{"id":16345,"date":"2019-10-03T04:18:54","date_gmt":"2019-10-03T04:18:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-add-and-subtract-rational-expressions-part-ii\/"},"modified":"2024-05-01T19:04:34","modified_gmt":"2024-05-01T19:04:34","slug":"read-add-and-subtract-rational-expressions-part-ii","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-add-and-subtract-rational-expressions-part-ii\/","title":{"raw":"Adding and Subtractracting Rational Expressions Part II","rendered":"Adding and Subtractracting Rational Expressions Part II"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add and subtract rational expressions that share no common factors<\/li>\r\n \t<li>Add and subtract more than two rational expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\nSo far all the rational expressions you've added and subtracted have shared some factors. What happens when they don't have factors in common?\r\n\r\n[caption id=\"attachment_4994\" align=\"aligncenter\" width=\"520\"]<img class=\"wp-image-4994 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/17224721\/Screen-Shot-2016-06-17-at-3.46.55-PM.png\" alt=\"fish, screwdriver, socks and soccer ball to show no common factors\" width=\"520\" height=\"402\" \/> No Common Factors[\/caption]\r\n<h2>Add and Subtract Rational Expressions with No Common Factor<\/h2>\r\nIn the next example, we show how to find a common denominator when there are no common factors in the expressions.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract\u00a0[latex] \\displaystyle \\frac{3y}{2y-1}-\\frac{4}{y-5}[\/latex], and give the domain.\r\n\r\nState the difference in simplest form.\r\n\r\n[reveal-answer q=\"699142\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"699142\"]\r\n\r\nNeither [latex]2y\u20131[\/latex] nor [latex]y\u20135[\/latex] can be factored. Because they<i> <\/i>have no common factors, the least common multiple, which will become the least common denominator, is the product of these denominators.\r\n<p style=\"text-align: center;\">[latex]\\text{LCM}=\\left(2y-1\\right)\\left(y-5\\right)[\/latex]<\/p>\r\nMultiply each expression by the equivalent of [latex]1[\/latex] that will give it the common denominator.\r\n\r\nThen rewrite the subtraction problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3y}{2y-1}\\cdot \\frac{y-5}{y-5}=\\frac{3y(y-5)}{(2y-1)(y-5)}\\\\\\\\\\frac{4}{y-5}\\cdot \\frac{2y-1}{2y-1}=\\frac{4(2y-1)}{(2y-1)(y-5)}\\\\\\\\\\frac{3y(y-5)}{(2y-1)(y-5)}-\\frac{4(2y-1)}{(2y-1)(y-5)}\\end{array}[\/latex]<\/p>\r\nSubtract and simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3{{y}^{2}}-15y}{(2y-1)(y-5)}-\\frac{8y-4}{(2y-1)(y-5)}\\\\\\\\\\frac{3{{y}^{2}}-15y-(8y-4)}{(2y-1)(y-5)}\\\\\\\\\\frac{3{{y}^{2}}-15y-8y+4}{(2y-1)(y-5)}\\end{array}[\/latex]<\/p>\r\nThe domain is found by setting the original denominators equal to zero.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2y-1=0\\text{ and }y-5=0\\\\\\,\\,\\,y=\\frac{1}{2}\\,\\,\\,\\,\\,\\,\\,\\text{ and }y=5\\end{array}[\/latex]<\/p>\r\nThe domain is [latex]y\\ne\\frac{1}{2}, y\\ne5[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{3y}{2y-1}-\\frac{4}{y-5}=\\frac{3{{y}^{2}}-23y+4}{2{{y}^{2}}-11y+5},y\\ne \\frac{1}{2},5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=CKGpiTE5vIg&amp;feature=youtu.be\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]40252[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Add and Subtract More Than Two Rational Expressions<\/h2>\r\nYou may need to combine more than two rational expressions. While this may seem pretty straightforward if they all have the same denominator, what happens if they do not?\r\n\r\nIn the example below, notice how a common denominator is found for three rational expressions. Once that is done, the addition and subtraction of the terms looks the same as earlier, when you were only dealing with two terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify[latex]\\frac{2{{x}^{2}}}{{{x}^{2}}-4}+\\frac{x}{x-2}-\\frac{1}{x+2}[\/latex], and give the domain.\r\n\r\nState the result in simplest form.\r\n\r\n[reveal-answer q=\"47691\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"47691\"]\r\n\r\nFind the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization. Remember that <i>x<\/i> cannot be [latex]2[\/latex] or [latex]-2[\/latex] because the denominators would be [latex]0[\/latex].\r\n\r\n[latex]\\left(x+2\\right)[\/latex] appears a maximum of one time, as does [latex]\\left(x\u20132\\right)[\/latex]. This means the LCM is [latex]\\left(x+2\\right)\\left(x\u20132\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}x^{2}-4=\\left(x+2\\right)\\left(x-2\\right)\\\\\\,\\,x-2=x-2\\\\\\,\\,x+2=x+2\\\\\\,\\,\\text{LCM}=\\left(x+2\\right)\\left(x-2\\right)\\end{array}[\/latex]<\/p>\r\nThe LCM becomes the common denominator.\r\n\r\nMultiply each expression by the equivalent of [latex]1[\/latex] that will give it the common denominator.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{2{{x}^{2}}}{{{x}^{2}}-4}=\\frac{2{{x}^{2}}}{(x+2)(x-2)}\\\\\\frac{x}{x-2}\\cdot \\frac{x+2}{x+2}=\\frac{x(x+2)}{(x+2)(x-2)}\\\\\\frac{1}{x+2}\\cdot \\frac{x-2}{x-2}=\\frac{1(x-2)}{(x+2)(x-2)}\\end{array}[\/latex]<\/p>\r\nRewrite the original problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{2{{x}^{2}}}{(x+2)(x-2)}+\\frac{x(x+2)}{(x+2)(x-2)}-\\frac{1(x-2)}{(x+2)(x-2)}[\/latex]<\/p>\r\nCombine the numerators.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{c}\\frac{2{{x}^{2}}+x(x+2)-1(x-2)}{(x+2)(x-2)}\\\\\\\\\\frac{2{{x}^{2}}+{{x}^{2}}+2x-x+2}{(x+2)(x-2)}\\end{array}[\/latex]<\/p>\r\nCheck for simplest form. Since neither [latex]\\left(x+2\\right)[\/latex] nor [latex]\\left(x-2\\right)[\/latex] is a factor of [latex]3{{x}^{2}}+x+2[\/latex], this expression is in simplest form.\r\n<p style=\"text-align: center;\">[latex]\\frac{3{{x}^{2}}+x+2}{(x+2)(x-2)}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{2{{x}^{2}}}{{{x}^{2}}-4}+\\frac{x}{x-2}-\\frac{1}{x+2}=\\frac{3{{x}^{2}}+x+2}{(x+2)(x-2)}[\/latex][latex] \\displaystyle x\\ne 2,-2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows we present an example of subtracting [latex]3[\/latex] rational expressions with unlike denominators. One of the terms being subtracted is a number, so the denominator is [latex]1[\/latex].\r\nhttps:\/\/www.youtube.com\/watch?v=c-8xQyU0ch0&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify[latex]\\frac{{{y}^{2}}}{3y}-\\frac{2}{x}-\\frac{15}{9}[\/latex], and give the domain.\r\n\r\nState the result in simplest form.\r\n\r\n[reveal-answer q=\"77199\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"77199\"]\r\n\r\nFind the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,3y=3\\cdot{y}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x=x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,9=3\\cdot3\\\\\\text{LCM}=3\\cdot3\\cdot{x}\\cdot{y}\\\\\\text{LCM}=9xy\\end{array}[\/latex]<\/p>\r\nThe LCM becomes the common denominator. Multiply each expression by the equivalent of [latex]1[\/latex] that will give it the common denominator.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{{{y}^{2}}}{3y}\\cdot \\frac{3x}{3x}=\\frac{3x{{y}^{2}}}{9xy}\\\\\\\\\\frac{2}{x}\\cdot \\frac{9y}{9y}=\\frac{18y}{9xy}\\,\\,\\\\\\\\\\frac{15}{9}\\cdot \\frac{xy}{xy}=\\frac{15xy}{9xy}\\end{array}[\/latex]<\/p>\r\nRewrite the original problem with the common denominator.\r\n<p style=\"text-align: center;\">[latex]\\frac{3x{{y}^{2}}}{9xy}-\\frac{18y}{9xy}-\\frac{15xy}{9xy}[\/latex]<\/p>\r\nCombine the numerators.\r\n<p style=\"text-align: center;\">[latex]\\frac{3x{{y}^{2}}-18y-15xy}{9xy}[\/latex]<\/p>\r\nCheck for simplest form.\r\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{3y(xy-6-5x)}{9xy}\\\\\\\\=\\frac{3y(xy-6-5x)}{3y(3x)}\\\\\\\\=\\frac{\\cancel{3y}(xy-6-5x)}{\\cancel{3y}(3x)}\\\\\\\\=\\frac{xy-6-5x}{3x}\\end{array}[\/latex]<\/p>\r\nThe domain is found by setting the denominators equal to zero. [latex]9=0[\/latex] is nonsense, so we don't need to worry about that denominator.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}3y=0\\text{ and }x=0\\\\y=0\\text{ and }x=0\\end{array}[\/latex]<\/p>\r\nThe domain is [latex]y\\ne0, x\\ne0[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{{{y}^{2}}}{3y}-\\frac{2}{x}-\\frac{15}{9}=\\frac{xy-5x-6}{3x},y\\ne 0,x\\ne 0[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this last video, we present another example of adding and subtracting three rational expressions with unlike denominators.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=43xPStLm39A&amp;feature=youtu.be\r\n\r\n[caption id=\"attachment_5017\" align=\"aligncenter\" width=\"270\"]<img class=\" wp-image-5017\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/20042339\/Screen-Shot-2016-06-19-at-9.23.12-PM.png\" alt=\"Add and Subtract\" width=\"270\" height=\"130\" \/> Add and Subtract[\/caption]\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]189261[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nThe methods shown here will help you when you are solving rational equations later on. \u00a0To add and subtract rational expressions that share common factors, you first identify which factors are missing from each expression, and build the LCD with them. To add and subtract rational expressions with no common factors, the LCD will be the product of all the factors of the denominators.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add and subtract rational expressions that share no common factors<\/li>\n<li>Add and subtract more than two rational expressions<\/li>\n<\/ul>\n<\/div>\n<p>So far all the rational expressions you&#8217;ve added and subtracted have shared some factors. What happens when they don&#8217;t have factors in common?<\/p>\n<div id=\"attachment_4994\" style=\"width: 530px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4994\" class=\"wp-image-4994 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/17224721\/Screen-Shot-2016-06-17-at-3.46.55-PM.png\" alt=\"fish, screwdriver, socks and soccer ball to show no common factors\" width=\"520\" height=\"402\" \/><\/p>\n<p id=\"caption-attachment-4994\" class=\"wp-caption-text\">No Common Factors<\/p>\n<\/div>\n<h2>Add and Subtract Rational Expressions with No Common Factor<\/h2>\n<p>In the next example, we show how to find a common denominator when there are no common factors in the expressions.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract\u00a0[latex]\\displaystyle \\frac{3y}{2y-1}-\\frac{4}{y-5}[\/latex], and give the domain.<\/p>\n<p>State the difference in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q699142\">Show Solution<\/span><\/p>\n<div id=\"q699142\" class=\"hidden-answer\" style=\"display: none\">\n<p>Neither [latex]2y\u20131[\/latex] nor [latex]y\u20135[\/latex] can be factored. Because they<i> <\/i>have no common factors, the least common multiple, which will become the least common denominator, is the product of these denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{LCM}=\\left(2y-1\\right)\\left(y-5\\right)[\/latex]<\/p>\n<p>Multiply each expression by the equivalent of [latex]1[\/latex] that will give it the common denominator.<\/p>\n<p>Then rewrite the subtraction problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3y}{2y-1}\\cdot \\frac{y-5}{y-5}=\\frac{3y(y-5)}{(2y-1)(y-5)}\\\\\\\\\\frac{4}{y-5}\\cdot \\frac{2y-1}{2y-1}=\\frac{4(2y-1)}{(2y-1)(y-5)}\\\\\\\\\\frac{3y(y-5)}{(2y-1)(y-5)}-\\frac{4(2y-1)}{(2y-1)(y-5)}\\end{array}[\/latex]<\/p>\n<p>Subtract and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{3{{y}^{2}}-15y}{(2y-1)(y-5)}-\\frac{8y-4}{(2y-1)(y-5)}\\\\\\\\\\frac{3{{y}^{2}}-15y-(8y-4)}{(2y-1)(y-5)}\\\\\\\\\\frac{3{{y}^{2}}-15y-8y+4}{(2y-1)(y-5)}\\end{array}[\/latex]<\/p>\n<p>The domain is found by setting the original denominators equal to zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2y-1=0\\text{ and }y-5=0\\\\\\,\\,\\,y=\\frac{1}{2}\\,\\,\\,\\,\\,\\,\\,\\text{ and }y=5\\end{array}[\/latex]<\/p>\n<p>The domain is [latex]y\\ne\\frac{1}{2}, y\\ne5[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{3y}{2y-1}-\\frac{4}{y-5}=\\frac{3{{y}^{2}}-23y+4}{2{{y}^{2}}-11y+5},y\\ne \\frac{1}{2},5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Add Rational Expressions with Unlike Denominators\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/CKGpiTE5vIg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm40252\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40252&theme=oea&iframe_resize_id=ohm40252&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Add and Subtract More Than Two Rational Expressions<\/h2>\n<p>You may need to combine more than two rational expressions. While this may seem pretty straightforward if they all have the same denominator, what happens if they do not?<\/p>\n<p>In the example below, notice how a common denominator is found for three rational expressions. Once that is done, the addition and subtraction of the terms looks the same as earlier, when you were only dealing with two terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify[latex]\\frac{2{{x}^{2}}}{{{x}^{2}}-4}+\\frac{x}{x-2}-\\frac{1}{x+2}[\/latex], and give the domain.<\/p>\n<p>State the result in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q47691\">Show Solution<\/span><\/p>\n<div id=\"q47691\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization. Remember that <i>x<\/i> cannot be [latex]2[\/latex] or [latex]-2[\/latex] because the denominators would be [latex]0[\/latex].<\/p>\n<p>[latex]\\left(x+2\\right)[\/latex] appears a maximum of one time, as does [latex]\\left(x\u20132\\right)[\/latex]. This means the LCM is [latex]\\left(x+2\\right)\\left(x\u20132\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}x^{2}-4=\\left(x+2\\right)\\left(x-2\\right)\\\\\\,\\,x-2=x-2\\\\\\,\\,x+2=x+2\\\\\\,\\,\\text{LCM}=\\left(x+2\\right)\\left(x-2\\right)\\end{array}[\/latex]<\/p>\n<p>The LCM becomes the common denominator.<\/p>\n<p>Multiply each expression by the equivalent of [latex]1[\/latex] that will give it the common denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{2{{x}^{2}}}{{{x}^{2}}-4}=\\frac{2{{x}^{2}}}{(x+2)(x-2)}\\\\\\frac{x}{x-2}\\cdot \\frac{x+2}{x+2}=\\frac{x(x+2)}{(x+2)(x-2)}\\\\\\frac{1}{x+2}\\cdot \\frac{x-2}{x-2}=\\frac{1(x-2)}{(x+2)(x-2)}\\end{array}[\/latex]<\/p>\n<p>Rewrite the original problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{2{{x}^{2}}}{(x+2)(x-2)}+\\frac{x(x+2)}{(x+2)(x-2)}-\\frac{1(x-2)}{(x+2)(x-2)}[\/latex]<\/p>\n<p>Combine the numerators.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{2{{x}^{2}}+x(x+2)-1(x-2)}{(x+2)(x-2)}\\\\\\\\\\frac{2{{x}^{2}}+{{x}^{2}}+2x-x+2}{(x+2)(x-2)}\\end{array}[\/latex]<\/p>\n<p>Check for simplest form. Since neither [latex]\\left(x+2\\right)[\/latex] nor [latex]\\left(x-2\\right)[\/latex] is a factor of [latex]3{{x}^{2}}+x+2[\/latex], this expression is in simplest form.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{3{{x}^{2}}+x+2}{(x+2)(x-2)}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{2{{x}^{2}}}{{{x}^{2}}-4}+\\frac{x}{x-2}-\\frac{1}{x+2}=\\frac{3{{x}^{2}}+x+2}{(x+2)(x-2)}[\/latex][latex]\\displaystyle x\\ne 2,-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video that follows we present an example of subtracting [latex]3[\/latex] rational expressions with unlike denominators. One of the terms being subtracted is a number, so the denominator is [latex]1[\/latex].<br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Subtract Rational Expressions with UnLike Denominators - 3 Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/c-8xQyU0ch0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify[latex]\\frac{{{y}^{2}}}{3y}-\\frac{2}{x}-\\frac{15}{9}[\/latex], and give the domain.<\/p>\n<p>State the result in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q77199\">Show Solution<\/span><\/p>\n<div id=\"q77199\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,3y=3\\cdot{y}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x=x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,9=3\\cdot3\\\\\\text{LCM}=3\\cdot3\\cdot{x}\\cdot{y}\\\\\\text{LCM}=9xy\\end{array}[\/latex]<\/p>\n<p>The LCM becomes the common denominator. Multiply each expression by the equivalent of [latex]1[\/latex] that will give it the common denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{{{y}^{2}}}{3y}\\cdot \\frac{3x}{3x}=\\frac{3x{{y}^{2}}}{9xy}\\\\\\\\\\frac{2}{x}\\cdot \\frac{9y}{9y}=\\frac{18y}{9xy}\\,\\,\\\\\\\\\\frac{15}{9}\\cdot \\frac{xy}{xy}=\\frac{15xy}{9xy}\\end{array}[\/latex]<\/p>\n<p>Rewrite the original problem with the common denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{3x{{y}^{2}}}{9xy}-\\frac{18y}{9xy}-\\frac{15xy}{9xy}[\/latex]<\/p>\n<p>Combine the numerators.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{3x{{y}^{2}}-18y-15xy}{9xy}[\/latex]<\/p>\n<p>Check for simplest form.<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{3y(xy-6-5x)}{9xy}\\\\\\\\=\\frac{3y(xy-6-5x)}{3y(3x)}\\\\\\\\=\\frac{\\cancel{3y}(xy-6-5x)}{\\cancel{3y}(3x)}\\\\\\\\=\\frac{xy-6-5x}{3x}\\end{array}[\/latex]<\/p>\n<p>The domain is found by setting the denominators equal to zero. [latex]9=0[\/latex] is nonsense, so we don&#8217;t need to worry about that denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}3y=0\\text{ and }x=0\\\\y=0\\text{ and }x=0\\end{array}[\/latex]<\/p>\n<p>The domain is [latex]y\\ne0, x\\ne0[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{{{y}^{2}}}{3y}-\\frac{2}{x}-\\frac{15}{9}=\\frac{xy-5x-6}{3x},y\\ne 0,x\\ne 0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In this last video, we present another example of adding and subtracting three rational expressions with unlike denominators.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Add and Subtract Rational Expressions with UnLike Denominators - 3 Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/43xPStLm39A?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"attachment_5017\" style=\"width: 280px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5017\" class=\"wp-image-5017\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/20042339\/Screen-Shot-2016-06-19-at-9.23.12-PM.png\" alt=\"Add and Subtract\" width=\"270\" height=\"130\" \/><\/p>\n<p id=\"caption-attachment-5017\" class=\"wp-caption-text\">Add and Subtract<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm189261\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=189261&theme=oea&iframe_resize_id=ohm189261&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Summary<\/h2>\n<p>The methods shown here will help you when you are solving rational equations later on. \u00a0To add and subtract rational expressions that share common factors, you first identify which factors are missing from each expression, and build the LCD with them. To add and subtract rational expressions with no common factors, the LCD will be the product of all the factors of the denominators.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-16345\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Image: No common factors.. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Add and subtract. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Subtract Rational Expressions with UnLike Denominators - 3 Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/c-8xQyU0ch0\">https:\/\/youtu.be\/c-8xQyU0ch0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Add and Subtract Rational Expressions with UnLike Denominators - 3 Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/43xPStLm39A\">https:\/\/youtu.be\/43xPStLm39A<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 15: Rational Expressions, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Add Rational Expressions with Unlike Denominators. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=CKGpiTE5vIg&#038;feature=youtu.be\">https:\/\/www.youtube.com\/watch?v=CKGpiTE5vIg&#038;feature=youtu.be<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Image: No common factors.\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: Add and subtract\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 15: Rational Expressions, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Add Rational Expressions with Unlike Denominators\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/www.youtube.com\/watch?v=CKGpiTE5vIg&feature=youtu.be\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Subtract Rational Expressions with UnLike Denominators - 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