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<div class="section" id="fwk-redden-ch04_s07" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">4.7</span> Solving Rational Equations</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch04_s07_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch04_s07_o01" numeration="arabic">
<li>Solve rational equations.</li>
<li>Solve literal equations, or formulas, involving rational expressions.</li>
<li>Solve applications involving the reciprocal of unknowns.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch04_s07_s01" version="5.0" lang="en">
<h2 class="title editable block">Solving Rational Equations</h2>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p01">A <span class="margin_term"><a class="glossterm">rational equation</a><span class="glossdef">An equation containing at least one rational expression.</span></span> is an equation containing at least one rational expression. Rational expressions typically contain a variable in the denominator. For this reason, we will take care to ensure that the denominator is not 0 by making note of restrictions and checking our solutions. Solving rational equations involves clearing fractions by multiplying both sides of the equation by the least common denominator (LCD).</p>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p02">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2154" display="inline"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p03">We first make a note of the restriction on <em class="emphasis">x</em>, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2155" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span> We then multiply both sides by the LCD, which in this case equals <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2156" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2157" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mspace width="1em"></mspace><mrow><mstyle color="#007fbf"><mrow><mi>M</mi><mi>u</mi><mi>l</mi><mi>t</mi><mi>i</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>L</mi><mi>C</mi><mi>D</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mstyle color="#007fbf"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mfrac><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mspace width="1em"></mspace><mrow><mstyle color="#007fbf"><mrow><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mspace width="1em"></mspace><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mi>n</mi><mtext> </mtext><mi>s</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p05">Check your answer. Substitute <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2158" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>5</mn></mrow></math></span> into the original equation and see if you obtain a true statement.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2159" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>O</mi><mi>r</mi><mi>i</mi><mi>g</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mrow><mstyle color="#007f3f"><mn>5</mn></mstyle></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><msup><mrow><mstyle color="#007f3f"><mn>5</mn></mstyle></mrow><mn>2</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mstyle color="#007f3f"><mn>5</mn></mstyle><mo>+</mo><mn>9</mn></mrow><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>5</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mn>5.</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mn>25</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>14</mn></mrow><mrow><mn>2</mn><mo>⋅</mo><mn>25</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>5</mn><mrow><mn>25</mn></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mn>25</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>7</mn><mrow><mn>25</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>7</mn><mrow><mn>25</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>7</mn><mrow><mn>25</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p07">Answer: The solution is 5.</p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p08">After multiplying both sides of the previous example by the LCD, we were left with a linear equation to solve. This is not always the case; sometimes we will be left with quadratic equation.</p>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p09">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2160" display="inline"><mrow><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p10">In this example, there are two restrictions, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2161" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>4</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2162" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>2</mn></mrow><mo>.</mo></math></span> Begin by multiplying both sides by the LCD, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2163" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p11"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2164" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></menclose></mrow></mfrac><mo>−</mo><mstyle color="#007fbf"><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></menclose></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></menclose></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>16</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p12">After distributing and simplifying both sides of the equation, a quadratic equation remains. To solve, rewrite the quadratic equation in standard form, factor, and then set each factor equal to 0.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p13"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2165" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p14">Check to see if these values solve the original equation.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p15"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2166" display="block"><mrow><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac></mrow></math></span></p>
<p class="para">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="left"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2167" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p></th>
<th align="left"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2168" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>4</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2169" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>0</mn><mtext> </mtext></mstyle><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mstyle color="#007fbf"><mn>0</mn><mtext> </mtext></mstyle><mo>−</mo><mn>4</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mstyle color="#007fbf"><mn>0</mn><mtext> </mtext></mstyle><mo>+</mo><mn>4</mn></mrow><mrow><mstyle color="#007fbf"><mn>0</mn><mtext> </mtext></mstyle><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mstyle color="#007fbf"><mn>0</mn><mtext> </mtext></mstyle><mo>−</mo><mn>2</mn></mrow><mrow><mstyle color="#007fbf"><mn>0</mn><mtext> </mtext></mstyle><mo>−</mo><mn>4</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>6</mn><mrow><mo>−</mo><mn>4</mn></mrow></mfrac><mo>−</mo><mfrac><mn>4</mn><mrow><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>2</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>4</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext> ✓</mtext></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2170" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mn>4</mn></mstyle><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mn>4</mn></mstyle><mo>−</mo><mn>4</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mn>4</mn></mstyle><mo>+</mo><mn>4</mn></mrow><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mn>4</mn></mstyle><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mn>4</mn></mstyle><mo>−</mo><mn>2</mn></mrow><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mn>4</mn></mstyle><mo>−</mo><mtext> </mtext><mn>4</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>8</mn></mrow></mfrac><mo>−</mo><mfrac><mn>0</mn><mrow><mo>−</mo><mn>6</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>6</mn></mrow><mrow><mo>−</mo><mn>8</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mo>−</mo><mn>6</mn></mrow><mrow><mo>−</mo><mn>8</mn></mrow></mfrac><mo>−</mo><mn>0</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext> ✓</mtext></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch04_s07_s01_p17">Answer: The solutions are 0 and −4.</p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p18">Up to this point, all of the possible solutions have solved the original equation. However, this may not always be the case. Multiplying both sides of an equation by variable factors may lead to <span class="margin_term"><a class="glossterm">extraneous solutions</a><span class="glossdef">A solution that does not solve the original equation.</span></span>, which are solutions that do not solve the original equation. A complete list of steps for solving a rational equation is outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p19">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2171" display="inline"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>14</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p20"><strong class="emphasis bold">Step 1:</strong> Factor all denominators and determine the LCD.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p21"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2172" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>14</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p22">The LCD is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2173" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p23"><strong class="emphasis bold">Step 2:</strong> Identify the restrictions. In this case, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2174" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2175" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>5</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p24"><strong class="emphasis bold">Step 3:</strong> Multiply both sides of the equation by the LCD. Distribute carefully and then simplify.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p25"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2176" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow></mfrac><mo>−</mo><mstyle color="#007fbf"><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p26"><strong class="emphasis bold">Step 4:</strong> Solve the resulting equation. Here the result is a quadratic equation. Rewrite it in standard form, factor, and then set each factor equal to 0.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p27"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2177" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>4</mn><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p28"><strong class="emphasis bold">Step 5:</strong> Check for extraneous solutions. Always substitute into the original equation, or the factored equivalent. In this case, choose the factored equivalent to check:</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p29"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2178" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span></p>
<p class="para">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="left"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2179" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mo>1</mo><mo>2</mo></mfrac></mrow></math></span></p></th>
<th align="left"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2180" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>5</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="left"><p class="para"><span class="inlineequation">
<math xml:id="fwk-redden-ch04_m2181" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mstyle scriptlevel="+1"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mstyle scriptlevel="+1"><mfrac><mrow><mn>11</mn></mrow><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mo>−</mo><mstyle scriptlevel="+1"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mstyle scriptlevel="+1"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mo>−</mo><mstyle scriptlevel="+1"><mfrac><mrow><mn>11</mn></mrow><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>2</mn><mrow><mn>11</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mo>−</mo><mn>6</mn></mrow><mrow><mrow><mo>(</mo><mrow><mstyle scriptlevel="+1"><mfrac><mrow><mn>11</mn></mrow><mn>4</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>2</mn><mrow><mn>11</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>24</mn></mrow><mrow><mn>11</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>22</mn></mrow><mrow><mn>11</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> ✓</mtext></mrow></mtd></mtr></mtable></mrow></math>
</span></p></td>
<td align="left"><p class="para"><span class="inlineequation">
<math xml:id="fwk-redden-ch04_m2182" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>2</mn><mo>⋅</mo><mstyle color="#007fbf"><mn>5</mn></mstyle></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>⋅</mo><mstyle color="#007fbf"><mn>5</mn></mstyle><mtext> </mtext><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>5</mn></mstyle><mtext> </mtext><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>5</mn></mstyle><mtext> </mtext><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>⋅</mo><mstyle color="#007fbf"><mn>5</mn></mstyle><mtext> </mtext><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>5</mn></mstyle><mtext> </mtext><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>0</mn></mfrac><mo>−</mo><mfrac><mrow><mn>16</mn></mrow><mrow><mrow><mo>(</mo><mrow><mn>16</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>10</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>0</mn></mfrac><mo>−</mo><mfrac><mrow><mn>16</mn></mrow><mn>0</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mtext> ✗</mtext></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p></td>
</tr>
<tr>
<td align="left"></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2183" display="inline"><mstyle color="#007fbf"><mi>U</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>f</mi><mi>i</mi><mi>n</mi><mi>e</mi><mi>d</mi><mtext> </mtext><mi>t</mi><mi>e</mi><mi>r</mi><mi>m</mi><mi>s</mi><mo>!</mo></mstyle></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch04_s07_s01_p31">Here 5 is an extraneous solution and is not included in the solution set. It is important to note that 5 is a restriction.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p32">Answer: The solution is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2184" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p33">If this process produces a solution that happens to be a restriction, then disregard it as a solution.</p>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p34"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2185" display="inline"><mrow><mfrac><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>36</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>6</mn><mo>−</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>6</mn><mo>+</mo><mi>x</mi></mrow></mfrac></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p35">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2186" display="inline"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/Mxln5c53J1M" condition="http://img.youtube.com/vi/Mxln5c53J1M/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/Mxln5c53J1M" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p37">Sometimes all potential solutions are extraneous, in which case we say that there is no solution to the original equation. In the next two examples, we demonstrate two ways in which rational equation can have no solutions.</p>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p38">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2187" display="inline"><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>22</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p39">To identify the LCD, first factor the denominators.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p40"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2188" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>1</mn><mo>+</mo><mfrac><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>22</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>1</mn><mo>+</mo><mfrac><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>22</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p41">Multiply both sides by the LCD, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2189" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>, distributing carefully.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p42"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2190" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mo stretchy="true">(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>22</mn></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="true">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mn>1</mn><mo>+</mo><mstyle color="#007fbf"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>22</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle><mo>⋅</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>22</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>4</mn><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>22</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>18</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>18</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>16</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#ff0000"><mrow><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p43">The equation is a contradiction and thus has no solution.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p44">Answer: No solution, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2191" display="inline"><mo>Ø</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n05">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p45">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2192" display="inline"><mrow><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn></mrow></mfrac><mo>=</mo><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p46">First, factor the denominators.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p47"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2193" display="block"><mrow><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mi>x</mi><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p48">Take note that the restrictions on the domain are <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2194" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>±</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>.</mo></math></span> To clear the fractions, multiply by the LCD, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2195" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p49"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2196" display="block"><mtable columnspacing="0.1em" columnalign="center"><mtr columnalign="center"><mtd columnalign="center"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>3</mn><mi>x</mi><mo>⋅</mo><mstyle color="#007fbf"><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mstyle></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mo stretchy="false">(</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>⋅</mo><mstyle color="#007fbf"><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mstyle></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>x</mi><mo>⋅</mo><mstyle color="#007fbf"><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mstyle></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>−</mo><mn>3</mn><mo stretchy="false">(</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>x</mi><mo>−</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mtable columnspacing="0.1em" columnalign="center"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p50">Both of these values are restrictions of the original equation; hence both are extraneous.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p51">Answer: No solution, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2197" display="inline"><mo>Ø</mo></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p52">It is important to point out that this technique for clearing algebraic fractions only works for equations. Do not try to clear algebraic fractions when simplifying expressions. As a reminder, an example of each is provided below.</p>
<p class="para block">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="left"><p class="para">Expression</p></th>
<th align="left"><p class="para">Equation</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2198" display="inline"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2199" display="inline"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch04_s07_s01_p54">Expressions are to be simplified and equations are to be solved. If we multiply the expression by the LCD, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2200" display="inline"><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>, we obtain another expression that is not equivalent.</p>
<p class="para block">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="left"><p class="para">Incorrect</p></th>
<th align="left"><p class="para">Correct</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2201" display="inline"><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>≠</mo><mstyle color="#ff0000"><mi>x</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo></mstyle><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mtext> ✗</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="left"><p class="para"><span class="inlineequation">
<math xml:id="fwk-redden-ch04_m2202" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mi>x</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo></mstyle><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>x</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo></mstyle><mn>0</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext> ✓</mtext></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p56">Rational equations are sometimes expressed using negative exponents.</p>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n06">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p57">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2203" display="inline"><mrow><mn>6</mn><mo>+</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p58">Begin by removing the negative exponents.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p59"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2204" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mn>6</mn><mo>+</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p60">Here we can see the restriction, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2205" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span> Next, multiply both sides by the LCD, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2206" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p61"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2207" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mn>6</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mn>6</mn><mo>+</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>o</mi><mi>r</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p62">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2208" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2209" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p63">A <span class="margin_term"><a class="glossterm">proportion</a><span class="glossdef">A statement of equality of two ratios.</span></span> is a statement of equality of two ratios.</p>
<p class="para block" id="fwk-redden-ch04_s07_s01_p64"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2210" display="block"><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>=</mo><mfrac><mi>c</mi><mi>d</mi></mfrac></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p65">This proportion is often read “<em class="emphasis">a</em> is to <em class="emphasis">b</em> as <em class="emphasis">c</em> is to <em class="emphasis">d</em>.” Given any nonzero real numbers <em class="emphasis">a</em>, <em class="emphasis">b</em>, <em class="emphasis">c</em>, and <em class="emphasis">d</em> that satisfy a proportion, multiply both sides by the product of the denominators to obtain the following:</p>
<p class="para block" id="fwk-redden-ch04_s07_s01_p66"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2211" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mi>c</mi><mi>d</mi></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>b</mi><mi>d</mi></mrow></mstyle><mo>⋅</mo><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>b</mi><mi>d</mi></mrow></mstyle><mo>⋅</mo><mfrac><mi>c</mi><mi>d</mi></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>a</mi><mi>d</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>b</mi><mi>c</mi></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch04_s07_s01_p67">This shows that cross products are equal, and is commonly referred to as <span class="margin_term"><a class="glossterm">cross multiplication</a><span class="glossdef">If <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2212" display="inline"><mtext> </mtext><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>=</mo><mfrac><mi>c</mi><mi>d</mi></mfrac></mrow></math></span> then <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2213" display="inline"><mrow><mi>a</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>c</mi></mrow><mo>.</mo></math></span></span></span>.</p>
<p class="para block" id="fwk-redden-ch04_s07_s01_p68"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2214" display="block"><mrow><mtext>If </mtext><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>=</mo><mfrac><mi>c</mi><mi>d</mi></mfrac><mtext> then </mtext><mi>a</mi><mi>d</mi><mo>=</mo><mi>b</mi><mi>c</mi></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p69">Cross multiply to solve proportions where terms are unknown.</p>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n07">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p70">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2215" display="inline"><mrow><mfrac><mrow><mn>5</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mn>5</mn></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mi>n</mi></mrow><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p71">When cross multiplying, be sure to group <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2216" display="inline"><mrow><mn>5</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large">
<img src="section_07/604ccfd417a0ed5b99b91e7f2a4220f5.png">
</div>
<p class="para" id="fwk-redden-ch04_s07_s01_p73"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2217" display="block"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>2</mn><mo>=</mo><mn>5</mn><mo>⋅</mo><mn>3</mn><mi>n</mi></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p74">Apply the distributive property in the next step.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p75"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2218" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mo stretchy="false">(</mo><mn>5</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mo>⋅</mo><mn>3</mn><mi>n</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>10</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>15</mn><mi>n</mi></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mi>n</mi></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mo>−</mo><mn>2</mn></mrow><mn>5</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>n</mi></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p76">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2219" display="inline"><mrow><mi>n</mi><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s07_s01_p77">Cross multiplication can be used as an alternate method for solving rational equations. The idea is to simplify each side of the equation to a single algebraic fraction and then cross multiply.</p>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n08">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p78">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2220" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>4</mn><mi>x</mi></mfrac><mo>=</mo><mo>−</mo><mfrac><mi>x</mi><mn>8</mn></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p79">Obtain a single algebraic fraction on the left side by subtracting the equivalent fractions with a common denominator.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p80"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2221" display="block"><mrow><mtable columnspacing="0.1em" columnalign="right"><mtr columnalign="right"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mi>x</mi><mi>x</mi></mfrac></mrow></mstyle><mo>−</mo><mfrac><mn>4</mn><mi>x</mi></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mn>2</mn><mn>2</mn></mfrac></mrow></mstyle></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="right"><mrow><mo>−</mo><mfrac><mi>x</mi><mn>8</mn></mfrac></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mn>8</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="right"><mrow><mo>−</mo><mfrac><mi>x</mi><mn>8</mn></mfrac></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="right"><mrow><mo>−</mo><mfrac><mi>x</mi><mn>8</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p81">Note that <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2222" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></math></span>, cross multiply, and then solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p82"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2223" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mi>x</mi></mrow><mn>8</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mn>8</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mi>x</mi><mo>⋅</mo><mn>2</mn><mi>x</mi></mtd></mtr><mtr><mtd columnalign="right"><mn>8</mn><mi>x</mi><mo>−</mo><mn>64</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>64</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>32</mn></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p83">Next, set each variable factor equal to zero.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p84"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2224" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p85">The check is left to the reader.</p>
<p class="para" id="fwk-redden-ch04_s07_s01_p86">Answer: −8, 4</p>
</div>
<div class="callout block" id="fwk-redden-ch04_s07_s01_n08a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s07_s01_p87"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2225" display="inline"><mrow><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s01_p88">Answer: 2, 3</p>
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</div>
<div class="section" id="fwk-redden-ch04_s07_s02" version="5.0" lang="en">
<h2 class="title editable block">Solving Literal Equations and Applications Involving Reciprocals</h2>
<p class="para editable block" id="fwk-redden-ch04_s07_s02_p01">Literal equations, or formulas, are often rational equations. Hence the techniques described in this section can be used to solve for particular variables. Assume that all variable expressions in the denominator are nonzero.</p>
<div class="callout block" id="fwk-redden-ch04_s07_s02_n01">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch04_s07_s02_p02">The reciprocal of the combined resistance <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2226" display="inline"><mi>R</mi></math></span> of two resistors <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2227" display="inline"><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2228" display="inline"><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow></math></span> in parallel is given by the formula <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2229" display="inline"><mrow><mfrac><mn>1</mn><mi>R</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow></mfrac></mrow><mo>.</mo></math></span> Solve for <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2230" display="inline"><mi>R</mi></math></span> in terms of <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2231" display="inline"><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2232" display="inline"><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large">
<img src="section_07/ac7ad8c0c038597c808c613f51bca7bd.png">
</div>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s02_p04">The goal is to isolate <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2233" display="inline"><mi>R</mi></math></span> on one side of the equation. Begin by multiplying both sides of the equation by the LCD, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2234" display="inline"><mrow><mi>R</mi><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s02_p05"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2235" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mi>R</mi><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mrow></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mi>R</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>R</mi><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mrow></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>1</mn></msub></mrow></mfrac><mo>+</mo><mstyle color="#007fbf"><mrow><mi>R</mi><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mrow></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mrow><msub><mi>R</mi><mn>2</mn></msub></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>R</mi><msub><mi>R</mi><mn>2</mn></msub><mo>+</mo><mi>R</mi><msub><mi>R</mi><mn>1</mn></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>R</mi><mrow><mo>(</mo><mrow><msub><mi>R</mi><mn>2</mn></msub><mo>+</mo><msub><mi>R</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mrow><mrow><msub><mi>R</mi><mn>2</mn></msub><mo>+</mo><msub><mi>R</mi><mn>1</mn></msub></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>R</mi></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s02_p06">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2236" display="inline"><mrow><mi>R</mi><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mn>1</mn></msub><msub><mi>R</mi><mn>2</mn></msub></mrow><mrow><msub><mi>R</mi><mn>1</mn></msub><mo>+</mo><msub><mi>R</mi><mn>2</mn></msub></mrow></mfrac></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s07_s02_n01a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s07_s02_p07"><strong class="emphasis bold">Try this!</strong> Solve for <em class="emphasis">y</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2237" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mrow><mi>y</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s02_p08">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2238" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span></p>
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<p class="para block" id="fwk-redden-ch04_s07_s02_p10">Recall that the reciprocal of a nonzero number <em class="emphasis">n</em> is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2239" display="inline"><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac></mrow><mo>.</mo></math></span> For example, the reciprocal of 5 is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2240" display="inline"><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2241" display="inline"><mrow><mn>5</mn><mo>⋅</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span> In this section, the applications will often involve the key word “reciprocal.” When this is the case, we will see that the algebraic setup results in a rational equation.</p>
<div class="callout block" id="fwk-redden-ch04_s07_s02_n02">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch04_s07_s02_p11">A positive integer is 3 less than another. If the reciprocal of the smaller integer is subtracted from twice the reciprocal of the larger, then the result is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2242" display="inline"><mrow><mfrac><mn>1</mn><mn>20</mn></mfrac></mrow><mo>.</mo></math></span> Find the two integers.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s07_s02_p12">Let <em class="emphasis">n</em> represent the larger positive integer.</p>
<p class="para" id="fwk-redden-ch04_s07_s02_p13">Let <em class="emphasis">n</em> − 3 represent the smaller positive integer.</p>
<p class="para" id="fwk-redden-ch04_s07_s02_p14">Set up an algebraic equation.</p>
<div class="informalfigure large">
<img src="section_07/eccee167d90ab6bcfed58a319551e4fc.png">
</div>
<p class="para" id="fwk-redden-ch04_s07_s02_p16">Solve this rational expression by multiplying both sides by the LCD. The LCD is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2243" display="inline"><mrow><mn>20</mn><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s02_p17"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2244" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>2</mn><mi>n</mi></mfrac><mi> </mi><mi> </mi><mi> </mi><mo>−</mo><mi> </mi><mi> </mi><mi> </mi><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mn>20</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mn>20</mn><mi>n</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mi>n</mi></mfrac><mi> </mi><mi> </mi><mi> </mi><mo>−</mo><mi> </mi><mi> </mi><mi> </mi><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mn>20</mn><mi>n</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>20</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mrow><mn>20</mn><mi>n</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mfrac><mn>2</mn><mi>n</mi></mfrac><mi> </mi><mi> </mi><mi> </mi><mo>−</mo><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mrow><mn>20</mn><mi>n</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mn>20</mn><mi>n</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>20</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s02_p18"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2245" display="block"><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>40</mn><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>20</mn><mi>n</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>n</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>40</mn><mi>n</mi><mo>−</mo><mn>120</mn><mo>−</mo><mn>20</mn><mi>n</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>n</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>20</mn><mi>n</mi><mo>−</mo><mn>120</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>n</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mn>23</mn><mi>n</mi><mo>+</mo><mn>120</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>15</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>n</mi><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mi>n</mi><mo>−</mo><mn>15</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>n</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>n</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>15</mn></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s02_p19">Here we have two viable possibilities for the larger integer <em class="emphasis">n</em>. For this reason, we will we have two solutions to this problem.</p>
<p class="para" id="fwk-redden-ch04_s07_s02_p20">If <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2246" display="inline"><mrow><mi>n</mi><mo>=</mo><mn>8</mn></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2247" display="inline"><mrow><mi>n</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>8</mn><mo>−</mo><mn>3</mn><mo>=</mo><mn>5</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s02_p21">If <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2248" display="inline"><mrow><mi>n</mi><mo>=</mo><mn>15</mn></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2249" display="inline"><mrow><mi>n</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>15</mn><mo>−</mo><mn>3</mn><mo>=</mo><mn>12</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s07_s02_p22">As a check, perform the operations indicated in the problem.</p>
<p class="para" id="fwk-redden-ch04_s07_s02_p23"><span class="informalequation"><math xml:id="fwk-redden-ch04_m2250" display="block"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac></mrow><mo>)</mo></mrow><mi> </mi><mi> </mi><mi> </mi><mo>−</mo><mi> </mi><mi> </mi><mi> </mi><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>20</mn></mrow></mfrac></mrow></math></span></p>
<p class="para">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="left"><p class="para">Check 8 and 5.</p></th>
<th align="left"><p class="para">Check 15 and 12.</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2251" display="inline"><mrow><mi> </mi><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mstyle color="#007fbf"><mn>8</mn></mstyle></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mn>1</mn><mstyle color="#007fbf"><mn>5</mn></mstyle></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>5</mn><mrow><mn>20</mn></mrow></mfrac><mo>−</mo><mfrac><mn>4</mn><mrow><mn>20</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mn>20</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext> ✓</mtext></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2252" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mstyle color="#007fbf"><mn>15</mn></mstyle></mrow></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mstyle color="#007fbf"><mn>12</mn></mstyle></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>2</mn><mrow><mn>15</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>8</mn><mrow><mn>60</mn></mrow></mfrac><mo>−</mo><mfrac><mn>5</mn><mrow><mn>60</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>3</mn><mrow><mn>60</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>20</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext> ✓</mtext></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch04_s07_s02_p25">Answer: Two sets of positive integers solve this problem: {5, 8} and {12, 15}.</p>
</div>
<div class="callout block" id="fwk-redden-ch04_s07_s02_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s07_s02_p26"><strong class="emphasis bold">Try this!</strong> When the reciprocal of the larger of two consecutive even integers is subtracted from 4 times the reciprocal of the smaller, the result is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2253" display="inline"><mrow><mfrac><mn>5</mn><mn>6</mn></mfrac></mrow><mo>.</mo></math></span> Find the integers.</p>
<p class="para" id="fwk-redden-ch04_s07_s02_p27">Answer: 4, 6</p>
<div class="mediaobject">
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</div>
</div>
<div class="key_takeaways editable block" id="fwk-redden-ch04_s07_s02_n03">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch04_s07_s02_l01" mark="bullet">
<li>Begin solving rational equations by multiplying both sides by the LCD. The resulting equivalent equation can be solved using the techniques learned up to this point.</li>
<li>Multiplying both sides of a rational equation by a variable expression introduces the possibility of extraneous solutions. Therefore, we must check the solutions against the set of restrictions. If a solution is a restriction, then it is not part of the domain and is extraneous.</li>
<li>When multiplying both sides of an equation by an expression, distribute carefully and multiply each term by that expression.</li>
<li>If all of the resulting solutions are extraneous, then the original equation has no solutions.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch04_s07_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd01">
<h3 class="title">Part A: Solving Rational Equations</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch04_s07_qs01_p01"><strong class="emphasis bold">Solve.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa01">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2254" display="block"><mrow><mfrac><mn>3</mn><mi>x</mi></mfrac><mo>+</mo><mn>2</mn><mo>=</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mi>x</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa02">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2256" display="block"><mrow><mn>5</mn><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa03">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2258" display="block"><mrow><mfrac><mn>7</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa04">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2259" display="block"><mrow><mfrac><mn>4</mn><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa05">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2260" display="block"><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mn>3</mn><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mn>7</mn><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa06">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2261" display="block"><mrow><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa07">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2262" display="block"><mrow><mn>2</mn><mo>+</mo><mfrac><mn>3</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>7</mn><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa08">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2264" display="block"><mrow><mfrac><mrow><mn>20</mn></mrow><mi>x</mi></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>44</mn></mrow><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mn>3</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa09">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2266" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>4</mn><mi>x</mi></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>18</mn></mrow><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa10">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2268" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa11">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2270" display="block"><mrow><mfrac><mn>4</mn><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa12">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2272" display="block"><mrow><mfrac><mn>5</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa13">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2273" display="block"><mrow><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>4</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa14">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2275" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>15</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>24</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa15">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2277" display="block"><mrow><mfrac><mi>x</mi><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow></mfrac><mo>−</mo><mfrac><mn>8</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>56</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa16">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2278" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>9</mn><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa17">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2281" display="block"><mrow><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>14</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa18">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2283" display="block"><mrow><mfrac><mi>x</mi><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo>−</mo><mfrac><mn>4</mn><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>4</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>20</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa19">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2284" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>5</mn><mo>+</mo><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>5</mn><mo>−</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa20">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2286" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mfrac><mn>6</mn><mrow><mn>9</mn><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa21">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2288" display="block"><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>8</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>16</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa22">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2289" display="block"><mrow><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa23">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2290" display="block"><mrow><mfrac><mi>x</mi><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>16</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa24">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2291" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>10</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>10</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>13</mn><mi>x</mi><mo>+</mo><mn>30</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa25">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2292" display="block"><mrow><mfrac><mn>5</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>18</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfrac><mo>=</mo><mfrac><mn>5</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa26">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2293" display="block"><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>60</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mo>+</mo><mn>60</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>36</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa27">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2294" display="block"><mrow><mfrac><mn>4</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>21</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa28">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2295" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>+</mo><mn>28</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa29">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2296" display="block"><mrow><mfrac><mn>5</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo>=</mo><mfrac><mn>5</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa30">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2297" display="block"><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>63</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>21</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>27</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa31">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2298" display="block"><mrow><mfrac><mn>4</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>12</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>12</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa32">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2299" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd01_qd02" start="33">
<p class="para" id="fwk-redden-ch04_s07_qs01_p66"><strong class="emphasis bold">Solve the following equations involving negative exponents.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2300" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2302" display="inline"><mrow><mn>3</mn><mo>+</mo><mi>x</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2304" display="inline"><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>64</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2306" display="inline"><mrow><mn>1</mn><mo>−</mo><mn>4</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2307" display="inline"><mrow><mi>x</mi><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2308" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2309" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p81"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2310" display="inline"><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd01_qd03" start="41">
<p class="para" id="fwk-redden-ch04_s07_qs01_p83"><strong class="emphasis bold">Solve by cross multiplying.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa41">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2312" display="block"><mrow><mfrac><mn>5</mn><mi>n</mi></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa42">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2314" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>n</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa43">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2316" display="block"><mrow><mo>−</mo><mn>3</mn><mo>=</mo><mfrac><mrow><mn>5</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>3</mn><mi>n</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa44">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2318" display="block"><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa45">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2319" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa46">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2320" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mi>x</mi></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa47">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2321" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa48">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2323" display="block"><mrow><mfrac><mrow><mn>6</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa49">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2325" display="block"><mrow><mfrac><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa50">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2326" display="block"><mrow><mfrac><mrow><mn>8</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn><mo>−</mo><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa51">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2327" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa52">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2328" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>8</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd01_qd04" start="53">
<p class="para" id="fwk-redden-ch04_s07_qs01_p108"><strong class="emphasis bold">Simplify or solve, whichever is appropriate.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa53">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2329" display="block"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa54">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2331" display="block"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa55">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2332" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn><mo>−</mo><mi>x</mi></mrow><mi>x</mi></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa56">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2334" display="block"><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa57">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2336" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa58">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2338" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa59">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2340" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mn>2</mn><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa60">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2342" display="block"><mrow><mn>5</mn><mo>−</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd01_qd05" start="61">
<p class="para" id="fwk-redden-ch04_s07_qs01_p125"><strong class="emphasis bold">Find the roots of the given function.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa61">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2344" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa62">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2346" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa63">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2348" display="block"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>81</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa64">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2349" display="block"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>20</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa65">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2350" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa66">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2352" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p138">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2354" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mn>5</mn></mrow></math></span>, find <em class="emphasis">x</em> when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2355" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p140">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2357" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac></mrow></math></span>, find <em class="emphasis">x</em> when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2358" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p142">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2360" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mn>2</mn></mrow></math></span>, find <em class="emphasis">x</em> when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2361" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p144">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2363" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mn>5</mn></mrow></math></span>, find <em class="emphasis">x</em> when <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2364" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd01_qd06" start="71">
<p class="para" id="fwk-redden-ch04_s07_qs01_p146"><strong class="emphasis bold">Find the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-intercepts.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p147"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2366" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p149"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2368" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>−</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p151"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2371" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p153"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2374" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p155"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2375" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p157"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2377" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd01_qd07" start="77">
<p class="para" id="fwk-redden-ch04_s07_qs01_p159"><strong class="emphasis bold">Find the points where the given functions coincide. (Hint: Find the points where</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2379" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>.</mo></math></span><strong class="emphasis bold">)</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2380" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2381" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p162"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2382" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2383" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p164"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2384" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mn>3</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2385" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p166"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2386" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>−</mo><mn>1</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2387" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd01_qd08" start="81">
<p class="para" id="fwk-redden-ch04_s07_qs01_p168"><strong class="emphasis bold">Recall that if</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2388" display="inline"><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mi>p</mi></mrow></math></span><strong class="emphasis bold">, then</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2389" display="inline"><mrow><mi>X</mi><mo>=</mo><mo>−</mo><mi>p</mi></mrow></math></span> <strong class="emphasis bold">or</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2390" display="inline"><mrow><mi>X</mi><mo>=</mo><mi>p</mi></mrow><mo>.</mo></math></span> <strong class="emphasis bold">Use this to solve the following absolute value equations.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa81">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2391" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa82">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2394" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa83">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2396" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa84">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2397" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa85">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2398" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa86">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2399" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>48</mn></mrow><mi>x</mi></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd02">
<h3 class="title">Part B: Solving Literal Equations</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd02_qd01" start="87">
<p class="para" id="fwk-redden-ch04_s07_qs01_p181"><strong class="emphasis bold">Solve for the given variable.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p182">Solve for <em class="emphasis">P</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2400" display="inline"><mrow><mi>w</mi><mo>=</mo><mfrac><mrow><mi>P</mi><mo>−</mo><mn>2</mn><mi>l</mi></mrow><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p184">Solve for <em class="emphasis">A</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2402" display="inline"><mrow><mi>t</mi><mo>=</mo><mfrac><mrow><mi>A</mi><mo>−</mo><mi>P</mi></mrow><mrow><mi>P</mi><mi>r</mi></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p186">Solve for <em class="emphasis">t</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2404" display="inline"><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mn>2</mn></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>t</mi></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p188">Solve for <em class="emphasis">n</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2406" display="inline"><mrow><mi>P</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mi>r</mi><mi>n</mi></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p190">Solve for <em class="emphasis">y</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2408" display="inline"><mrow><mi>m</mi><mo>=</mo><mfrac><mrow><mi>y</mi><mo>−</mo><msub><mi>y</mi><mn>0</mn></msub></mrow><mrow><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p192">Solve for <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2410" display="inline"><mrow><msub><mi>m</mi><mn>1</mn></msub></mrow></math></span>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2411" display="inline"><mrow><mi>F</mi><mo>=</mo><mi>G</mi><mfrac><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mi>m</mi><mn>2</mn></msub></mrow><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p194">Solve for <em class="emphasis">y</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2413" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p196">Solve for <em class="emphasis">y</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2415" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>y</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p198">Solve for <em class="emphasis">y</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2417" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>y</mi></mrow><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>5</mn></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p200">Solve for <em class="emphasis">y</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2419" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>y</mi></mrow></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p202">Solve for <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2421" display="inline"><mrow><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>c</mi><mi>b</mi></mfrac><mo>=</mo><mfrac><mi>a</mi><mi>c</mi></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p204">Solve for <em class="emphasis">y</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2423" display="inline"><mrow><mfrac><mi>a</mi><mi>y</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mo>=</mo><mi>b</mi></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd02_qd02" start="99">
<p class="para" id="fwk-redden-ch04_s07_qs01_p206"><strong class="emphasis bold">Use algebra to solve the following applications.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p207">The value in dollars of a tablet computer is given by the function <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2425" display="inline"><mrow><mi>V</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>460</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>, where <em class="emphasis">t</em> represents the age of the tablet. Determine the age of the tablet if it is now worth $100.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p209">The value in dollars of a car is given by the function <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2426" display="inline"><mrow><mi>V</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>24,000</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>0.5</mn><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span>, where <em class="emphasis">t</em> represents the age of the car. Determine the age of the car if it is now worth $6,000.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd02_qd03" start="101">
<p class="para" id="fwk-redden-ch04_s07_qs01_p211"><em class="emphasis"><strong class="emphasis bold">Solve for the unknowns.</strong></em></p>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p212">When 2 is added to 5 times the reciprocal of a number, the result is 12. Find the number.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p214">When 1 is subtracted from 4 times the reciprocal of a number, the result is 11. Find the number.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p216">The sum of the reciprocals of two consecutive odd integers is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2429" display="inline"><mrow><mfrac><mn>12</mn><mn>35</mn></mfrac></mrow><mo>.</mo></math></span> Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p218">The sum of the reciprocals of two consecutive even integers is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2430" display="inline"><mrow><mfrac><mn>9</mn><mn>40</mn></mfrac></mrow><mo>.</mo></math></span> Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p220">An integer is 4 more than another. If 2 times the reciprocal of the larger is subtracted from 3 times the reciprocal of the smaller, then the result is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2431" display="inline"><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac></mrow><mo>.</mo></math></span> Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p222">An integer is 2 more than twice another. If 2 times the reciprocal of the larger is subtracted from 3 times the reciprocal of the smaller, then the result is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2432" display="inline"><mrow><mfrac><mn>5</mn><mn>14</mn></mfrac></mrow><mo>.</mo></math></span> Find the integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p224">If 3 times the reciprocal of the larger of two consecutive integers is subtracted from 2 times the reciprocal of the smaller, then the result is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2433" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>.</mo></math></span> Find the two integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p226">If 3 times the reciprocal of the smaller of two consecutive integers is subtracted from 7 times the reciprocal of the larger, then the result is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2434" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>.</mo></math></span> Find the two integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p228">A positive integer is 5 less than another. If the reciprocal of the smaller integer is subtracted from 3 times the reciprocal of the larger, then the result is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2435" display="inline"><mrow><mfrac><mn>1</mn><mn>12</mn></mfrac></mrow><mo>.</mo></math></span> Find the two integers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p230">A positive integer is 6 less than another. If the reciprocal of the smaller integer is subtracted from 10 times the reciprocal of the larger, then the result is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2436" display="inline"><mrow><mfrac><mn>3</mn><mn>7</mn></mfrac></mrow><mo>.</mo></math></span> Find the two integers.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s07_qs01_qd03_qd01" start="111">
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa111">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p232">Explain how we can tell the difference between a rational expression and a rational equation. How do we treat them differently? Give an example of each.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa112">
<div class="question">
<p class="para" id="fwk-redden-ch04_s07_qs01_p233">Research and discuss reasons why multiplying both sides of a rational equation by the LCD sometimes produces extraneous solutions.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch04_s07_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s07_qs01_p03_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2255" display="inline"><mrow><mo>−</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s07_qs01_p07_ans">−4</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s07_qs01_p11_ans">−7, 3</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s07_qs01_p15_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2263" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>, 2</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s07_qs01_p19_ans">−2, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2267" display="inline"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s07_qs01_p23_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2271" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s07_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s07_qs01_p27_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2274" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span></p>
</div>
</li>