 8 times the quantity x plus 1 can be simplified to 8 x plus 24. The numerator 3 x times the quantity x minus 1 can be simplified to 3 x squared minus 9 x. Now add the rational expression 8 x plus 24 divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3 and the rational expression 3 x squared minus 9 x divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. The result is the quantity 8 x plus the quantity 24 plus the quantity 3 x squared minus the quantity 9 x, divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. Simplify the numerator by combining like terms. The result is the quantity 3 x squared minus the quantity x plus the quantity 24, divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. The numerator is prime, so there are no common factors.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833385764\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Do the expressions have a common denominator?<\/td><td data-valign=\"top\" data-align=\"center\">No.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite each expression with the LCD.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{}\\\\ \\\\ \\text{Find the LCD.}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\phantom{\\rule{1.3em}{0ex}}{x}^{2}-2x-3=\\left(x+1\\right)\\left(x-3\\right)\\hfill \\\\ \\underset{______________________________}{{x}^{2}+4x+3=\\left(x+1\\right)\\phantom{\\rule{1em}{0ex}}\\left(x+3\\right)}\\phantom{\\rule{1em}{0ex}}\\hfill \\\\ \\\\ \\phantom{\\rule{3.75em}{0ex}}\\text{LCD}\\phantom{\\rule{0.2em}{0ex}}=\\left(x+1\\right)\\left(x-3\\right)\\left(x+3\\right)\\hfill \\end{array}\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<div data-type=\"newline\"><br><\/div>equivalent rational expression with the LCD.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836549343\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836526365\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add the rational expressions.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836447539\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The numerator is prime, so there are<div data-type=\"newline\"><br><\/div>no common factors.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826025196\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829719758\"><div data-type=\"problem\" id=\"fs-id1167833053581\"><p id=\"fs-id1167836487061\">Add: \\(\\frac{1}{{m}^{2}-m-2}+\\frac{5m}{{m}^{2}+3m+2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833057174\"><p id=\"fs-id1167836440102\">\\(\\frac{5{m}^{2}-9m+2}{\\left(m+1\\right)\\left(m-2\\right)\\left(m+2\\right)}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836618958\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824732931\"><div data-type=\"problem\" id=\"fs-id1167836625629\"><p id=\"fs-id1167829809775\">Add:\\(\\frac{2n}{{n}^{2}-3n-10}+\\frac{6}{{n}^{2}+5n+6}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824735756\"><p id=\"fs-id1167833335669\">\\(\\frac{2{n}^{2}+12n-30}{\\left(n+2\\right)\\left(n-5\\right)\\left(n+3\\right)}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836508157\">The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.<\/p><div data-type=\"example\" id=\"fs-id1167829620874\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829620876\"><div data-type=\"problem\" id=\"fs-id1167836295546\"><p id=\"fs-id1167833021096\">Subtract: \\(\\frac{8y}{{y}^{2}-16}-\\frac{4}{y-4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829739352\"><table id=\"fs-id1167836731518\" class=\"unnumbered unstyled\" summary=\"Subtract the rational expressions 8y divided by the quantity y squared minus 16 and 4 divided by the quantity y minus 4. Notice the denominators y squared minus 16 and y minus 4 do not have a common denominator. Rewrite each rational expression using the least common denominator. To find the least common denominator, factor y squared minus 16 and y minus 4, lining their common factors up in columns. The denominator y squared minus 16 is equal to the quantity y minus 4 times the quantity y plus 4. The denominator y minus 4 cannot be factored. The least common denominator is the product of each factor from each column, the quantity y minus 4 times the quantity y plus 4. Rewrite the rationale expression as an equivalent rational expression using the least common denominator, the quantity y minus 4 times the quantity y plus 4. The first denominator of the rational expression, the quantity 8 y divided by the quantity y squared minus 16 minus 4 divided by the quantity y plus 4, can be factored into the quantity y minus 4 times the quantity y plus 4, which is the least common denominator. The equivalent rational expression for 8 y divided by the quantity y squared minus 16 is 8 y divided by the quantity y minus 4 times the quantity y plus 4. Multiply the numerator and the denominator of the rational expression, 4 divided by the quantity y minus 4, by y plus 4 to write its equivalent rational expression. The result is 4 times the quantity y plus 4 all divided by the quantity y minus 4 times the quantity y plus 4. Simplify the numerators. Notice that the numerator of 8 y divided by the quantity y minus 4 times the quantity y plus 4 is already simplified. The numerator 4 times the quantity y plus 4 can be simplified to 4 y plus 16. Now subtract the rational expressions, 8 y divided by the quantity y minus 4 times the quantity y plus 4 and the quantity 4 y plus 16 all divided by the quantity y minus 4 times the quantity y plus 4. The result is the quantity 8 y minus 4 y minus 16 all divided by the quantity y minus 4 times the quantity y plus 4. Simplify the numerator by combining like terms. The result is the quantity 4 y minus 16 all divided by the quantity y minus 4 times the quantity y plus 4. Factor the numerator, 4 y minus 16, to look for common factors. 4 y minus 16 factors into 4 times the quantity y minus 4. Notice that y minus 4 is a common factor in the numerator and denominator so it can be removed. Once simplified, the result is 4 divided by the quantity y plus 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833021924\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Do the expressions have a common denominator?<\/td><td data-valign=\"top\" data-align=\"center\">No.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite each expression with the LCD.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{cccc}\\text{Find the LCD.}\\hfill &amp; &amp; &amp; \\begin{array}{c}{y}^{2}-16=\\left(y-4\\right)\\left(y+4\\right)\\hfill \\\\ \\phantom{\\rule{0.55em}{0ex}}\\underset{____________}{\\text{\u200b}y-4\\phantom{\\rule{0.2em}{0ex}}=y-4}\\hfill \\\\ \\phantom{\\rule{0.55em}{0ex}}\\text{LCD}\\phantom{\\rule{0.5em}{0ex}}=\\left(y-4\\right)\\left(y+4\\right)\\hfill \\end{array}\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<div data-type=\"newline\"><br><\/div>equivalent rational expression with the LCD.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824734398\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836364150\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract the rational expressions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829720811\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836608058\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the numerator to look for common factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836504084\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Remove common factors<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836415575\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836390532\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836732354\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836732358\"><div data-type=\"problem\"><p id=\"fs-id1167829614620\">Subtract: \\(\\frac{2x}{{x}^{2}-4}-\\frac{1}{x+2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829936504\"><p id=\"fs-id1167829936506\">\\(\\frac{1}{x-2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829717023\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829717026\"><div data-type=\"problem\" id=\"fs-id1167829893774\"><p id=\"fs-id1167829893776\">Subtract: \\(\\frac{3}{z+3}-\\frac{6z}{{z}^{2}-9}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829683638\"><p id=\"fs-id1167836691345\">\\(\\frac{-3}{z-3}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167833356152\">There are lots of negative signs in the next example. Be extra careful.<\/p><div data-type=\"example\" id=\"fs-id1167836508951\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836508953\"><div data-type=\"problem\" id=\"fs-id1167836363982\"><p id=\"fs-id1167836363984\">Subtract:\\(\\frac{-3n-9}{{n}^{2}+n-6}-\\frac{n+3}{2-n}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829593986\"><table id=\"fs-id1167829593989\" class=\"unnumbered unstyled\" summary=\"Subtract the rational expressions the quantity negative 3 n minus 9 all divided by the quantity n squared plus n minus 6 and the quantity n plus 3 divided by the quantity 2 minus n. Factor the denominator of the first rational expression, n squared plus n minus 6. The result is the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 minus the quantity n plus 3 divided by the quantity 2 minus n. Notice that n minus 2 and 2 minus n are opposites. Multiply the numerator and denominator of the second rational expression, n plus 3 divided by the quantity 2 minus n, by negative 1. Write its denominator, negative 1 times the quantity 2 minus n, as n minus 2. The result is the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 minus negative one times the quantity n plus 3 all divided by n minus 2. Simplify the numerator of the second rational expression, remembering that a minus negative b is a plus b. Write the numerator, negative times the quantity n plus 3, as n plus 3. The result is the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 minus the quantity n plus 3 all divided by n minus 2. Notice that the denominators do not have any common factors. Find the lowest common denominator. The denominator, n squared plus n minus 6, factors into the quantity n minus 2 times the quantity n plus 3. The denominator, n minus 2, cannot be factored. Line the denominators up by common factors. The least common denominator is the product of a factor from each column, the quantity n minus 2 times the quantity n plus 3. Rewrite each rational expression as an equivalent rational expression using the lowest common denominator. The first rational expression, the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 , is already written using the LCD. Multiply the numerator and denominator of the second rational expression, the quantity n plus 3 divided by the quantity n plus 2, by n plus 3. Simplify the numerators. The result is the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 minus the quantity n squared plus 6 n plus 9 all divided by the quantity n minus 2 times the quantity n plus 3. Add the rational expressions. The result is the quantity negative 3 n minus 9 plus n squared plus 6 n plus 9 all divided by the quantity n minus 2 times the quantity n plus 3. Simplify the numerator by combining like terms. The result is the quantity n squared plus 3 n all divided by the quantity n minus 2 times the quantity n plus 3. Factoring the numerator, results in n times the quantity n plus 3 all divided by the quantity n minus 2 times the quantity n plus 3. Notice that n plus 3 is a common factor in the numerator and denominator. Remove it and simplify. The result is n divided by the quantity n minus 2.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836626594\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(n-2\\) and \\(2-n\\) are opposites, we will<div data-type=\"newline\"><br><\/div>multiply the second rational expression by \\(\\frac{-1}{-1}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829586348\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836626907\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify. Remember, \\(a-\\left(\\text{\u2212}b\\right)=a+b.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833407431\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Do the rational expressions have a<div data-type=\"newline\"><br><\/div>common denominator? No.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{cccc}\\text{Find the LCD.}\\hfill &amp; &amp; &amp; \\begin{array}{c}{n}^{2}+n-6=\\left(n-2\\right)\\left(n+3\\right)\\hfill \\\\ \\phantom{\\rule{0.95em}{0ex}}\\underset{_________________}{n-2=\\left(n-2\\right)}\\hfill \\\\ \\phantom{\\rule{0.5em}{0ex}}\\text{LCD}\\phantom{\\rule{1.75em}{0ex}}=\\left(n-2\\right)\\left(n+3\\right)\\hfill \\end{array}\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<div data-type=\"newline\"><br><\/div>equivalent rational expression with the LCD. <\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836533932\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836606028\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add the rational expressions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836728972\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833049923\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the numerator to look for common factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836287998\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829908132\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836611144\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836611148\"><div data-type=\"problem\" id=\"fs-id1167836689139\"><p id=\"fs-id1167836689141\">Subtract :\\(\\frac{3x-1}{{x}^{2}-5x-6}-\\frac{2}{6-x}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829683969\"><p id=\"fs-id1167829683971\">\\(\\frac{5x+1}{\\left(x-6\\right)\\left(x+1\\right)}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833311026\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833412704\"><div data-type=\"problem\" id=\"fs-id1167833412706\"><p id=\"fs-id1167833412708\">Subtract: \\(\\frac{-2y-2}{{y}^{2}+2y-8}-\\frac{y-1}{2-y}.\\)<\/p><\/div><div data-type=\"solution\"><p>\\(\\frac{y+3}{y+4}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165927773319\">Things can get very messy when both fractions must be multiplied by a binomial to get the common denominator.<\/p><div data-type=\"example\" id=\"fs-id1167829905216\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167833381669\"><p id=\"fs-id1167833381671\">Subtract: \\(\\frac{4}{{a}^{2}+6a+5}-\\frac{3}{{a}^{2}+7a+10}.\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167824734719\" class=\"unnumbered unstyled\" summary=\"Subtract the rational expressions, 4 divided by the quantity a squared plus 6 a plus 5 and 3 divided by the quantity a squared plus 7 a plus 10. Factor each denominator. The result is 4 divided by the quantity a plus 1 times a plus 5 minus 3 divided by the quantity a plus 2 times a plus 5. The expressions do not have a common denominator. Line the denominators up by common factors in columns. The least common denominator is the product of the factors from each column, the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2. Rewrite each rational expression as an equivalent rational expression using the least common denominator. Multiply the numerator and denominator of the first rational expression by a plus 2. Multiply the numerator and denominator of the second rational expression by a plus 1. Simplifying each numerator, the result is the quantity 4 a plus 8 divided by the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2 minus 3 divided by the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2. Subtract the rational expressions. The result is the quantity 4 a plus 8 minus the quantity 3 a plus 3 all divided by the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2. Simplify the numerator by combining like terms. The result is the quantity a plus 5 all divided by the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2. Notice a plus 5 is a common factor in the numerator and denominator. Remove it and simplify. The result is 1 divided by the quantity a plus 1 times the quantity a plus 2.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829644596\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the denominators.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836662670\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Do the rational expressions have a<div data-type=\"newline\"><br><\/div>common denominator? No.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{cccc}\\text{Find the LCD.}\\hfill &amp; &amp; &amp; \\begin{array}{c}\\phantom{\\rule{0.1em}{0ex}}{a}^{2}+6a+5\\phantom{\\rule{0.75em}{0ex}}=\\left(a+1\\right)\\left(a+5\\right)\\hfill \\\\ \\phantom{\\rule{0.2em}{0ex}}\\underset{____________________________}{{a}^{2}+7a+10\\phantom{\\rule{0.2em}{0ex}}=\\phantom{\\rule{3em}{0ex}}\\left(a+5\\right)\\left(a+2\\right)}\\hfill \\\\ \\phantom{\\rule{3.7em}{0ex}}\\text{LCD}=\\left(a+1\\right)\\left(a+5\\right)\\left(a+2\\right)\\hfill \\end{array}\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<div data-type=\"newline\"><br><\/div>equivalent rational expression with the LCD.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836508350\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829844002\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract the rational expressions.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836665473\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832978265\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833047259\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Look for common factors.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836529296\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836392589\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836596323\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836612453\"><div data-type=\"problem\" id=\"fs-id1167836612456\"><p id=\"fs-id1167836692540\">Subtract: \\(\\frac{3}{{b}^{2}-4b-5}-\\frac{2}{{b}^{2}-6b+5}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836310949\"><p id=\"fs-id1167836310951\">\\(\\frac{1}{\\left(b+1\\right)\\left(b-1\\right)}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829741840\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836356000\"><div data-type=\"problem\" id=\"fs-id1167836356002\"><p id=\"fs-id1167836356004\">Subtract: \\(\\frac{4}{{x}^{2}-4}-\\frac{3}{{x}^{2}-x-2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833223781\"><p id=\"fs-id1167833223783\">\\(\\frac{1}{\\left(x+2\\right)\\left(x+1\\right)}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167829721097\">We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example, we will start by factoring all three denominators to find their LCD.<\/p><div data-type=\"example\" id=\"fs-id1167836510837\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836510840\"><div data-type=\"problem\" id=\"fs-id1167833386586\"><p id=\"fs-id1167833386588\">Simplify: \\(\\frac{2u}{u-1}+\\frac{1}{u}-\\frac{2u-1}{{u}^{2}-u}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826146770\"><table id=\"fs-id1167836556634\" class=\"unnumbered unstyled can-break\" summary=\"Simplify 2 u divided by the quantity u minus 1 plus 1 divided by u minus the quantity 2 u minus 1 all divided by u squared minus u. Notice that the expressions do not have any common denominators. Find the least common denominator. Notice that u minus 1 and u cannot be factored, but u squared minus u factors into u times the quantity u minus 1. The least common denominator is u times the quantity u minus 1. Rewrite each rational expression as an equivalent rational expression using the least common denominator. Multiply the numerator and denominator of the first rational expression 2 u divided the quantity u minus 1 by u. Multiply the numerator and denominator of the second rational expression, 1 divided by u, by u minus 1. The denominator of the third rational expression is already written using the least common denominator. Simplifying, the result is 2 u squared divided by the quantity u minus 1 times u plus the quantity u minus 1 divided by u times the quantity u minus 1 minus the quantity 2 u minus 1 divided by u times the quantity u minus 1. Write the expressions as one rational expression by adding and subtracting the numerators. The result is the quantity 2 u squared plus u minus 1 minus 2 u plus 1 all divided by u times the quantity u minus 1. Simplify by combining like terms in the numerator. The result is 2 u squared minus u all divided by u times the quantity u minus 1. Factor the numerator. The result is u times the quantity 2 u minus 1 all divided by u times the quantity u minus 1. Notice that u is a common factor in the numerator and denominator. Remove it and simplify. The result is the quantity 2 u minus 1 divided by the quantity u minus 1.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416633\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Do the expressions have a common denominator? No.<div data-type=\"newline\"><br><\/div>Rewrite each expression with the LCD.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{cccc}\\text{Find the LCD.}\\hfill &amp; &amp; &amp; \\begin{array}{c}u-1\\phantom{\\rule{0.65em}{0ex}}=\\phantom{\\rule{0.5em}{0ex}}\\left(u-1\\right)\\hfill \\\\ u\\phantom{\\rule{2.3em}{0ex}}=u\\hfill \\\\ \\underset{_______________}{{u}^{2}-u=u\\left(u-1\\right)}\\hfill \\\\ \\text{LCD}\\phantom{\\rule{0.8em}{0ex}}=u\\left(u-1\\right)\\hfill \\end{array}\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<div data-type=\"newline\"><br><\/div>equivalent rational expression with the LCD.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836531565\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836548004\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write as one rational expression.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833339385\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833350333\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the numerator, and remove<div data-type=\"newline\"><br><\/div>common factors.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836447474\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836521831\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833175384\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832945632\"><div data-type=\"problem\" id=\"fs-id1167832945634\"><p id=\"fs-id1167833356109\">Simplify: \\(\\frac{v}{v+1}+\\frac{3}{v-1}-\\frac{6}{{v}^{2}-1}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836298137\"><p id=\"fs-id1167836298139\">\\(\\frac{v+3}{v+1}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829784997\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836486350\"><div data-type=\"problem\" id=\"fs-id1167836486352\"><p>Simplify: \\(\\frac{3w}{w+2}+\\frac{2}{w+7}-\\frac{17w+4}{{w}^{2}+9w+14}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829811785\"><p id=\"fs-id1167829811787\">\\(\\frac{3w}{w+7}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829712902\"><h3 data-type=\"title\">Add and subtract rational functions<\/h3><p id=\"fs-id1167829843979\">To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829717361\"><div data-type=\"problem\"><p id=\"fs-id1167836557016\">Find \\(R\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)\\) where \\(f\\left(x\\right)=\\frac{x+5}{x-2}\\) and \\(g\\left(x\\right)=\\frac{5x+18}{{x}^{2}-4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836732726\"><table id=\"fs-id1167836732729\" class=\"unnumbered unstyled can-break\" summary=\"The function R is equal to the function f minus the function g. Substitute the quantity x plus 5 divided by the quantity x minus 2 for the function f and the quantity 5 x plus 18 divided by the quantity x squared minus 4. Factor the denominators. Notice that x minus 2 cannot be factored, but x squared minus 4 can be factored into the quantity x minus 2 times the quantity x plus 2. The result is the function R is equal to the quantity x plus 5 all divided by the quantity x minus 2 minus the quantity 5 x plus 18 all divided by the quantity x minus 2 times the quantity x plus 2. The expressions do have a common denominator. Rewrite the expressions as equivalent expressions using the least common denominator. Line the factors for each denominator up by common factors in columns. The least common denominator is the product of the factors for each column, the quantity x minus 2 times the quantity x plus 2. In the function R, multiply the numerator and denominator of the first rational expression by x plus 2. The second rational expression is already written with the least common denominator. Write the rational expressions as one. The result is the quantity x plus 5 times the quantity x plus 2 minus the quantity 5 x plus 18 all divided by the quantity x minus 2 times the quantity x plus 2. Simplify the numerator. The result is the quantity x squared plus 7 x plus 10 minus 5 x minus 18 all divided by the quantity x minus 2 times the quantity x plus 2, which further simplifies to the quantity x squared plus 2 x minus 8 all divided by the quantity x minus 2 times the quantity x plus 2. Factoring the numerator, the result is the quantity x plus 4 times the quantity x minus 2 all divided by the quantity x minus 2 times the quantity x plus 2. Notice that x minus 2 is a common factor in the numerator and denominator. Remove it, and then simplify. The result is R is equal to the quantity x plus 4 divided by the quantity x plus 2.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756084\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute in the functions \\(f\\left(x\\right),\\)\\(g\\left(x\\right).\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829788706\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the denominators.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829594854\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Do the expressions have a common denominator? No.<div data-type=\"newline\"><br><\/div>Rewrite each expression with the LCD.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{cccc}\\text{Find the LCD.}\\hfill &amp; &amp; &amp; \\begin{array}{c}x-2\\phantom{\\rule{0.5em}{0ex}}=\\left(x-2\\right)\\hfill \\\\ \\underset{___________________}{{x}^{2}-4=\\left(x-2\\right)\\left(x+2\\right)}\\hfill \\\\ \\phantom{\\rule{0.7em}{0ex}}\\text{LCD}=\\left(x-2\\right)\\left(x+2\\right)\\hfill \\end{array}\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<div data-type=\"newline\"><br><\/div>equivalent rational expression with the LCD.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836289631\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write as one rational expression.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547177\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833279815\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833407475\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the numerator, and remove<div data-type=\"newline\"><br><\/div>common factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832950920\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829702024\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829719644\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167836295476\">Find \\(R\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)\\) where \\(f\\left(x\\right)=\\frac{x+1}{x+3}\\) and \\(g\\left(x\\right)=\\frac{x+17}{{x}^{2}-x-12}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833340016\"><p id=\"fs-id1167833340018\">\\(\\frac{x-7}{x-4}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836534519\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836508918\"><p id=\"fs-id1167836508920\">Find \\(R\\left(x\\right)=f\\left(x\\right)+g\\left(x\\right)\\) where \\(f\\left(x\\right)=\\frac{x-4}{x+3}\\) and \\(g\\left(x\\right)=\\frac{4x+6}{{x}^{2}-9}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836528213\"><p id=\"fs-id1167836528215\">\\(\\frac{{x}^{2}-3x+18}{\\left(x+3\\right)\\left(x-3\\right)}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826102760\" class=\"media-2\"><p id=\"fs-id1167826102765\">Access this online resource for additional instruction and practice with adding and subtracting rational expressions.<\/p><ul id=\"fs-id1171792804576\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37AddSubRatExp\">Add and Subtract Rational Expressions- Unlike Denominators<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836518508\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167836391201\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Rational Expression Addition and Subtraction<\/strong><div data-type=\"newline\"><br><\/div> If <em data-effect=\"italics\">p<\/em>, <em data-effect=\"italics\">q<\/em>, and <em data-effect=\"italics\">r<\/em> are polynomials where \\(r\\ne 0,\\) then<div data-type=\"newline\"><br><\/div> \\(\\phantom{\\rule{8em}{0ex}}\\frac{p}{r}+\\frac{q}{r}=\\frac{p+q}{r}\\) and \\(\\frac{p}{r}-\\frac{q}{r}=\\frac{p-q}{r}\\)<\/li><li><strong data-effect=\"bold\">How to find the least common denominator of rational expressions.<\/strong><ol id=\"fs-id1167833071624\" type=\"1\" class=\"stepwise\"><li>Factor each expression completely.<\/li><li>List the factors of each expression. Match factors vertically when possible.<\/li><li>Bring down the columns.<\/li><li>Write the LCD as the product of the factors.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to add or subtract rational expressions.<\/strong><ol id=\"fs-id1167824735388\" type=\"1\" class=\"stepwise\"><li>Determine if the expressions have a common denominator. <ul id=\"fs-id1167833339764\" data-bullet-style=\"bullet\"><li>Yes \u2013 go to step 2.<\/li><li>No \u2013 Rewrite each rational expression with the LCD. <ul id=\"fs-id1167836688152\" data-bullet-style=\"bullet\"><li>Find the LCD.<\/li><li>Rewrite each rational expression as an equivalent rational expression with the LCD.<\/li><\/ul><\/li><\/ul><\/li><li>Add or subtract the rational expressions.<\/li><li>Simplify, if possible.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829811067\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829811070\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167836319644\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions with a Common Denominator<\/strong><\/p><p id=\"fs-id1167836609419\">In the following exercises, add.<\/p><div data-type=\"exercise\" id=\"fs-id1167836609422\"><div data-type=\"problem\" id=\"fs-id1167836328949\"><p id=\"fs-id1167836328951\">\\(\\frac{2}{15}+\\frac{7}{15}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836738177\"><p id=\"fs-id1167829579790\">\\(\\frac{3}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833339785\"><div data-type=\"problem\" id=\"fs-id1167833138215\"><p id=\"fs-id1167833138217\">\\(\\frac{7}{24}+\\frac{11}{24}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836508792\"><div data-type=\"problem\" id=\"fs-id1167836508794\"><p id=\"fs-id1167836619480\">\\(\\frac{3c}{4c-5}+\\frac{5}{4c-5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836440093\"><p id=\"fs-id1167836665068\">\\(\\frac{3c+5}{4c-5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836300389\"><div data-type=\"problem\" id=\"fs-id1167832940534\"><p id=\"fs-id1167832940536\">\\(\\frac{7m}{2m+n}+\\frac{4}{2m+n}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836521456\"><p id=\"fs-id1167836521458\">\\(\\frac{2{r}^{2}}{2r-1}+\\frac{15r-8}{2r-1}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833137093\"><p id=\"fs-id1167833066700\">\\(r+8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836440414\"><div data-type=\"problem\" id=\"fs-id1167836440612\"><p id=\"fs-id1167836440614\">\\(\\frac{3{s}^{2}}{3s-2}+\\frac{13s-10}{3s-2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829720099\"><div data-type=\"problem\" id=\"fs-id1167829720101\"><p id=\"fs-id1167829720104\">\\(\\frac{2{w}^{2}}{{w}^{2}-16}+\\frac{8w}{{w}^{2}-16}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836487053\"><p id=\"fs-id1167832936236\">\\(\\frac{2w}{w-4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832936278\"><div data-type=\"problem\" id=\"fs-id1167832936280\"><p id=\"fs-id1167833053565\">\\(\\frac{7{x}^{2}}{{x}^{2}-9}+\\frac{21x}{{x}^{2}-9}\\)<\/p><\/div><\/div><p id=\"fs-id1167836629242\">In the following exercises, subtract.<\/p><div data-type=\"exercise\" id=\"fs-id1167829590371\"><div data-type=\"problem\" id=\"fs-id1167829590373\"><p id=\"fs-id1167829590375\">\\(\\frac{9{a}^{2}}{3a-7}-\\frac{49}{3a-7}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836743433\"><p id=\"fs-id1167836743435\">\\(3a+7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829634217\"><div data-type=\"problem\" id=\"fs-id1167829634219\"><p id=\"fs-id1167829749871\">\\(\\frac{25{b}^{2}}{5b-6}-\\frac{36}{5b-6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833135505\"><div data-type=\"problem\" id=\"fs-id1167829743925\"><p id=\"fs-id1167829743927\">\\(\\frac{3{m}^{2}}{6m-30}-\\frac{21m-30}{6m-30}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836729645\"><p id=\"fs-id1167836662865\">\\(\\frac{m-2}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836535828\"><div data-type=\"problem\" id=\"fs-id1167836535830\"><p id=\"fs-id1167836535833\">\\(\\frac{2{n}^{2}}{4n-32}-\\frac{18n-16}{4n-32}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836697382\"><div data-type=\"problem\" id=\"fs-id1167836697384\"><p id=\"fs-id1167836697386\">\\(\\frac{6{p}^{2}+3p+4}{{p}^{2}+4p-5}-\\frac{5{p}^{2}+p+7}{{p}^{2}+4p-5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836727800\"><p id=\"fs-id1167836727803\">\\(\\frac{p+3}{p+5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829743945\"><div data-type=\"problem\" id=\"fs-id1167829743948\"><p id=\"fs-id1167829743950\">\\(\\frac{5{q}^{2}+3q-9}{{q}^{2}+6q+8}-\\frac{4{q}^{2}+9q+7}{{q}^{2}+6q+8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836664177\"><div data-type=\"problem\" id=\"fs-id1167836664179\"><p id=\"fs-id1167836664182\">\\(\\frac{5{r}^{2}+7r-33}{{r}^{2}-49}-\\frac{4{r}^{2}+5r+30}{{r}^{2}-49}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829808586\"><p id=\"fs-id1167829808588\">\\(\\frac{r+9}{r+7}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829732204\"><div data-type=\"problem\" id=\"fs-id1167829732206\"><p id=\"fs-id1167836630357\">\\(\\frac{7{t}^{2}-t-4}{{t}^{2}-25}-\\frac{6{t}^{2}+12t-44}{{t}^{2}-25}\\)<\/p><\/div><\/div><p id=\"fs-id1167836613288\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions whose Denominators are Opposites<\/strong><\/p><p id=\"fs-id1167833381688\">In the following exercises, add or subtract.<\/p><div data-type=\"exercise\" id=\"fs-id1167833381691\"><div data-type=\"problem\" id=\"fs-id1167836574708\"><p id=\"fs-id1167836574710\">\\(\\frac{10v}{2v-1}+\\frac{2v+4}{1-2v}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829589753\"><p id=\"fs-id1167829589755\">\\(4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836557054\"><div data-type=\"problem\" id=\"fs-id1167836557056\"><p id=\"fs-id1167829850189\">\\(\\frac{20w}{5w-2}+\\frac{5w+6}{2-5w}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833339004\"><div data-type=\"problem\" id=\"fs-id1167833339006\"><p id=\"fs-id1167833339008\">\\(\\frac{10{x}^{2}+16x-7}{8x-3}+\\frac{2{x}^{2}+3x-1}{3-8x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829806996\"><p id=\"fs-id1167829806998\">\\(x+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832926886\"><div data-type=\"problem\" id=\"fs-id1167832926888\"><p id=\"fs-id1167832926890\">\\(\\frac{6{y}^{2}+2y-11}{3y-7}+\\frac{3{y}^{2}-3y+17}{7-3y}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833097182\"><div data-type=\"problem\" id=\"fs-id1167833097184\"><p id=\"fs-id1167829743862\">\\(\\frac{{z}^{2}+6z}{{z}^{2}-25}-\\frac{3z+20}{25-{z}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833082458\"><p id=\"fs-id1167833129499\">\\(\\frac{z+4}{z-5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833047549\"><div data-type=\"problem\" id=\"fs-id1167833047551\"><p>\\(\\frac{{a}^{2}+3a}{{a}^{2}-9}-\\frac{3a-27}{9-{a}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829783563\"><div data-type=\"problem\" id=\"fs-id1167829579106\"><p id=\"fs-id1167829579108\">\\(\\frac{2{b}^{2}+30b-13}{{b}^{2}-49}-\\frac{2{b}^{2}-5b-8}{49-{b}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826025455\"><p id=\"fs-id1167826025457\">\\(\\frac{4b-3}{b-7}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824732175\"><div data-type=\"problem\" id=\"fs-id1167824732178\"><p id=\"fs-id1167824732180\">\\(\\frac{{c}^{2}+5c-10}{{c}^{2}-16}-\\frac{{c}^{2}-8c-10}{16-{c}^{2}}\\)<\/p><\/div><\/div><p id=\"fs-id1167836686848\"><strong data-effect=\"bold\">Find the Least Common Denominator of Rational Expressions<\/strong><\/p><p id=\"fs-id1167829614424\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the LCD for the given rational expressions <span class=\"token\">\u24d1<\/span> rewrite them as equivalent rational expressions with the lowest common denominator.<\/p><div data-type=\"exercise\" id=\"fs-id1167836576082\"><div data-type=\"problem\" id=\"fs-id1167836576084\"><p id=\"fs-id1167836576086\">\\(\\frac{5}{{x}^{2}-2x-8},\\frac{2x}{{x}^{2}-x-12}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836699862\"><p id=\"fs-id1167836699864\"><span class=\"token\">\u24d0<\/span>\\(\\left(x+2\\right)\\left(x-4\\right)\\left(x+3\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{5x+15}{\\left(x+2\\right)\\left(x-4\\right)\\left(x+3\\right)}\\),<div data-type=\"newline\"><br><\/div>\\(\\frac{2{x}^{2}+4x}{\\left(x+2\\right)\\left(x-4\\right)\\left(x+3\\right)}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829579658\"><div data-type=\"problem\" id=\"fs-id1167829789346\"><p id=\"fs-id1167829789349\">\\(\\frac{8}{{y}^{2}+12y+35},\\frac{3y}{{y}^{2}+y-42}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829740447\"><div data-type=\"problem\" id=\"fs-id1167833365223\"><p id=\"fs-id1167833365225\">\\(\\frac{9}{{z}^{2}+2z-8},\\frac{4z}{{z}^{2}-4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829828588\"><p id=\"fs-id1167829828590\"><span class=\"token\">\u24d0<\/span>\\(\\left(z-2\\right)\\left(z+4\\right)\\left(z-4\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{9z-36}{\\left(z-2\\right)\\left(z+4\\right)\\left(z-4\\right)}\\),<div data-type=\"newline\"><br><\/div>\\(\\frac{4{z}^{2}-8z}{\\left(z-2\\right)\\left(z+4\\right)\\left(z-4\\right)}\\)<p id=\"fs-id1167829714694\"><\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829714698\"><div data-type=\"problem\" id=\"fs-id1167829714701\"><p id=\"fs-id1167833386222\">\\(\\frac{6}{{a}^{2}+14a+45},\\frac{5a}{{a}^{2}-81}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829894030\"><div data-type=\"problem\" id=\"fs-id1167829894032\"><p id=\"fs-id1167832999769\">\\(\\frac{4}{{b}^{2}+6b+9},\\frac{2b}{{b}^{2}-2b-15}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829718970\"><p id=\"fs-id1167829718972\"><span class=\"token\">\u24d0<\/span>\\(\\left(b+3\\right)\\left(b+3\\right)\\left(b-5\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{4b-20}{\\left(b+3\\right)\\left(b+3\\right)\\left(b-5\\right)}\\),<div data-type=\"newline\"><br><\/div>\\(\\frac{2{b}^{2}+6b}{\\left(b+3\\right)\\left(b+3\\right)\\left(b-5\\right)}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829624352\"><div data-type=\"problem\" id=\"fs-id1167829624354\"><p id=\"fs-id1167829624357\">\\(\\frac{5}{{c}^{2}-4c+4},\\frac{3c}{{c}^{2}-7c+10}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833021546\"><div data-type=\"problem\" id=\"fs-id1167829609144\"><p id=\"fs-id1167829609146\">\\(\\frac{2}{3{d}^{2}+14d-5},\\frac{5d}{3{d}^{2}-19d+6}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833086390\"><p id=\"fs-id1167836535846\"><span class=\"token\">\u24d0<\/span>\\(\\left(d+5\\right)\\left(3d-1\\right)\\left(d-6\\right)\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{2d-12}{\\left(d+5\\right)\\left(3d-1\\right)\\left(d-6\\right)}\\),<div data-type=\"newline\"><br><\/div>\\(\\frac{5{d}^{2}+25d}{\\left(d+5\\right)\\left(3d-1\\right)\\left(d-6\\right)}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829828600\"><div data-type=\"problem\" id=\"fs-id1167829828602\"><p id=\"fs-id1167829828604\">\\(\\frac{3}{5{m}^{2}-3m-2},\\frac{6m}{5{m}^{2}+17m+6}\\)<\/p><\/div><\/div><p id=\"fs-id1167833255974\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions with Unlike Denominators<\/strong><\/p><p id=\"fs-id1167833224739\">In the following exercises, perform the indicated operations.<\/p><div data-type=\"exercise\" id=\"fs-id1167833224742\"><div data-type=\"problem\" id=\"fs-id1167833224744\"><p id=\"fs-id1167833224746\">\\(\\frac{7}{10{x}^{2}y}+\\frac{4}{15x{y}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836292506\"><p id=\"fs-id1167836292508\">\\(\\frac{21y+8x}{30{x}^{2}{y}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829833103\"><div data-type=\"problem\" id=\"fs-id1167829833106\"><p id=\"fs-id1167829833108\">\\(\\frac{1}{12{a}^{3}{b}^{2}}+\\frac{5}{9{a}^{2}{b}^{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829853067\"><div data-type=\"problem\" id=\"fs-id1167829853069\"><p id=\"fs-id1167829853071\">\\(\\frac{3}{r+4}+\\frac{2}{r-5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829596476\"><p id=\"fs-id1167829596479\">\\(\\frac{5r-7}{\\left(r+4\\right)\\left(r-5\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829651378\"><div data-type=\"problem\" id=\"fs-id1167829651380\"><p id=\"fs-id1167829651382\">\\(\\frac{4}{s-7}+\\frac{5}{s+3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833369992\"><div data-type=\"problem\" id=\"fs-id1167833369994\"><p id=\"fs-id1167833369996\">\\(\\frac{5}{3w-2}+\\frac{2}{w+1}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836409813\"><p id=\"fs-id1167836409815\">\\(\\frac{11w+1}{\\left(3w-2\\right)\\left(w+1\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829745143\"><div data-type=\"problem\" id=\"fs-id1167829745145\"><p id=\"fs-id1167829745147\">\\(\\frac{4}{2x+5}+\\frac{2}{x-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829585698\"><div data-type=\"problem\" id=\"fs-id1167829585701\"><p id=\"fs-id1167829585703\">\\(\\frac{2y}{y+3}+\\frac{3}{y-1}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833009019\"><p id=\"fs-id1167833009021\">\\(\\frac{2{y}^{2}+y+9}{\\left(y+3\\right)\\left(y-1\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829859736\"><div data-type=\"problem\" id=\"fs-id1167829859738\"><p id=\"fs-id1167829859740\">\\(\\frac{3z}{z-2}+\\frac{1}{z+5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833022388\"><div data-type=\"problem\"><p>\\(\\frac{5b}{{a}^{2}b-2{a}^{2}}+\\frac{2b}{{b}^{2}-4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833338764\"><p id=\"fs-id1167833338766\">\\(\\frac{b\\left(5b+10+2{a}^{2}\\right)}{{a}^{2}\\left(b-2\\right)\\left(b+2\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742750\"><div data-type=\"problem\"><p id=\"fs-id1167829742755\">\\(\\frac{4}{cd+3c}+\\frac{1}{{d}^{2}-9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829696169\"><div data-type=\"problem\" id=\"fs-id1167829696171\"><p id=\"fs-id1167829696173\">\\(\\frac{-3m}{3m-3}+\\frac{5m}{{m}^{2}+3m-4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829786207\"><p id=\"fs-id1167829786210\">\\(\\text{\u2212}\\frac{m}{m+4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829786229\"><div data-type=\"problem\"><p id=\"fs-id1167824732508\">\\(\\frac{8}{4n+4}+\\frac{6}{{n}^{2}-n-2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836585278\"><div data-type=\"problem\" id=\"fs-id1167836585280\"><p id=\"fs-id1167836585282\">\\(\\frac{3r}{{r}^{2}+7r+6}+\\frac{9}{{r}^{2}+4r+3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833022457\"><p id=\"fs-id1167833022459\">\\(\\frac{3\\left({r}^{2}+6r+18\\right)}{\\left(r+1\\right)\\left(r+6\\right)\\left(r+3\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833239647\"><div data-type=\"problem\" id=\"fs-id1167833239649\"><p id=\"fs-id1167833239651\">\\(\\frac{2s}{{s}^{2}+2s-8}+\\frac{4}{{s}^{2}+3s-10}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836606012\"><div data-type=\"problem\" id=\"fs-id1167836606014\"><p id=\"fs-id1167836606016\">\\(\\frac{t}{t-6}-\\frac{t-2}{t+6}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829783002\"><p id=\"fs-id1167829783006\">\\(\\frac{2\\left(7t-6\\right)}{\\left(t-6\\right)\\left(t+6\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829782601\"><div data-type=\"problem\" id=\"fs-id1167829782603\"><p id=\"fs-id1167833364714\">\\(\\frac{x-3}{x+6}-\\frac{x}{x+3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836574205\"><div data-type=\"problem\" id=\"fs-id1167836574207\"><p id=\"fs-id1167836574209\">\\(\\frac{5a}{a+3}-\\frac{a+2}{a+6}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836614555\"><p id=\"fs-id1167836614557\">\\(\\frac{4{a}^{2}+25a-6}{\\left(a+3\\right)\\left(a+6\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833129050\"><div data-type=\"problem\" id=\"fs-id1167833129052\"><p id=\"fs-id1167833129054\">\\(\\frac{3b}{b-2}-\\frac{b-6}{b-8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829689013\"><div data-type=\"problem\" id=\"fs-id1167829689015\"><p id=\"fs-id1167829689018\">\\(\\frac{6}{m+6}-\\frac{12m}{{m}^{2}-36}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167825885849\"><p id=\"fs-id1167825885851\">\\(\\frac{-6}{m-6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826129923\"><div data-type=\"problem\" id=\"fs-id1167826129925\"><p id=\"fs-id1167826129928\">\\(\\frac{4}{n+4}-\\frac{8n}{{n}^{2}-16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836768930\"><div data-type=\"problem\" id=\"fs-id1167836768932\"><p id=\"fs-id1167836768934\">\\(\\frac{-9p-17}{{p}^{2}-4p-21}-\\frac{p+1}{7-p}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836485767\"><p id=\"fs-id1167836485769\">\\(\\frac{p+2}{p+3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836485791\"><div data-type=\"problem\" id=\"fs-id1167836485793\"><p id=\"fs-id1167836485796\">\\(\\frac{-13q-8}{{q}^{2}+2q-24}-\\frac{q+2}{4-q}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167825703893\"><div data-type=\"problem\" id=\"fs-id1167825703895\"><p id=\"fs-id1167825703898\">\\(\\frac{-2r-16}{{r}^{2}+6r-16}-\\frac{5}{2-r}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833364221\"><p id=\"fs-id1167833364224\">\\(\\frac{3}{r-2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833364241\"><div data-type=\"problem\" id=\"fs-id1167833364243\"><p id=\"fs-id1167833364245\">\\(\\frac{2t-30}{{t}^{2}+6t-27}-\\frac{2}{3-t}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829783655\"><div data-type=\"problem\" id=\"fs-id1167829783657\"><p id=\"fs-id1167829783659\">\\(\\frac{2x+7}{10x-1}+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836439597\"><p id=\"fs-id1167836439599\">\\(\\frac{4\\left(8x+1\\right)}{10x-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833364483\"><div data-type=\"problem\" id=\"fs-id1167833364485\"><p id=\"fs-id1167833364487\">\\(\\frac{8y-4}{5y+2}-6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829808222\"><div data-type=\"problem\" id=\"fs-id1167829808224\"><p id=\"fs-id1167829808226\">\\(\\frac{3}{{x}^{2}-3x-4}-\\frac{2}{{x}^{2}-5x+4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829651393\"><p id=\"fs-id1167829651395\">\\(\\frac{x-5}{\\left(x-4\\right)\\left(x+1\\right)\\left(x-1\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836664672\"><div data-type=\"problem\"><p>\\(\\frac{4}{{x}^{2}-6x+5}-\\frac{3}{{x}^{2}-7x+10}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829831823\"><div data-type=\"problem\" id=\"fs-id1167829831825\"><p id=\"fs-id1167829831827\">\\(\\frac{5}{{x}^{2}+8x-9}-\\frac{4}{{x}^{2}+10x+9}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836440776\"><p id=\"fs-id1167836440778\">\\(\\frac{1}{\\left(x-1\\right)\\left(x+1\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836440815\"><div data-type=\"problem\" id=\"fs-id1167836440817\"><p id=\"fs-id1167836440819\">\\(\\frac{3}{2{x}^{2}+5x+2}-\\frac{1}{2{x}^{2}+3x+1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833085118\"><div data-type=\"problem\" id=\"fs-id1167833085120\"><p id=\"fs-id1167833085122\">\\(\\frac{5a}{a-2}+\\frac{9}{a}-\\frac{2a+18}{{a}^{2}-2a}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167825703513\"><p id=\"fs-id1167825703516\">\\(\\frac{5{a}^{2}+7a-36}{a\\left(a-2\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167825703558\"><div data-type=\"problem\" id=\"fs-id1167825703560\"><p id=\"fs-id1167825703562\">\\(\\frac{2b}{b-5}+\\frac{3}{2b}-\\frac{2b-15}{2{b}^{2}-10b}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826130332\"><div data-type=\"problem\" id=\"fs-id1167826130334\"><p id=\"fs-id1167826130336\">\\(\\frac{c}{c+2}+\\frac{5}{c-2}-\\frac{10c}{{c}^{2}-4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830095264\"><p id=\"fs-id1167830095266\">\\(\\frac{c-5}{c+2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830095289\"><div data-type=\"problem\" id=\"fs-id1167830095291\"><p id=\"fs-id1167830095293\">\\(\\frac{6d}{d-5}+\\frac{1}{d+4}-\\frac{7d-5}{{d}^{2}-d-20}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829829429\"><div data-type=\"problem\" id=\"fs-id1167829829432\"><p id=\"fs-id1167833364561\">\\(\\frac{3d}{d+2}+\\frac{4}{d}-\\frac{d+8}{{d}^{2}+2d}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833364611\"><p id=\"fs-id1167833364614\">\\(\\frac{3\\left(d+1\\right)}{d+2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829782365\"><div data-type=\"problem\" id=\"fs-id1167829782367\"><p id=\"fs-id1167829782370\">\\(\\frac{2q}{q+5}+\\frac{3}{q-3}-\\frac{13q+15}{{q}^{2}+2q-15}\\)<\/p><\/div><\/div><p id=\"fs-id1167829650515\"><strong data-effect=\"bold\">Add and Subtract Rational Functions<\/strong><\/p><p id=\"fs-id1167829650521\">In the following exercises, find <span class=\"token\">\u24d0<\/span> \\(R\\left(x\\right)=f\\left(x\\right)+g\\left(x\\right)\\) <span class=\"token\">\u24d1<\/span> \\(R\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right).\\)<\/p><div data-type=\"exercise\" id=\"fs-id1167826129980\"><div data-type=\"problem\" id=\"fs-id1167826129983\"><p id=\"fs-id1167826129985\">\\(f\\left(x\\right)=\\frac{-5x-5}{{x}^{2}+x-6}\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{1.8em}{0ex}}g\\left(x\\right)=\\frac{x+1}{2-x}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836399038\"><p id=\"fs-id1167836399040\"><span class=\"token\">\u24d0<\/span>\\(R\\left(x\\right)=-\\frac{\\left(x+8\\right)\\left(x+1\\right)}{\\left(x-2\\right)\\left(x+3\\right)}\\)<span class=\"token\">\u24d1<\/span>\\(R\\left(x\\right)=\\frac{x+1}{x+3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833256034\"><div data-type=\"problem\"><p id=\"fs-id1167833256039\">\\(f\\left(x\\right)=\\frac{-4x-24}{{x}^{2}+x-30}\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{1.8em}{0ex}}g\\left(x\\right)=\\frac{x+7}{5-x}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833369660\"><div data-type=\"problem\" id=\"fs-id1167833369662\"><p id=\"fs-id1167833369664\">\\(f\\left(x\\right)=\\frac{6x}{{x}^{2}-64}\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{1.8em}{0ex}}g\\left(x\\right)=\\frac{3}{x-8}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167833369724\"><p id=\"fs-id1167833369726\"><span class=\"token\">\u24d0<\/span>\\(\\frac{3\\left(3x+8\\right)}{\\left(x-8\\right)\\left(x+8\\right)}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(R\\left(x\\right)=\\frac{3}{x+8}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829651530\"><div data-type=\"problem\" id=\"fs-id1167829651532\"><p id=\"fs-id1167829651534\">\\(f\\left(x\\right)=\\frac{5}{x+7}\\) and<\/p><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{1.8em}{0ex}}g\\left(x\\right)=\\frac{10x}{{x}^{2}-49}\\)<\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836570221\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167836570229\"><div data-type=\"problem\" id=\"fs-id1167836570231\"><p id=\"fs-id1167836570233\">Donald thinks that \\(\\frac{3}{x}+\\frac{4}{x}\\) is \\(\\frac{7}{2x}.\\) Is Donald correct? Explain.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836570267\"><p id=\"fs-id1167836570269\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836570275\"><div data-type=\"problem\" id=\"fs-id1167836570277\"><p id=\"fs-id1167836570279\">Explain how you find the Least Common Denominator of \\({x}^{2}+5x+4\\) and \\({x}^{2}-16.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833086811\"><div data-type=\"problem\" id=\"fs-id1167833086813\"><p id=\"fs-id1167833086815\">Felipe thinks \\(\\frac{1}{x}+\\frac{1}{y}\\) is \\(\\frac{2}{x+y}.\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> Choose numerical values for <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> and evaluate \\(\\frac{1}{x}+\\frac{1}{y}.\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Evaluate \\(\\frac{2}{x+y}\\) for the same values of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> you used in part <span class=\"token\">\u24d0<\/span>.<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Explain why Felipe is wrong.<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> Find the correct expression for \\(\\frac{1}{x}+\\frac{1}{y}.\\)<\/div><div data-type=\"solution\" id=\"fs-id1167829651079\"><p id=\"fs-id1167829651081\"><span class=\"token\">\u24d0<\/span> Answers will vary.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Answers will vary.<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Answers will vary.<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d3<\/span> \\(\\frac{x+y}{xy}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836510855\"><div data-type=\"problem\" id=\"fs-id1167836510857\"><p id=\"fs-id1167836510860\">Simplify the expression \\(\\frac{4}{{n}^{2}+6n+9}-\\frac{1}{{n}^{2}-9}\\) and explain all your steps.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836510912\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167836510917\"><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-id1167836510932\" data-alt=\"This table has four columns and six 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 add and subtract rational expressions with a common denominator. In row 3, the I can was add and subtract rational expressions with denominators that are opposites. In row 4, the I can find the least common denominator of rational expressions. In row 5, the I can was add and subtract rational expressions with unlike denominators. In row 6, the I can was add or subtract rational functions. There is the nothing in the other columns.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and six 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 add and subtract rational expressions with a common denominator. In row 3, the I can was add and subtract rational expressions with denominators that are opposites. In row 4, the I can find the least common denominator of rational expressions. In row 5, the I can was add and subtract rational expressions with unlike denominators. In row 6, the I can was add or subtract rational functions. There is the nothing in the other columns.\"><\/span><p id=\"fs-id1167836510928\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p><\/div><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Add and subtract rational expressions with a common denominator<\/li>\n<li>Add and subtract rational expressions whose denominators are opposites<\/li>\n<li>Find the least common denominator of rational expressions<\/li>\n<li>Add and subtract rational expressions with unlike denominators<\/li>\n<li>Add and subtract rational functions<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836622513\" class=\"be-prepared\">\n<p id=\"fs-id1167829644630\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167836605367\" type=\"1\">\n<li>Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d30bb4bb4aae25b3f3833ab19c9d589_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"59\" style=\"vertical-align: -7px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836553755\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-851dce11994345f2454da38d1118f5ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"53\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836518722\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51120878c84644c1ad7aed8d68777bde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#52;&#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=\"170\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/987da0d0-2366-47d6-aa25-904e24991866#fs-id1167829791835\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836516393\">\n<h3 data-type=\"title\">Add and Subtract Rational Expressions with a Common Denominator<\/h3>\n<p id=\"fs-id1167833328016\">What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.<\/p>\n<p id=\"fs-id1167836306628\">It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836628310\">\n<div data-type=\"title\">Rational Expression Addition and Subtraction<\/div>\n<p id=\"fs-id1167836514001\">If <em data-effect=\"italics\">p<\/em>, <em data-effect=\"italics\">q<\/em>, and <em data-effect=\"italics\">r<\/em> are polynomials where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39ebc9dc80feddbfd4064c205be07e25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/> then<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836312458\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d60a9b3a596269df5657433bc1910a2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#114;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#114;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#43;&#113;&#125;&#123;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#114;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#114;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#45;&#113;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"251\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p>To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result over the common denominator.<\/p>\n<p id=\"fs-id1167836551229\">We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.<\/p>\n<p>Remember, too, we do not allow values that would make the denominator zero. What value of <em data-effect=\"italics\">x<\/em> should be excluded in the next example?<\/p>\n<div data-type=\"example\" id=\"fs-id1167836544834\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829690445\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce9d925ea5f494bdfffbbb34ecccc4c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#120;&#43;&#50;&#56;&#125;&#123;&#120;&#43;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#120;&#43;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"103\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829908743\">\n<p id=\"fs-id1167836524115\">Since the denominator is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e99b50d3eea4e9f83417d1d8a3c75cdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"45\" style=\"vertical-align: -4px;\" \/> we must exclude the value <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de80293e0b5a1d0b98c28c8b715d136d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1167836689465\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08ce3ca0a3e08ff1d75ef5ae3ff2785d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#120;&#43;&#50;&#56;&#125;&#123;&#120;&#43;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#120;&#43;&#52;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#110;&#101;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#115;&#32;&#104;&#97;&#118;&#101;&#32;&#97;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#32;&#97;&#100;&#100;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#115;&#32;&#97;&#110;&#100;&#32;&#112;&#108;&#97;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#117;&#109;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#118;&#101;&#114;&#32;&#116;&#104;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#120;&#43;&#50;&#56;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#120;&#43;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#103;&#114;&#101;&#101;&#115;&#32;&#105;&#110;&#32;&#100;&#101;&#115;&#99;&#101;&#110;&#100;&#105;&#110;&#103;&#32;&#111;&#114;&#100;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#43;&#50;&#56;&#125;&#123;&#120;&#43;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#43;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#98;&#121;&#32;&#114;&#101;&#109;&#111;&#118;&#105;&#110;&#103;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#120;&#43;&#52;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#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;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#43;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"370\" width=\"671\" style=\"vertical-align: -179px;\" \/><\/p>\n<p>The expression simplifies to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc75694098649a22421bc1ec912c0b09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: -2px;\" \/> but the original expression had a denominator of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cd392abb5a6fd3a660e3b5912589956_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/> so <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>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836519066\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836524224\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-435eaad530d8d3ef27aeef8787b2eac2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#120;&#43;&#49;&#52;&#125;&#123;&#120;&#43;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#120;&#43;&#55;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"96\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836393292\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836393168\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836694802\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2de0d6b9ea7bac7662dd7eefe1e1ac6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#125;&#123;&#120;&#43;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#120;&#43;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"97\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836448476\">\n<p id=\"fs-id1167836286266\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e012de22392b339bc76f22e4ddf59f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836791982\">To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator. Be careful of the signs when you subtract a binomial or trinomial.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836503887\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p>Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-099de3752b8e3d54df3ee265856b462d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#57;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"160\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836615100\">\n<p id=\"fs-id1167829746839\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18a1d7b0db4ea1245bbb41d0f6c36909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#57;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#56;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#115;&#32;&#97;&#110;&#100;&#32;&#112;&#108;&#97;&#99;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#118;&#101;&#114;&#32;&#116;&#104;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#56;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#115;&#116;&#114;&#105;&#98;&#117;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#105;&#103;&#110;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#51;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#43;&#57;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#49;&#56;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#109;&#98;&#105;&#110;&#101;&#32;&#108;&#105;&#107;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#49;&#56;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#32;&#97;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#98;&#121;&#32;&#114;&#101;&#109;&#111;&#118;&#105;&#110;&#103;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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=\"475\" width=\"581\" style=\"vertical-align: -233px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836386277\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836293673\">\n<div data-type=\"problem\" id=\"fs-id1167836697800\">\n<p>Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bfd9a1bc84e27bf336e67af27c5fb23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#120;&#43;&#56;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"160\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836481451\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41bb5cce20bcd6c0df4d812f8fa7b277_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#49;&#49;&#125;&#123;&#120;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"33\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836515050\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836516618\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829695628\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4bd8f2417f20f038bacc1b268fe93bc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#43;&#50;&#48;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#45;&#55;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"167\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824733285\">\n<p id=\"fs-id1167829788807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-805cf0bcb22d145a673929a5430ae93d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#51;&#125;&#123;&#120;&#43;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"26\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Add and Subtract Rational Expressions Whose Denominators are Opposites<\/h3>\n<p id=\"fs-id1167829718460\">When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e72c0eef54be1191c4a5f9feaced8dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#125;&#123;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"23\" style=\"vertical-align: -7px;\" \/><\/p>\n<p id=\"fs-id1167833051123\">Let\u2019s see how this works.<\/p>\n<table id=\"fs-id1167836662756\" class=\"unnumbered unstyled\" summary=\"7 divided by d plus 5 divided by negative d. Multiply 5 divided by d times the fraction, negative 1 over negative 1. The result is 7 divided by d plus the quantity negative 1 times 5 all divided by the quantity negative 1times negative d. Notice that the denominators are the same. The denominators are d. 7 divided by d plus negative 5 divided by d. Simplify by adding the numerators. The result is 2 divided by d.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836319593\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the second fraction by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e72c0eef54be1191c4a5f9feaced8dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#125;&#123;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"23\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829746072\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The denominators are the same.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836296174\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165927912335\">Be careful with the signs as you work with the opposites when the fractions are being subtracted.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836728843\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829692333\">\n<div data-type=\"problem\">\n<p>Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2fe7b49f2e6a245e9d80c352ec0984b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#109;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#109;&#43;&#50;&#125;&#123;&#49;&#45;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"117\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836360555\">\n<table id=\"fs-id1167833339440\" class=\"unnumbered unstyled can-break\" summary=\"The rational expression the quantity m squared minus 6 divided by the quantity m squared minus 1 minus the rational expression the quantity 3 m plus 2 divided by the quantity 1 minus m squared. The denominators m squared minus 1 and 1 minus m squared are opposites so multiply the second rational expression by negative 1 over negative 1. The result is the quantity m squared minus 6 divided by the quantity m squared minus 1 minus negative 1 times the quantity 3 m plus 2 divided by negative 1 times the quantity 1 minus m squared. Simplify the second rational expression. The result is the rational expression, the quantity m squared minus 6 divided by the quantity m squared minus 1 minus the rational expression, the quantity negative 3 m minus 2 divided by the quantity m squared minus 1. The denominators of both rational expressions are the same, m squared minus 1, so subtract the numerators. The result is m squared minus 6 m minus the quantity negative 3 m minus 2 all divided by the quantity m squared minus 1. Combine the like terms in the numerator. The result is the quantity m squared minus 3 m plus 2 all divided by m squared minus 1. The numerator, the quantity m squared minus 3 m plus 2, factors into the quantity m minus 1 times the quantity m minus 2, and the denominator, m squared minus 1, factors into the quantity m minus 1 times the quantity m plus 1. The result is the quantity m minus 1 times the quantity m minus 2 all divided by the quantity m minus 1 times the quantity m plus 1. Simplify the expression by removing the common factor, m minus 1, from the numerator and denominator. The result is the quantity m minus 2 divided by the quantity m plus 1.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The denominators are opposites, so multiply the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>second fraction by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e72c0eef54be1191c4a5f9feaced8dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#125;&#123;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"23\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836530198\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the second fraction.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836376367\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The denominators are the same. Subtract the numerators.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826170180\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1165927925224\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829809610\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the numerator and denominator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836609915\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify by removing common factors.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836550878\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836620617\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_002h_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-id1167836558518\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836362962\">\n<div data-type=\"problem\" id=\"fs-id1167829717704\">\n<p id=\"fs-id1167836418052\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a26c9ab29458f086558ce62c08bdcd3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#121;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#121;&#45;&#54;&#125;&#123;&#52;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"102\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-debc34f2adfcad2f83168ae3d1ad3a97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#51;&#125;&#123;&#121;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"25\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836728750\">\n<div data-type=\"problem\" id=\"fs-id1167836699650\">\n<p id=\"fs-id1167836558293\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fdb75ffc63872118f727876796f8916_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#110;&#45;&#49;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#110;&#45;&#49;&#125;&#123;&#49;&#45;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"156\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836309417\">\n<p id=\"fs-id1167829936836\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4694708cce18b58002d0499bd76aa1f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#110;&#45;&#50;&#125;&#123;&#110;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"33\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836288812\">\n<h3 data-type=\"title\">Find the Least Common Denominator of Rational Expressions<\/h3>\n<p id=\"fs-id1167836652759\">When we add or subtract rational expressions with unlike denominators, we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.<\/p>\n<p id=\"fs-id1167836399264\">Let\u2019s look at this example: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41d8215aa80e5a9edb60969e59a1919f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"59\" style=\"vertical-align: -7px;\" \/> Since the denominators are not the same, the first step was to find the least common denominator (LCD).<\/p>\n<p>To find the LCD of the fractions, we factored 12 and 18 into primes, lining up any common primes in columns. Then we \u201cbrought down\u201d one prime from each column. Finally, we multiplied the factors to find the LCD.<\/p>\n<p id=\"fs-id1167836630479\">When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836299691\" data-alt=\"Seven-twelfths plus five-eighteenths. Write the prime factorizations of each denominator and line up the common factors. The denominator of the first fraction is 12. The prime factorization of 12 is 2 times 2 times 3. The denominator of the second fraction is 18. The prime factorization of 18 is 2 times 3 times 3. Bringing down a factor from each column, the lowest common denominator of 12 and 18 is 2 times 2 times 3 times 3, which is 36. Write both fractions using the lowest common denominator. To do this multiply the numerator and denominator of the first fraction by 3 and multiply the numerator and denominator of the second fraction by 2. The result is 7 times 3 all divided by 12 times 3 plus 5 times 2 all divided by 18 times 2. Simplify each fraction. 7 times 3 is 21 and 12 times 3 is 36. 5 times 2 is 10 and 18 times 2 is 36. The result is twenty-one thirty-sixths plus ten thirty-sixths.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Seven-twelfths plus five-eighteenths. Write the prime factorizations of each denominator and line up the common factors. The denominator of the first fraction is 12. The prime factorization of 12 is 2 times 2 times 3. The denominator of the second fraction is 18. The prime factorization of 18 is 2 times 3 times 3. Bringing down a factor from each column, the lowest common denominator of 12 and 18 is 2 times 2 times 3 times 3, which is 36. Write both fractions using the lowest common denominator. To do this multiply the numerator and denominator of the first fraction by 3 and multiply the numerator and denominator of the second fraction by 2. The result is 7 times 3 all divided by 12 times 3 plus 5 times 2 all divided by 18 times 2. Simplify each fraction. 7 times 3 is 21 and 12 times 3 is 36. 5 times 2 is 10 and 18 times 2 is 36. The result is twenty-one thirty-sixths plus ten thirty-sixths.\" \/><\/span><\/p>\n<p id=\"fs-id1167829580440\">We do the same thing for rational expressions. However, we leave the LCD in factored form.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833239730\" class=\"howto\">\n<div data-type=\"title\">Find the least common denominator of rational expressions.<\/div>\n<ol type=\"1\" class=\"stepwise\">\n<li>Factor each denominator completely.<\/li>\n<li>List the factors of each denominator. Match factors vertically when possible.<\/li>\n<li>Bring down the columns by including all factors, but do not include common factors twice.<\/li>\n<li>Write the LCD as the product of the factors.<\/li>\n<\/ol>\n<\/div>\n<p>Remember, we always exclude values that would make the denominator zero. What values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> should we exclude in this next example?<\/p>\n<div data-type=\"example\" id=\"fs-id1167836331076\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833386366\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836612766\"><span class=\"token\">\u24d0<\/span> Find the LCD for the expressions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a571945d0802b123a9a9281359b845b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"128\" style=\"vertical-align: -9px;\" \/> and <span class=\"token\">\u24d1<\/span> rewrite them as equivalent rational expressions with the lowest common denominator.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829694661\">\n<p id=\"fs-id1167836493853\"><span class=\"token\">\u24d0<\/span><\/p>\n<table id=\"fs-id1167826130067\" class=\"unnumbered unstyled\" summary=\"Find the lowest common denominator of the rational expressions 8 divided by the quantity x squared minus 2 x minus 3 and 3 x divided by the quantity x squared plus 4 x plus 3. Line up the denominators and factor them completely. Line up the common factors. The denominator x squared minus 2 x minus 3 is equal to the quantity x plus 1 times the quantity x minus 3. The denominator x squared plus 4 x plus 3 is equal to the quantity x plus 1 times the quantity x plus 3. The quantity x plus 1 is a factor of both denominators. The quantity x minus 3 is only a factor of the denominator x squared minus 2 x minus 3, so it is in a column alone. The quantity x plus 3 is only a factor of the denominator x squared plus 4 x plus 3, so it is in a column alone. The lowest common denominator is the product of the primes from each column, the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the LCD for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c2485e94b0364a861194267eb9fa41b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"132\" style=\"vertical-align: -9px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor each denominator completely, lining up common factors.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Bring down the columns.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the LCD as the product of the factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829594568\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836287401\"><span class=\"token\">\u24d1<\/span><\/p>\n<table id=\"fs-id1167836692693\" class=\"unnumbered unstyled\" summary=\"Factor the denominators of the rational expressions, 8 divided by the quantity x squared minus 2 x minus 3 and 3 x divided by the quantity x squared plus 4 x plus 3. The denominator, x squared minus 2 x minus 3, factors into the quantity x plus 1 times the quantity x minus 3. The denominator, x squared plus 4 x plus 3, factors into the quantity x plus 1 times the quantity x plus 3. Write each rational expression with a factored denominator. The first rational expression is 8 divided by the quantity x plus 1 times the quantity x minus 3. The second rational expression is 3 x divided by the quantity x plus 1 times the quantity x plus 3. Multiply the numerator and denominator of each expression by the missing lowest common denominator factor. The results are 8 times the quantity x plus 3 all divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3 and the quantity 3 x times the quantity x minus 3 all divided by the quantity x plus 1 times the quantity x plus 3 times the quantity x minus 3. Simplify the numerators of the rational expressions and keep the denominators. 8 times the quantity x plus 3 is 8 x plus 24. 3 x times the quantity x minus 3 is 3 x squared minus 9. The results are 8 times the quantity x plus 3 all divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3 and the quantity 3 x squared minus 9 x all divided by the quantity x plus 1 times x plus 3 times the quantity x minus 3.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor each denominator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836532070\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply each denominator by the \u2018missing\u2019<\/p>\n<div data-type=\"newline\"><\/div>\n<p>LCD factor and multiply each numerator by the same factor.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836512927\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836527650\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_005d_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\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836524675\">\n<div data-type=\"problem\" id=\"fs-id1167836288975\">\n<p id=\"fs-id1167829691761\"><span class=\"token\">\u24d0<\/span> Find the LCD for the expressions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-210b78dc5cbf62881c96f4fcbee4311e_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;&#45;&#120;&#45;&#49;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"109\" style=\"vertical-align: -8px;\" \/> <span class=\"token\">\u24d1<\/span> rewrite them as equivalent rational expressions with the lowest common denominator.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829712961\">\n<p id=\"fs-id1167833021001\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a04ef37280210f8d73b3ac0751802159_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"168\" 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-7b026d6d2913f6418419a6ae90291a1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#43;&#56;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"109\" style=\"vertical-align: -9px;\" \/>,<\/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-bb37d6dd70b3125c7f236e56382037f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#51;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"109\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836683527\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836362198\">\n<div data-type=\"problem\" id=\"fs-id1167836534817\">\n<p id=\"fs-id1167836619996\"><span class=\"token\">\u24d0<\/span> Find the LCD for the expressions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-611951094fc86e01336eadc191be3a84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#48;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"134\" style=\"vertical-align: -9px;\" \/> <span class=\"token\">\u24d1<\/span> rewrite them as equivalent rational expressions with the lowest common denominator.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833058297\">\n<p id=\"fs-id1167836321317\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfdc1f86300bc7b92be93a5a35c6c59a_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;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"168\" 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-3678792feb581bbf0d9f10db051fac2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"109\" style=\"vertical-align: -9px;\" \/>,<\/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-949d8ddcbffa6ba60b30f0f2c76aeffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#45;&#50;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"109\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832936445\">\n<h3 data-type=\"title\">Add and Subtract Rational Expressions with Unlike Denominators<\/h3>\n<p id=\"fs-id1167833378257\">Now we have all the steps we need to add or subtract rational expressions with unlike denominators.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836730276\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Add Rational Expressions with Unlike Denominators<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836439678\">\n<div data-type=\"problem\" id=\"fs-id1167836578933\">\n<p id=\"fs-id1167833396962\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71c451a1ebe990224a30ae8557cb33d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#45;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829753077\"><span data-type=\"media\" id=\"fs-id1167836622324\" data-alt=\"Step 1 is to determine if the rational expressions 3 divided by the quantity x minus 3 and 2 divided by the quantity x minus 2 have a common factors. The denominators x minus 3 and x minus 2 do not have any common factors, which means the lowest common denominator of the rational expressions is the quantity x minus 3 times the quantity x minus 2. Rewrite each rational expression with the least common denominator. Multiply the numerator and denominator of 3 divided by the quantity x minus 3 by the quantity x minus 2. Multiply the numerator and denominator of 2 divided by the quantity x minus 2 by the quantity x minus 2. The result is the rational expression 3 times the quantity x minus 2 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression 2 times the quantity x minus 3 divided by the quantity x minus 2 times the quantity x minus 3. Simplify the numerators and keep the denominators factored. The numerator of the first rational expression, 3 times the quantity x minus 2, simplifies to 3 x minus 6. The numerator of the second rational expression, 2 times the quantity x minus 3, simplifies to 2 x minus 6. The result is the rational expression the quantity 3 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression, the quantity 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_006a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to determine if the rational expressions 3 divided by the quantity x minus 3 and 2 divided by the quantity x minus 2 have a common factors. The denominators x minus 3 and x minus 2 do not have any common factors, which means the lowest common denominator of the rational expressions is the quantity x minus 3 times the quantity x minus 2. Rewrite each rational expression with the least common denominator. Multiply the numerator and denominator of 3 divided by the quantity x minus 3 by the quantity x minus 2. Multiply the numerator and denominator of 2 divided by the quantity x minus 2 by the quantity x minus 2. The result is the rational expression 3 times the quantity x minus 2 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression 2 times the quantity x minus 3 divided by the quantity x minus 2 times the quantity x minus 3. Simplify the numerators and keep the denominators factored. The numerator of the first rational expression, 3 times the quantity x minus 2, simplifies to 3 x minus 6. The numerator of the second rational expression, 2 times the quantity x minus 3, simplifies to 2 x minus 6. The result is the rational expression the quantity 3 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2 plus the rational expression, the quantity 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829746788\" data-alt=\"Step 2 is to add or subtract the rational expressions by adding the numerators, the quantity 3 x minus 6 and the quantity 2 x minus 6, and placing the sum over the denominator, the quantity x minus 3 times the quantity x minus 2. The result is the quantity 3 x minus 6 plus 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2. Simplify the numerator by combining like terms. The result is the quantity 5 x minus 12 all divided by the quantity x minus 3 times the quantity x minus 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_006b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to add or subtract the rational expressions by adding the numerators, the quantity 3 x minus 6 and the quantity 2 x minus 6, and placing the sum over the denominator, the quantity x minus 3 times the quantity x minus 2. The result is the quantity 3 x minus 6 plus 2 x minus 6 all divided by the quantity x minus 3 times the quantity x minus 2. Simplify the numerator by combining like terms. The result is the quantity 5 x minus 12 all divided by the quantity x minus 3 times the quantity x minus 2.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836728342\" data-alt=\"Step 3. Notice that 5 x minus 12 cannot be factored, so the answer is simplified.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_006c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. Notice that 5 x minus 12 cannot be factored, so the answer is simplified.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836712359\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836520469\">\n<div data-type=\"problem\" id=\"fs-id1167836432782\">\n<p id=\"fs-id1167833408057\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b32e521214fd7cd78eac4b7f9249978_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#45;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#120;&#43;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"82\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836609856\">\n<p id=\"fs-id1167829614414\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19801361dd44bafbac88471a2cbefd9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#120;&#45;&#52;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"73\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836529491\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829931412\">\n<div data-type=\"problem\" id=\"fs-id1167833383016\">\n<p id=\"fs-id1167836477545\">Add:<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-269d343b2b2629c91c00d6944f69e316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#109;&#43;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#109;&#43;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"91\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836628945\">\n<p id=\"fs-id1167836528220\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ceff93abf0c2a0d7ef9527ec5ef81da1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#109;&#43;&#50;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"81\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836630117\">The steps used to add rational expressions are summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836493535\" class=\"howto\">\n<div data-type=\"title\">Add or subtract rational expressions.<\/div>\n<ol id=\"fs-id1167836294894\" type=\"1\" class=\"stepwise\">\n<li>Determine if the expressions have a common denominator.\n<ul id=\"fs-id1167836508187\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Yes<\/strong> \u2013 go to step 2.<\/li>\n<li><strong data-effect=\"bold\">No<\/strong> \u2013 Rewrite each rational expression with the LCD.\n<ul id=\"fs-id1167836609976\" data-bullet-style=\"bullet\">\n<li>Find the LCD.<\/li>\n<li>Rewrite each rational expression as an equivalent rational expression with the LCD.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Add or subtract the rational expressions.<\/li>\n<li>Simplify, if possible.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836524712\">Avoid the temptation to simplify too soon. In the example above, we must leave the first rational expression as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53cf7bbe73fc5385083c4c6cf0e962e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#45;&#54;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"73\" style=\"vertical-align: -9px;\" \/> to be able to add it to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83b4c95a4f1c3f605d2efb6c5e95e4e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#45;&#54;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"78\" style=\"vertical-align: -9px;\" \/> Simplify <em data-effect=\"italics\">only<\/em> after you have combined the numerators.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836541935\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836296452\">\n<div data-type=\"problem\" id=\"fs-id1167836300182\">\n<p id=\"fs-id1167836409414\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-187a71832dc08626a0c3fe629a4bd573_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"146\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829872177\">\n<table id=\"fs-id1167836289072\" class=\"unnumbered unstyled\" summary=\"Add the rational expressions 8 divided by the quantity x squared minus 2 x minus 3 and 3 x divided by the quantity x squared plus 4 x plus 3. Notice the denominators x squared minus 2 x minus 3 and x squared plus 4 x plus 3 do not have a common denominator. Rewrite each rational expression using the least common denominator. To find the least common denominator, factor x squared minus 2 x minus 3 and x squared plus 4 x plus 3, lining their common factors up in columns. The denominator x squared minus 2 x minus 3 is equal to the quantity x plus 1 times the quantity x minus 3. The denominator x squared plus 4 x plus 3 is equal to the quantity x plus 1 times the quantity x plus 3. The least common denominator is the product of each factor from each column, the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. Rewrite the rationale expression as an equivalent rational expression using the least common denominator, quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. The first denominator of the rational expression, 8 divided by the quantity x squared minus 2 x minus 3, can be factored into the quantity x plus 1 times the quantity x minus 3. Multiply the numerator and the denominator of the rational expression by the quantity x plus three to create the least common denominator. The equivalent rational expression is the quantity 8 times the quantity x plus one, divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. The first denominator of the rational expression, 3 x divided by the quantity x squared plus 4 x plus 3, can be factored into the quantity x plus 1 times the quantity x plus 3. Multiply the numerator and the denominator of the rational expression by the quantity x minus 3 to create the least common denominator. The equivalent rational expression is the quantity 3 x times the quantity x minus 1, divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. Simplify the numerators. The numerator 8 times the quantity x plus 1 can be simplified to 8 x plus 24. The numerator 3 x times the quantity x minus 1 can be simplified to 3 x squared minus 9 x. Now add the rational expression 8 x plus 24 divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3 and the rational expression 3 x squared minus 9 x divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. The result is the quantity 8 x plus the quantity 24 plus the quantity 3 x squared minus the quantity 9 x, divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. Simplify the numerator by combining like terms. The result is the quantity 3 x squared minus the quantity x plus the quantity 24, divided by the quantity x plus 1 times the quantity x minus 3 times the quantity x plus 3. The numerator is prime, so there are no common factors.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833385764\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Do the expressions have a common denominator?<\/td>\n<td data-valign=\"top\" data-align=\"center\">No.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite each expression with the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-055f6ef6cc40a2eca69e1b5ee5229f2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#76;&#67;&#68;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;&#61;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#51;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#55;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#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;&#61;&#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;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"207\" width=\"308\" style=\"vertical-align: -121px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equivalent rational expression with the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836549343\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836526365\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add the rational expressions.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836447539\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The numerator is prime, so there are<\/p>\n<div data-type=\"newline\"><\/div>\n<p>no common factors.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826025196\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829719758\">\n<div data-type=\"problem\" id=\"fs-id1167833053581\">\n<p id=\"fs-id1167836487061\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06fdf9e7cac44bb9922e9881b979cf72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#109;&#45;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#109;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#109;&#43;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"157\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833057174\">\n<p id=\"fs-id1167836440102\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd5c0a7258b7d5e5c290fbdb8facd0d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#109;&#43;&#50;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"122\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836618958\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824732931\">\n<div data-type=\"problem\" id=\"fs-id1167836625629\">\n<p id=\"fs-id1167829809775\">Add:<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a10dbe15dd8fd82adfa2f93c153c945_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#110;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#110;&#45;&#49;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#110;&#43;&#54;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"156\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824735756\">\n<p id=\"fs-id1167833335669\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-661ea19152a5947148f306a2939d4afd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#110;&#45;&#51;&#48;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"110\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836508157\">The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829620874\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829620876\">\n<div data-type=\"problem\" id=\"fs-id1167836295546\">\n<p id=\"fs-id1167833021096\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e361dd494fc6eae0af0209578b5a30f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#121;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#121;&#45;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"95\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829739352\">\n<table id=\"fs-id1167836731518\" class=\"unnumbered unstyled\" summary=\"Subtract the rational expressions 8y divided by the quantity y squared minus 16 and 4 divided by the quantity y minus 4. Notice the denominators y squared minus 16 and y minus 4 do not have a common denominator. Rewrite each rational expression using the least common denominator. To find the least common denominator, factor y squared minus 16 and y minus 4, lining their common factors up in columns. The denominator y squared minus 16 is equal to the quantity y minus 4 times the quantity y plus 4. The denominator y minus 4 cannot be factored. The least common denominator is the product of each factor from each column, the quantity y minus 4 times the quantity y plus 4. Rewrite the rationale expression as an equivalent rational expression using the least common denominator, the quantity y minus 4 times the quantity y plus 4. The first denominator of the rational expression, the quantity 8 y divided by the quantity y squared minus 16 minus 4 divided by the quantity y plus 4, can be factored into the quantity y minus 4 times the quantity y plus 4, which is the least common denominator. The equivalent rational expression for 8 y divided by the quantity y squared minus 16 is 8 y divided by the quantity y minus 4 times the quantity y plus 4. Multiply the numerator and the denominator of the rational expression, 4 divided by the quantity y minus 4, by y plus 4 to write its equivalent rational expression. The result is 4 times the quantity y plus 4 all divided by the quantity y minus 4 times the quantity y plus 4. Simplify the numerators. Notice that the numerator of 8 y divided by the quantity y minus 4 times the quantity y plus 4 is already simplified. The numerator 4 times the quantity y plus 4 can be simplified to 4 y plus 16. Now subtract the rational expressions, 8 y divided by the quantity y minus 4 times the quantity y plus 4 and the quantity 4 y plus 16 all divided by the quantity y minus 4 times the quantity y plus 4. The result is the quantity 8 y minus 4 y minus 16 all divided by the quantity y minus 4 times the quantity y plus 4. Simplify the numerator by combining like terms. The result is the quantity 4 y minus 16 all divided by the quantity y minus 4 times the quantity y plus 4. Factor the numerator, 4 y minus 16, to look for common factors. 4 y minus 16 factors into 4 times the quantity y minus 4. Notice that y minus 4 is a common factor in the numerator and denominator so it can be removed. Once simplified, the result is 4 divided by the quantity y plus 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833021924\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Do the expressions have a common denominator?<\/td>\n<td data-valign=\"top\" data-align=\"center\">No.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite each expression with the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ccd95fd40f70dda5628713549938273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#76;&#67;&#68;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#121;&#45;&#52;&#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;&#61;&#121;&#45;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"92\" width=\"361\" style=\"vertical-align: -40px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equivalent rational expression with the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824734398\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836364150\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract the rational expressions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829720811\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836608058\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the numerator to look for common factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836504084\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Remove common factors<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836415575\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836390532\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_008h_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-id1167836732354\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836732358\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829614620\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e11fcd34ee149d8b93aadb325b17af2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#43;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"89\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829936504\">\n<p id=\"fs-id1167829936506\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2de7f36d4739d938b268bdded92038d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"26\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829717023\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829717026\">\n<div data-type=\"problem\" id=\"fs-id1167829893774\">\n<p id=\"fs-id1167829893776\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-130f1df3be1a21192f8d60910b0937a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#122;&#43;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#122;&#125;&#123;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"87\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829683638\">\n<p id=\"fs-id1167836691345\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9758018dee9cb67b3ceb24f15562f2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#125;&#123;&#122;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833356152\">There are lots of negative signs in the next example. Be extra careful.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836508951\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836508953\">\n<div data-type=\"problem\" id=\"fs-id1167836363982\">\n<p id=\"fs-id1167836363984\">Subtract:<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ee7c83e71e4ad84668569944cf009e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#110;&#45;&#57;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#110;&#45;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#43;&#51;&#125;&#123;&#50;&#45;&#110;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"109\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829593986\">\n<table id=\"fs-id1167829593989\" class=\"unnumbered unstyled\" summary=\"Subtract the rational expressions the quantity negative 3 n minus 9 all divided by the quantity n squared plus n minus 6 and the quantity n plus 3 divided by the quantity 2 minus n. Factor the denominator of the first rational expression, n squared plus n minus 6. The result is the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 minus the quantity n plus 3 divided by the quantity 2 minus n. Notice that n minus 2 and 2 minus n are opposites. Multiply the numerator and denominator of the second rational expression, n plus 3 divided by the quantity 2 minus n, by negative 1. Write its denominator, negative 1 times the quantity 2 minus n, as n minus 2. The result is the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 minus negative one times the quantity n plus 3 all divided by n minus 2. Simplify the numerator of the second rational expression, remembering that a minus negative b is a plus b. Write the numerator, negative times the quantity n plus 3, as n plus 3. The result is the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 minus the quantity n plus 3 all divided by n minus 2. Notice that the denominators do not have any common factors. Find the lowest common denominator. The denominator, n squared plus n minus 6, factors into the quantity n minus 2 times the quantity n plus 3. The denominator, n minus 2, cannot be factored. Line the denominators up by common factors. The least common denominator is the product of a factor from each column, the quantity n minus 2 times the quantity n plus 3. Rewrite each rational expression as an equivalent rational expression using the lowest common denominator. The first rational expression, the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 , is already written using the LCD. Multiply the numerator and denominator of the second rational expression, the quantity n plus 3 divided by the quantity n plus 2, by n plus 3. Simplify the numerators. The result is the quantity negative 3 n minus 9 all divided by the quantity n minus 2 times the quantity n plus 3 minus the quantity n squared plus 6 n plus 9 all divided by the quantity n minus 2 times the quantity n plus 3. Add the rational expressions. The result is the quantity negative 3 n minus 9 plus n squared plus 6 n plus 9 all divided by the quantity n minus 2 times the quantity n plus 3. Simplify the numerator by combining like terms. The result is the quantity n squared plus 3 n all divided by the quantity n minus 2 times the quantity n plus 3. Factoring the numerator, results in n times the quantity n plus 3 all divided by the quantity n minus 2 times the quantity n plus 3. Notice that n plus 3 is a common factor in the numerator and denominator. Remove it and simplify. The result is n divided by the quantity n minus 2.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836626594\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-376421fcb29272bc9eaf7a0fe5d80c73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-255804b4c2c022cebca8045de7ee91d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#45;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> are opposites, we will<\/p>\n<div data-type=\"newline\"><\/div>\n<p>multiply the second rational expression by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e72c0eef54be1191c4a5f9feaced8dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#125;&#123;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"23\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829586348\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836626907\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify. Remember, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5330c3cfe75a48a97d2d442ab69e34b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#43;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833407431\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Do the rational expressions have a<\/p>\n<div data-type=\"newline\"><\/div>\n<p>common denominator? No.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ba5ed746495774299f9a4a60a8f3546_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#76;&#67;&#68;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#110;&#45;&#54;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#110;&#45;&#50;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#55;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#51;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"389\" style=\"vertical-align: -45px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equivalent rational expression with the LCD. <\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836533932\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836606028\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add the rational expressions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836728972\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833049923\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the numerator to look for common factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836287998\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829908132\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_009m_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-id1167836611144\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836611148\">\n<div data-type=\"problem\" id=\"fs-id1167836689139\">\n<p id=\"fs-id1167836689141\">Subtract :<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10a230339e0a5d34104943d4b8e6bf2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#45;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#54;&#45;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"114\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829683969\">\n<p id=\"fs-id1167829683971\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9e541430c1362af7d7264df611343b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#43;&#49;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"73\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833311026\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833412704\">\n<div data-type=\"problem\" id=\"fs-id1167833412706\">\n<p id=\"fs-id1167833412708\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-515f6141c86b3e3658af7416b1704b51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#121;&#45;&#50;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#45;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#49;&#125;&#123;&#50;&#45;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"113\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ef749221b6133da99637f4c6ff2138a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#51;&#125;&#123;&#121;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"25\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165927773319\">Things can get very messy when both fractions must be multiplied by a binomial to get the common denominator.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829905216\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833381669\">\n<p id=\"fs-id1167833381671\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06fa07ffa16fe8ff5430b00c4707654c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#97;&#43;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#97;&#43;&#49;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"151\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167824734719\" class=\"unnumbered unstyled\" summary=\"Subtract the rational expressions, 4 divided by the quantity a squared plus 6 a plus 5 and 3 divided by the quantity a squared plus 7 a plus 10. Factor each denominator. The result is 4 divided by the quantity a plus 1 times a plus 5 minus 3 divided by the quantity a plus 2 times a plus 5. The expressions do not have a common denominator. Line the denominators up by common factors in columns. The least common denominator is the product of the factors from each column, the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2. Rewrite each rational expression as an equivalent rational expression using the least common denominator. Multiply the numerator and denominator of the first rational expression by a plus 2. Multiply the numerator and denominator of the second rational expression by a plus 1. Simplifying each numerator, the result is the quantity 4 a plus 8 divided by the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2 minus 3 divided by the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2. Subtract the rational expressions. The result is the quantity 4 a plus 8 minus the quantity 3 a plus 3 all divided by the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2. Simplify the numerator by combining like terms. The result is the quantity a plus 5 all divided by the quantity a plus 1 times the quantity a plus 5 times the quantity a plus 2. Notice a plus 5 is a common factor in the numerator and denominator. Remove it and simplify. The result is 1 divided by the quantity a plus 1 times the quantity a plus 2.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829644596\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the denominators.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836662670\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Do the rational expressions have a<\/p>\n<div data-type=\"newline\"><\/div>\n<p>common denominator? No.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-532d365fed571fcfeb927e910f82b275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#76;&#67;&#68;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#97;&#43;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#97;&#43;&#49;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#50;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"119\" width=\"465\" style=\"vertical-align: -54px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equivalent rational expression with the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836508350\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829844002\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract the rational expressions.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836665473\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832978265\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833047259\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Look for common factors.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836529296\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836392589\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_010j_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-id1167836596323\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836612453\">\n<div data-type=\"problem\" id=\"fs-id1167836612456\">\n<p id=\"fs-id1167836692540\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be18991b77aedffa47044f8380b11f55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#98;&#45;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#98;&#43;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"139\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836310949\">\n<p id=\"fs-id1167836310951\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45f47ae5c790abe2f0e6e3ac904e8fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"69\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829741840\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836356000\">\n<div data-type=\"problem\" id=\"fs-id1167836356002\">\n<p id=\"fs-id1167836356004\">Subtract: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3f061db253ab99d5956e89ea4707baf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"114\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833223781\">\n<p id=\"fs-id1167833223783\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-222bf388f8119d0cebc48f22f0c5896b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"73\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829721097\">We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example, we will start by factoring all three denominators to find their LCD.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836510837\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836510840\">\n<div data-type=\"problem\" id=\"fs-id1167833386586\">\n<p id=\"fs-id1167833386588\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64866fcae44adbb9f988aac8c46ebfd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#117;&#125;&#123;&#117;&#45;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#117;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#117;&#45;&#49;&#125;&#123;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#117;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"124\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826146770\">\n<table id=\"fs-id1167836556634\" class=\"unnumbered unstyled can-break\" summary=\"Simplify 2 u divided by the quantity u minus 1 plus 1 divided by u minus the quantity 2 u minus 1 all divided by u squared minus u. Notice that the expressions do not have any common denominators. Find the least common denominator. Notice that u minus 1 and u cannot be factored, but u squared minus u factors into u times the quantity u minus 1. The least common denominator is u times the quantity u minus 1. Rewrite each rational expression as an equivalent rational expression using the least common denominator. Multiply the numerator and denominator of the first rational expression 2 u divided the quantity u minus 1 by u. Multiply the numerator and denominator of the second rational expression, 1 divided by u, by u minus 1. The denominator of the third rational expression is already written using the least common denominator. Simplifying, the result is 2 u squared divided by the quantity u minus 1 times u plus the quantity u minus 1 divided by u times the quantity u minus 1 minus the quantity 2 u minus 1 divided by u times the quantity u minus 1. Write the expressions as one rational expression by adding and subtracting the numerators. The result is the quantity 2 u squared plus u minus 1 minus 2 u plus 1 all divided by u times the quantity u minus 1. Simplify by combining like terms in the numerator. The result is 2 u squared minus u all divided by u times the quantity u minus 1. Factor the numerator. The result is u times the quantity 2 u minus 1 all divided by u times the quantity u minus 1. Notice that u is a common factor in the numerator and denominator. Remove it and simplify. The result is the quantity 2 u minus 1 divided by the quantity u minus 1.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836416633\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Do the expressions have a common denominator? No.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Rewrite each expression with the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fa1882894d7803e91d6ba70bee5c1c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#76;&#67;&#68;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#117;&#45;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#117;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#117;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#117;&#61;&#117;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#117;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#49;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"118\" width=\"314\" style=\"vertical-align: -54px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equivalent rational expression with the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836531565\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836548004\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write as one rational expression.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833339385\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833350333\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the numerator, and remove<\/p>\n<div data-type=\"newline\"><\/div>\n<p>common factors.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836447474\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836521831\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_011h_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-id1167833175384\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832945632\">\n<div data-type=\"problem\" id=\"fs-id1167832945634\">\n<p id=\"fs-id1167833356109\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea63c498689f57a319fc1ee4f3d61f0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#125;&#123;&#118;&#43;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#118;&#45;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"138\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836298137\">\n<p id=\"fs-id1167836298139\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cff636a1b1bdd8d0edaa3082164ef07c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#43;&#51;&#125;&#123;&#118;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"25\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829784997\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836486350\">\n<div data-type=\"problem\" id=\"fs-id1167836486352\">\n<p>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52b47d48c4732315716dd37108fa81dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#119;&#125;&#123;&#119;&#43;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#119;&#43;&#55;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#119;&#43;&#52;&#125;&#123;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#119;&#43;&#49;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"182\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829811785\">\n<p id=\"fs-id1167829811787\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9dabac80386cb58c186b326d8d79cf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#119;&#125;&#123;&#119;&#43;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"28\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829712902\">\n<h3 data-type=\"title\">Add and subtract rational functions<\/h3>\n<p id=\"fs-id1167829843979\">To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829717361\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836557016\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7de960a332c98ab624de84fc3632a5b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71880e87f8eec428c583edd3bb585360_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;&#53;&#125;&#123;&#120;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5100013f0a6d5a04ef5c875adce491f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#43;&#49;&#56;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"107\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836732726\">\n<table id=\"fs-id1167836732729\" class=\"unnumbered unstyled can-break\" summary=\"The function R is equal to the function f minus the function g. Substitute the quantity x plus 5 divided by the quantity x minus 2 for the function f and the quantity 5 x plus 18 divided by the quantity x squared minus 4. Factor the denominators. Notice that x minus 2 cannot be factored, but x squared minus 4 can be factored into the quantity x minus 2 times the quantity x plus 2. The result is the function R is equal to the quantity x plus 5 all divided by the quantity x minus 2 minus the quantity 5 x plus 18 all divided by the quantity x minus 2 times the quantity x plus 2. The expressions do have a common denominator. Rewrite the expressions as equivalent expressions using the least common denominator. Line the factors for each denominator up by common factors in columns. The least common denominator is the product of the factors for each column, the quantity x minus 2 times the quantity x plus 2. In the function R, multiply the numerator and denominator of the first rational expression by x plus 2. The second rational expression is already written with the least common denominator. Write the rational expressions as one. The result is the quantity x plus 5 times the quantity x plus 2 minus the quantity 5 x plus 18 all divided by the quantity x minus 2 times the quantity x plus 2. Simplify the numerator. The result is the quantity x squared plus 7 x plus 10 minus 5 x minus 18 all divided by the quantity x minus 2 times the quantity x plus 2, which further simplifies to the quantity x squared plus 2 x minus 8 all divided by the quantity x minus 2 times the quantity x plus 2. Factoring the numerator, the result is the quantity x plus 4 times the quantity x minus 2 all divided by the quantity x minus 2 times the quantity x plus 2. Notice that x minus 2 is a common factor in the numerator and denominator. Remove it, and then simplify. The result is R is equal to the quantity x plus 4 divided by the quantity x plus 2.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756084\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute in the functions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccd462e56046be46a7c722c8d562e055_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;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9d4a03a388c1012a45f49710076105e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829788706\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the denominators.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829594854\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Do the expressions have a common denominator? No.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Rewrite each expression with the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-616d5190f859b51c5c2ef0cedaf911a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#76;&#67;&#68;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"103\" width=\"356\" style=\"vertical-align: -46px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite each rational expression as an<\/p>\n<div data-type=\"newline\"><\/div>\n<p>equivalent rational expression with the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836289631\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write as one rational expression.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547177\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833279815\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833407475\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the numerator, and remove<\/p>\n<div data-type=\"newline\"><\/div>\n<p>common factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832950920\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829702024\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_012j_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-id1167829719644\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836295476\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7de960a332c98ab624de84fc3632a5b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ef685304c8b209d4af9dc519a95d6e5_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;&#120;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"89\" style=\"vertical-align: -8px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fd5cb1a474a9004195573fe7b11bdec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#55;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#49;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"125\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833340016\">\n<p id=\"fs-id1167833340018\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43f915cf54e5eb2d6078ab7592fa67d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#55;&#125;&#123;&#120;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"26\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836534519\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836508918\">\n<p id=\"fs-id1167836508920\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c9ba76d92ef455d911f821204f4be95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4473cd432fe1ff9550423ce7078a5c6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#52;&#125;&#123;&#120;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"89\" style=\"vertical-align: -8px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-339792a3d7eda371223feb9ee368558a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#43;&#54;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"100\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836528213\">\n<p id=\"fs-id1167836528215\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb84da1ccb2748d42d96c9e3be2f5e89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#56;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"73\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826102760\" class=\"media-2\">\n<p id=\"fs-id1167826102765\">Access this online resource for additional instruction and practice with adding and subtracting rational expressions.<\/p>\n<ul id=\"fs-id1171792804576\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37AddSubRatExp\">Add and Subtract Rational Expressions- Unlike Denominators<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836518508\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836391201\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Rational Expression Addition and Subtraction<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> If <em data-effect=\"italics\">p<\/em>, <em data-effect=\"italics\">q<\/em>, and <em data-effect=\"italics\">r<\/em> are polynomials where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39ebc9dc80feddbfd4064c205be07e25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/> then<\/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-a0408df37f7dcf5e570d7e48d481150f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#114;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#114;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#43;&#113;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"92\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ceb641191a58fffd4cba7ba2accfb60f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#114;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#114;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#45;&#113;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><strong data-effect=\"bold\">How to find the least common denominator of rational expressions.<\/strong>\n<ol id=\"fs-id1167833071624\" type=\"1\" class=\"stepwise\">\n<li>Factor each expression completely.<\/li>\n<li>List the factors of each expression. Match factors vertically when possible.<\/li>\n<li>Bring down the columns.<\/li>\n<li>Write the LCD as the product of the factors.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to add or subtract rational expressions.<\/strong>\n<ol id=\"fs-id1167824735388\" type=\"1\" class=\"stepwise\">\n<li>Determine if the expressions have a common denominator.\n<ul id=\"fs-id1167833339764\" data-bullet-style=\"bullet\">\n<li>Yes \u2013 go to step 2.<\/li>\n<li>No \u2013 Rewrite each rational expression with the LCD.\n<ul id=\"fs-id1167836688152\" data-bullet-style=\"bullet\">\n<li>Find the LCD.<\/li>\n<li>Rewrite each rational expression as an equivalent rational expression with the LCD.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Add or subtract the rational expressions.<\/li>\n<li>Simplify, if possible.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829811067\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829811070\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167836319644\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions with a Common Denominator<\/strong><\/p>\n<p id=\"fs-id1167836609419\">In the following exercises, add.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836609422\">\n<div data-type=\"problem\" id=\"fs-id1167836328949\">\n<p id=\"fs-id1167836328951\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0757e2ab6a3de854f06f8a901b38e86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#49;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"53\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836738177\">\n<p id=\"fs-id1167829579790\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56fd13ae94393ab95b21e4d3651f1ec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833339785\">\n<div data-type=\"problem\" id=\"fs-id1167833138215\">\n<p id=\"fs-id1167833138217\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3813ac607b96e30ec2e811b653e87c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"53\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836508792\">\n<div data-type=\"problem\" id=\"fs-id1167836508794\">\n<p id=\"fs-id1167836619480\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a126e6336360a6ceb77105800c7063f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#99;&#125;&#123;&#52;&#99;&#45;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#99;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"87\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836440093\">\n<p id=\"fs-id1167836665068\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e194a6e77fd592c8fc420214b5e3565a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#99;&#43;&#53;&#125;&#123;&#52;&#99;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"31\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836300389\">\n<div data-type=\"problem\" id=\"fs-id1167832940534\">\n<p id=\"fs-id1167832940536\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28becd1b8e203d89195643ebba6adba5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#109;&#125;&#123;&#50;&#109;&#43;&#110;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#50;&#109;&#43;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"103\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836521456\">\n<p id=\"fs-id1167836521458\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a134adaf076263326448e51b23893b51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#114;&#45;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#114;&#45;&#56;&#125;&#123;&#50;&#114;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"95\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833137093\">\n<p id=\"fs-id1167833066700\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe72a34144993c6d29d9de5c74119f11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836440414\">\n<div data-type=\"problem\" id=\"fs-id1167836440612\">\n<p id=\"fs-id1167836440614\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55b03ec283649b76d917338f3822cdab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#115;&#45;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#115;&#45;&#49;&#48;&#125;&#123;&#51;&#115;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"101\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829720099\">\n<div data-type=\"problem\" id=\"fs-id1167829720101\">\n<p id=\"fs-id1167829720104\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c4beb0c1a2cdd196ecff498017bc4c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#119;&#125;&#123;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"109\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836487053\">\n<p id=\"fs-id1167832936236\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-094360f2ca326320e599ccbec8a3ef4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#119;&#125;&#123;&#119;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"28\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832936278\">\n<div data-type=\"problem\" id=\"fs-id1167832936280\">\n<p id=\"fs-id1167833053565\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e3c4c610e2bb6b6a751e6ad68353d00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836629242\">In the following exercises, subtract.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829590371\">\n<div data-type=\"problem\" id=\"fs-id1167829590373\">\n<p id=\"fs-id1167829590375\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bb218a6f073e941fe8dabca5532ee93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#97;&#45;&#55;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#57;&#125;&#123;&#51;&#97;&#45;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836743433\">\n<p id=\"fs-id1167836743435\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f366df8a7372e71f09f8a3829f29a8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#97;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829634217\">\n<div data-type=\"problem\" id=\"fs-id1167829634219\">\n<p id=\"fs-id1167829749871\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30d44711f453c5018a1e53dedf148a3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#53;&#98;&#45;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#54;&#125;&#123;&#53;&#98;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"87\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833135505\">\n<div data-type=\"problem\" id=\"fs-id1167829743925\">\n<p id=\"fs-id1167829743927\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2be314774f810bb603b4091d9bde7d71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#54;&#109;&#45;&#51;&#48;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#109;&#45;&#51;&#48;&#125;&#123;&#54;&#109;&#45;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"120\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836729645\">\n<p id=\"fs-id1167836662865\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b3adfd5e1b9e29a285a83952961928a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#45;&#50;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836535828\">\n<div data-type=\"problem\" id=\"fs-id1167836535830\">\n<p id=\"fs-id1167836535833\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33e7bb8e736de92acf4a5132353b044a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#110;&#45;&#51;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#110;&#45;&#49;&#54;&#125;&#123;&#52;&#110;&#45;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"112\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836697382\">\n<div data-type=\"problem\" id=\"fs-id1167836697384\">\n<p id=\"fs-id1167836697386\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3788a875e80d9f480b014f4e3a6727fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#112;&#43;&#52;&#125;&#123;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#112;&#45;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#112;&#43;&#55;&#125;&#123;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#112;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"145\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836727800\">\n<p id=\"fs-id1167836727803\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28bee05ef4172b4192ac2d3fa704dc12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#43;&#51;&#125;&#123;&#112;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"26\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829743945\">\n<div data-type=\"problem\" id=\"fs-id1167829743948\">\n<p id=\"fs-id1167829743950\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e939e2c180dadd5d891dfe2a8d90a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#113;&#45;&#57;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#113;&#43;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#113;&#43;&#55;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#113;&#43;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"151\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836664177\">\n<div data-type=\"problem\" id=\"fs-id1167836664179\">\n<p id=\"fs-id1167836664182\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11fccbed267cd0d6f175d1a474b51e28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#114;&#45;&#51;&#51;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#114;&#43;&#51;&#48;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"164\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829808586\">\n<p id=\"fs-id1167829808588\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7479ec3c89975a80ff9d375cebf32c29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#43;&#57;&#125;&#123;&#114;&#43;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"25\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829732204\">\n<div data-type=\"problem\" id=\"fs-id1167829732206\">\n<p id=\"fs-id1167836630357\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c92348dae2ec3ae473d7dfe1af9efaf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#116;&#45;&#52;&#125;&#123;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#116;&#45;&#52;&#52;&#125;&#123;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"151\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836613288\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions whose Denominators are Opposites<\/strong><\/p>\n<p id=\"fs-id1167833381688\">In the following exercises, add or subtract.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833381691\">\n<div data-type=\"problem\" id=\"fs-id1167836574708\">\n<p id=\"fs-id1167836574710\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-546e24ee87be053f4cf0093b26fbbb19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#118;&#125;&#123;&#50;&#118;&#45;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#118;&#43;&#52;&#125;&#123;&#49;&#45;&#50;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"89\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829589753\">\n<p id=\"fs-id1167829589755\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836557054\">\n<div data-type=\"problem\" id=\"fs-id1167836557056\">\n<p id=\"fs-id1167829850189\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ba99c268c997406e6f36abd7e8887e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#119;&#125;&#123;&#53;&#119;&#45;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#119;&#43;&#54;&#125;&#123;&#50;&#45;&#53;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833339004\">\n<div data-type=\"problem\" id=\"fs-id1167833339006\">\n<p id=\"fs-id1167833339008\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b034c20a49f95f40a350854427b96ee4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#120;&#45;&#55;&#125;&#123;&#56;&#120;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#49;&#125;&#123;&#51;&#45;&#56;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"169\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829806996\">\n<p id=\"fs-id1167829806998\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4baf3c092ee3bd5ae188f77cf55175b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832926886\">\n<div data-type=\"problem\" id=\"fs-id1167832926888\">\n<p id=\"fs-id1167832926890\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7d164a3009bb497014e18cf45c0f151_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#45;&#49;&#49;&#125;&#123;&#51;&#121;&#45;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#121;&#43;&#49;&#55;&#125;&#123;&#55;&#45;&#51;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"167\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833097182\">\n<div data-type=\"problem\" id=\"fs-id1167833097184\">\n<p id=\"fs-id1167829743862\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9786ff13d7fe7d1a10330ae589150440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#122;&#125;&#123;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#122;&#43;&#50;&#48;&#125;&#123;&#50;&#53;&#45;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"103\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833082458\">\n<p id=\"fs-id1167833129499\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3f80ece7dea6510fbe0cc542925b2a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#122;&#43;&#52;&#125;&#123;&#122;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833047549\">\n<div data-type=\"problem\" id=\"fs-id1167833047551\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4dd8450ead6e649357e33746c94d2345_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#97;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#97;&#45;&#50;&#55;&#125;&#123;&#57;&#45;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"103\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829783563\">\n<div data-type=\"problem\" id=\"fs-id1167829579106\">\n<p id=\"fs-id1167829579108\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f98ff92be48a544458cc5a3e774811b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#48;&#98;&#45;&#49;&#51;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#98;&#45;&#56;&#125;&#123;&#52;&#57;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"161\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826025455\">\n<p id=\"fs-id1167826025457\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27ddc9fa54c3773499882a9424b5ad2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#98;&#45;&#51;&#125;&#123;&#98;&#45;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"31\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824732175\">\n<div data-type=\"problem\" id=\"fs-id1167824732178\">\n<p id=\"fs-id1167824732180\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43bab7191c4dbab417aa3aae3686265d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#99;&#45;&#49;&#48;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#99;&#45;&#49;&#48;&#125;&#123;&#49;&#54;&#45;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"148\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836686848\"><strong data-effect=\"bold\">Find the Least Common Denominator of Rational Expressions<\/strong><\/p>\n<p id=\"fs-id1167829614424\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the LCD for the given rational expressions <span class=\"token\">\u24d1<\/span> rewrite them as equivalent rational expressions with the lowest common denominator.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836576082\">\n<div data-type=\"problem\" id=\"fs-id1167836576084\">\n<p id=\"fs-id1167836576086\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8868bbdc1544161364cdd19efc4ab0d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#56;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"128\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836699862\">\n<p id=\"fs-id1167836699864\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50092cbdc28da954c54354163b55d792_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;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"168\" 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-72befb61e90fa77eaba942fe47ecce3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#43;&#49;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"109\" style=\"vertical-align: -9px;\" \/>,<\/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-0839516607672ce668d74b1da3a425d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"109\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829579658\">\n<div data-type=\"problem\" id=\"fs-id1167829789346\">\n<p id=\"fs-id1167829789349\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86df69b8da5973ab3c3e73ec39a984fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#43;&#51;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#121;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#121;&#45;&#52;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"139\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829740447\">\n<div data-type=\"problem\" id=\"fs-id1167833365223\">\n<p id=\"fs-id1167833365225\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3478981c98e61e3d0cf797b408cb2f40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#122;&#45;&#56;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#122;&#125;&#123;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"100\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829828588\">\n<p id=\"fs-id1167829828590\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ceaf362262b7eaa2460b00048c33d028_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"164\" 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-252c5eb2c2bb3a4c2522ddd76b9145d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#122;&#45;&#51;&#54;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"106\" style=\"vertical-align: -9px;\" \/>,<\/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-99109ca08c49d6d7b67a703de6176123_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#122;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"106\" style=\"vertical-align: -9px;\" \/><\/p>\n<p id=\"fs-id1167829714694\">\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829714698\">\n<div data-type=\"problem\" id=\"fs-id1167829714701\">\n<p id=\"fs-id1167833386222\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e09280c15166a4550385d9f1fdc4c106_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#97;&#43;&#52;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#97;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"121\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829894030\">\n<div data-type=\"problem\" id=\"fs-id1167829894032\">\n<p id=\"fs-id1167832999769\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48e01def064c8156855638acb992e5fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#98;&#43;&#57;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#98;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#98;&#45;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"127\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829718970\">\n<p id=\"fs-id1167829718972\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcbacd986c377232ee179762ec3cbf07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"160\" 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-5a3517a5eb663957eeaa0a5d30578a7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#98;&#45;&#50;&#48;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"103\" style=\"vertical-align: -9px;\" \/>,<\/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-0d8666ea576804f7a24978efc0eac78a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#98;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"103\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624352\">\n<div data-type=\"problem\" id=\"fs-id1167829624354\">\n<p id=\"fs-id1167829624357\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71f64950a88a332f02dadd7165917a81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#99;&#43;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#99;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#99;&#43;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"127\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833021546\">\n<div data-type=\"problem\" id=\"fs-id1167829609144\">\n<p id=\"fs-id1167829609146\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e426e943a4e272c785d0cae0edc7c49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#100;&#45;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#100;&#125;&#123;&#51;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#57;&#100;&#43;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"152\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833086390\">\n<p id=\"fs-id1167836535846\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16a6df03beb0c919a4fdedb43b1d696e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#100;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"174\" 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-5c92c1645c7470307a9932180b512f1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#100;&#45;&#49;&#50;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#100;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"113\" style=\"vertical-align: -9px;\" \/>,<\/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-56f3bf3834a759c28132013fd7f74c71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#100;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#100;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"113\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829828600\">\n<div data-type=\"problem\" id=\"fs-id1167829828602\">\n<p id=\"fs-id1167829828604\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e819174a70db8aa748ec4afb15817f0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#109;&#45;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#109;&#125;&#123;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#55;&#109;&#43;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"166\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833255974\"><strong data-effect=\"bold\">Add and Subtract Rational Expressions with Unlike Denominators<\/strong><\/p>\n<p id=\"fs-id1167833224739\">In the following exercises, perform the indicated operations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833224742\">\n<div data-type=\"problem\" id=\"fs-id1167833224744\">\n<p id=\"fs-id1167833224746\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e16ce446b9ac2c00c1c3818f4ded541d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#53;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"97\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836292506\">\n<p id=\"fs-id1167836292508\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6273b5a19ee18ff1df9c5f7cf575d0d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#121;&#43;&#56;&#120;&#125;&#123;&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"47\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829833103\">\n<div data-type=\"problem\" id=\"fs-id1167829833106\">\n<p id=\"fs-id1167829833108\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8acc0d2d0b89682d10f37feff948936_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"100\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829853067\">\n<div data-type=\"problem\" id=\"fs-id1167829853069\">\n<p id=\"fs-id1167829853071\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-180ca6ab4804668aac924aa65fd1da79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#114;&#43;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#114;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"75\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829596476\">\n<p id=\"fs-id1167829596479\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69d21b89f52a69836a3ddf83487e081b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#114;&#45;&#55;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"70\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829651378\">\n<div data-type=\"problem\" id=\"fs-id1167829651380\">\n<p id=\"fs-id1167829651382\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c38814297fd8827bb050d23f4c69feb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#115;&#45;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#115;&#43;&#51;&#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-id1167833369992\">\n<div data-type=\"problem\" id=\"fs-id1167833369994\">\n<p id=\"fs-id1167833369996\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf65929c5eaba108973eb87e2d05b0db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#119;&#45;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#119;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"88\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836409813\">\n<p id=\"fs-id1167836409815\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13c6427047b969fb04457756c4a8eb07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#119;&#43;&#49;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#119;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"84\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829745143\">\n<div data-type=\"problem\" id=\"fs-id1167829745145\">\n<p id=\"fs-id1167829745147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fdb1416f20522097c4f8ca5a68932c90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#50;&#120;&#43;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"84\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829585698\">\n<div data-type=\"problem\" id=\"fs-id1167829585701\">\n<p id=\"fs-id1167829585703\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8f2a3ac04764c86433846f77ff95185_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#121;&#125;&#123;&#121;&#43;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#121;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"75\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833009019\">\n<p id=\"fs-id1167833009021\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ecbdd696dd66ddacf7ba8a856c266de6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#121;&#43;&#57;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"72\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829859736\">\n<div data-type=\"problem\" id=\"fs-id1167829859738\">\n<p id=\"fs-id1167829859740\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f7ab2432bc6bae8e865b09169efe670_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#122;&#125;&#123;&#122;&#45;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#122;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"75\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833022388\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-501867795e0491c91693ad0d94e76d75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#98;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#98;&#45;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#98;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"108\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833338764\">\n<p id=\"fs-id1167833338766\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-993a1046a61e3027dca8df859b1fbb8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#98;&#43;&#49;&#48;&#43;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"89\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829742750\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829742755\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75cbec19a507688f9baaa1fa6ccf6de6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#99;&#100;&#43;&#51;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"94\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829696169\">\n<div data-type=\"problem\" id=\"fs-id1167829696171\">\n<p id=\"fs-id1167829696173\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e769b22090542e9bdbd77129167103c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#109;&#125;&#123;&#51;&#109;&#45;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#109;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#109;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"129\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829786207\">\n<p id=\"fs-id1167829786210\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b2f5e91bd607faf05f35ba706c3cd8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#109;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"30\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829786229\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167824732508\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be7b792534e2717ff5374b6d1a13e8ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#52;&#110;&#43;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#110;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"110\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836585278\">\n<div data-type=\"problem\" id=\"fs-id1167836585280\">\n<p id=\"fs-id1167836585282\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d10b1d39475f4fccfbe6185d5febc7a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#114;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#114;&#43;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#114;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"137\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833022457\">\n<p id=\"fs-id1167833022459\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e99caef014ff859e5e8837c232072a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#114;&#43;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"105\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833239647\">\n<div data-type=\"problem\" id=\"fs-id1167833239649\">\n<p id=\"fs-id1167833239651\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0552aea4c0a0f5eadfa175f6937b8953_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#115;&#125;&#123;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#115;&#45;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#115;&#45;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"142\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836606012\">\n<div data-type=\"problem\" id=\"fs-id1167836606014\">\n<p id=\"fs-id1167836606016\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e07be085c40d7c0167d63cfde7bc82f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#116;&#45;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#45;&#50;&#125;&#123;&#116;&#43;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"71\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829783002\">\n<p id=\"fs-id1167829783006\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5016c38c4ea502bb891107ab58c487c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#116;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"67\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829782601\">\n<div data-type=\"problem\" id=\"fs-id1167829782603\">\n<p id=\"fs-id1167833364714\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3020d204c1bf67e749dd9a9307fab3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#51;&#125;&#123;&#120;&#43;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#120;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"77\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836574205\">\n<div data-type=\"problem\" id=\"fs-id1167836574207\">\n<p id=\"fs-id1167836574209\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e2a814d28dbeb5f59ac70d8efe878f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#97;&#125;&#123;&#97;&#43;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#50;&#125;&#123;&#97;&#43;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"75\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836614555\">\n<p id=\"fs-id1167836614557\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-741aa513f64ce54114c1f38641b0341a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#97;&#45;&#54;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"72\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833129050\">\n<div data-type=\"problem\" id=\"fs-id1167833129052\">\n<p id=\"fs-id1167833129054\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab30b7a2e37adf2ff96d4a2ea830e600_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#98;&#125;&#123;&#98;&#45;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#45;&#54;&#125;&#123;&#98;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829689013\">\n<div data-type=\"problem\" id=\"fs-id1167829689015\">\n<p id=\"fs-id1167829689018\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3b8e4cb5b9a7bef306189c9f8dd5029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#109;&#43;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#109;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"99\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825885849\">\n<p id=\"fs-id1167825885851\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-982cede5c2ca520b1cf0a906634ca819_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#125;&#123;&#109;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826129923\">\n<div data-type=\"problem\" id=\"fs-id1167826129925\">\n<p id=\"fs-id1167826129928\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b357320b976db10873c37a0f47b6fa58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#110;&#43;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#110;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"92\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836768930\">\n<div data-type=\"problem\" id=\"fs-id1167836768932\">\n<p id=\"fs-id1167836768934\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ff4fc6d81282f4191e41bf79949dc81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#57;&#112;&#45;&#49;&#55;&#125;&#123;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#112;&#45;&#50;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#43;&#49;&#125;&#123;&#55;&#45;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"114\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836485767\">\n<p id=\"fs-id1167836485769\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe903b1243f403656432d894544106bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#43;&#50;&#125;&#123;&#112;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"26\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836485791\">\n<div data-type=\"problem\" id=\"fs-id1167836485793\">\n<p id=\"fs-id1167836485796\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cdbbd0549e3f39d4c4e9ffcf7dd62b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#51;&#113;&#45;&#56;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#113;&#45;&#50;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#43;&#50;&#125;&#123;&#52;&#45;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"113\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825703893\">\n<div data-type=\"problem\" id=\"fs-id1167825703895\">\n<p id=\"fs-id1167825703898\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc32a50b1bc71a1946d6cfdd9e781111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#114;&#45;&#49;&#54;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#114;&#45;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#45;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"112\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833364221\">\n<p id=\"fs-id1167833364224\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f6563eb42ebebbceb50eae2bb2345bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#114;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833364241\">\n<div data-type=\"problem\" id=\"fs-id1167833364243\">\n<p id=\"fs-id1167833364245\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1eeb3a23fc1551bfe0e7c77808d8de92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#116;&#45;&#51;&#48;&#125;&#123;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#116;&#45;&#50;&#55;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#45;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"107\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829783655\">\n<div data-type=\"problem\" id=\"fs-id1167829783657\">\n<p id=\"fs-id1167829783659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4485459e116356e122334cf91b467d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#43;&#55;&#125;&#123;&#49;&#48;&#120;&#45;&#49;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836439597\">\n<p id=\"fs-id1167836439599\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ab9d949689a97ea2ba28e709d0bb372_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#49;&#48;&#120;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"50\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833364483\">\n<div data-type=\"problem\" id=\"fs-id1167833364485\">\n<p id=\"fs-id1167833364487\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4116237b0ce13a3c0991dc669788b4b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#121;&#45;&#52;&#125;&#123;&#53;&#121;&#43;&#50;&#125;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"64\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829808222\">\n<div data-type=\"problem\" id=\"fs-id1167829808224\">\n<p id=\"fs-id1167829808226\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d078963f4812beb54bb76720ee68b6e_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;&#51;&#120;&#45;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"141\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829651393\">\n<p id=\"fs-id1167829651395\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b2bc2ad9a1f1b3d7389c3e65421d0f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#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;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"109\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836664672\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7e2997039cd2085724694e8fc016391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"148\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829831823\">\n<div data-type=\"problem\" id=\"fs-id1167829831825\">\n<p id=\"fs-id1167829831827\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c87330a7dc7a744a8d2f22feb96801c_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;&#56;&#120;&#45;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"148\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836440776\">\n<p id=\"fs-id1167836440778\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb638afac26e984c8a7136ced5830cfa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#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;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"73\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836440815\">\n<div data-type=\"problem\" id=\"fs-id1167836440817\">\n<p id=\"fs-id1167836440819\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-331b28996581d2cd1c5386f97266c1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"155\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833085118\">\n<div data-type=\"problem\" id=\"fs-id1167833085120\">\n<p id=\"fs-id1167833085122\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d68e80d071636ff10ccdf67390b0106_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#97;&#125;&#123;&#97;&#45;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#97;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#97;&#43;&#49;&#56;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"122\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825703513\">\n<p id=\"fs-id1167825703516\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e91e404bf19e115e7d948e102c82d9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#97;&#45;&#51;&#54;&#125;&#123;&#97;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"71\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825703558\">\n<div data-type=\"problem\" id=\"fs-id1167825703560\">\n<p id=\"fs-id1167825703562\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e046dc7b895110f553ed2c6ba4d22276_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#98;&#125;&#123;&#98;&#45;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#98;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#98;&#45;&#49;&#53;&#125;&#123;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"138\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826130332\">\n<div data-type=\"problem\" id=\"fs-id1167826130334\">\n<p id=\"fs-id1167826130336\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-954d512e1350bb52acc30c6260615909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#99;&#43;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#99;&#45;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#99;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"129\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830095264\">\n<p id=\"fs-id1167830095266\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2af1b8488f03798cb750cca71a18d00c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#45;&#53;&#125;&#123;&#99;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"24\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830095289\">\n<div data-type=\"problem\" id=\"fs-id1167830095291\">\n<p id=\"fs-id1167830095293\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8409d62d9fade0c80fe3694fa1312f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#100;&#125;&#123;&#100;&#45;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#100;&#43;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#100;&#45;&#53;&#125;&#123;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#100;&#45;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"157\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829829429\">\n<div data-type=\"problem\" id=\"fs-id1167829829432\">\n<p id=\"fs-id1167833364561\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45cdfbcde42e6b6463e844f17cf374e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#100;&#125;&#123;&#100;&#43;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#100;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#100;&#43;&#56;&#125;&#123;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"123\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833364611\">\n<p id=\"fs-id1167833364614\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3322952febbecdb807e70f67778e03b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#100;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"43\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829782365\">\n<div data-type=\"problem\" id=\"fs-id1167829782367\">\n<p id=\"fs-id1167829782370\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6cb61027924a85196a53285788d4940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#113;&#125;&#123;&#113;&#43;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#113;&#45;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#113;&#43;&#49;&#53;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#113;&#45;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"162\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829650515\"><strong data-effect=\"bold\">Add and Subtract Rational Functions<\/strong><\/p>\n<p id=\"fs-id1167829650521\">In the following exercises, find <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c9ba76d92ef455d911f821204f4be95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1526d013df81a0fdffb11b5c18617226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826129980\">\n<div data-type=\"problem\" id=\"fs-id1167826129983\">\n<p id=\"fs-id1167826129985\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2c04afaac69e83b4a88365a7c4471c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#53;&#120;&#45;&#53;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"115\" style=\"vertical-align: -9px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3a57b97844194ae187ad1a7863bd230_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#50;&#45;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"87\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836399038\">\n<p id=\"fs-id1167836399040\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ecb4eb6c6050795548249794993924f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"153\" style=\"vertical-align: -9px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4784fae360b586afd20588373724b61d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#49;&#125;&#123;&#120;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"92\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833256034\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833256039\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce42a1e786922c1740eee0f24c02d6f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#52;&#120;&#45;&#50;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"122\" style=\"vertical-align: -9px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-239bf3975818c862fdbce2ec688897b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#55;&#125;&#123;&#53;&#45;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"87\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833369660\">\n<div data-type=\"problem\" id=\"fs-id1167833369662\">\n<p id=\"fs-id1167833369664\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e05d406d7cfee74cef808e54442e5313_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;&#54;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"103\" style=\"vertical-align: -7px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb2a6050caa02d34cac66330fa22d90a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"87\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833369724\">\n<p id=\"fs-id1167833369726\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bd001a679259c260ae732bb8c61f0cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"73\" style=\"vertical-align: -9px;\" \/><\/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-264fa6134805cd53ab97e10c9b120cab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"92\" style=\"vertical-align: -8px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829651530\">\n<div data-type=\"problem\" id=\"fs-id1167829651532\">\n<p id=\"fs-id1167829651534\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-680364708d79a9290f5fe3e6b60a249d_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;&#125;&#123;&#120;&#43;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"89\" style=\"vertical-align: -8px;\" \/> and<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae7349e36ea9517475ee34ca270e1ed1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"101\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836570221\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167836570229\">\n<div data-type=\"problem\" id=\"fs-id1167836570231\">\n<p id=\"fs-id1167836570233\">Donald thinks that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ec917966b595ee1a6a6ffff283b49cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7cc33be11d7a487dfe31ba961ff9b247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"21\" style=\"vertical-align: -6px;\" \/> Is Donald correct? Explain.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836570267\">\n<p id=\"fs-id1167836570269\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836570275\">\n<div data-type=\"problem\" id=\"fs-id1167836570277\">\n<p id=\"fs-id1167836570279\">Explain how you find the Least Common Denominator of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c619f0083ff54a4b80fd49a488dfdfc5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-013b79156dd2fd6b22037c7b63c9b39b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833086811\">\n<div data-type=\"problem\" id=\"fs-id1167833086813\">\n<p id=\"fs-id1167833086815\">Felipe thinks <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3da7345b677e6e73ec7eb8b523498a39_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;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"41\" style=\"vertical-align: -9px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce2509ce1aacd16a667359522fd7773d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#43;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"32\" style=\"vertical-align: -9px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> Choose numerical values for <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> and evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-452cfa3d14e0af918ae2be1673f187b2_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;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"46\" style=\"vertical-align: -9px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb56b81cde315969a95b70cb34aa9d2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#43;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" style=\"vertical-align: -9px;\" \/> for the same values of <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> you used in part <span class=\"token\">\u24d0<\/span>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Explain why Felipe is wrong.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> Find the correct expression for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-452cfa3d14e0af918ae2be1673f187b2_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;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"46\" style=\"vertical-align: -9px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829651079\">\n<p id=\"fs-id1167829651081\"><span class=\"token\">\u24d0<\/span> Answers will vary.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Answers will vary.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Answers will vary.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b73f70a8e04b13414e9b54be33bdfe59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#121;&#125;&#123;&#120;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"26\" style=\"vertical-align: -9px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836510855\">\n<div data-type=\"problem\" id=\"fs-id1167836510857\">\n<p id=\"fs-id1167836510860\">Simplify the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55d1c8d3370e0e0c5a424c86524573cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#110;&#43;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"117\" style=\"vertical-align: -9px;\" \/> and explain all your steps.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836510912\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167836510917\"><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-id1167836510932\" data-alt=\"This table has four columns and six 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 add and subtract rational expressions with a common denominator. In row 3, the I can was add and subtract rational expressions with denominators that are opposites. In row 4, the I can find the least common denominator of rational expressions. In row 5, the I can was add and subtract rational expressions with unlike denominators. In row 6, the I can was add or subtract rational functions. There is the nothing in the other columns.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_02_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and six 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 add and subtract rational expressions with a common denominator. In row 3, the I can was add and subtract rational expressions with denominators that are opposites. In row 4, the I can find the least common denominator of rational expressions. In row 5, the I can was add and subtract rational expressions with unlike denominators. In row 6, the I can was add or subtract rational functions. There is the nothing in the other columns.\" \/><\/span><\/p>\n<p id=\"fs-id1167836510928\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":103,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3228","chapter","type-chapter","status-publish","hentry"],"part":3130,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3228","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":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3228\/revisions"}],"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\/3228\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3228"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3228"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3228"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3228"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}