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256 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Solving Rational Inequalities</h1>
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<h2>Rational</h2>
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<p>A <a href="rational-expression.html">Rational Expression</a> looks like:</p>
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<p class="center"><img src="images/rational-expression.svg" alt="Rational Expression"></p>
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<h2>Inequalities</h2>
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<p>Sometimes we need to solve rational <a href="inequality.html">inequalities</a> like these:</p>
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr>
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<th> <div class="center"><b>Symbol</b></div></th>
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<th> <div class="center"><b>Words</b></div></th>
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<th> <div class="center"><b>Example</b></div></th>
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</tr>
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<tr height="6">
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td class="large">
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<div class="center">></div></td>
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<td>
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<div class="center">greater than</div></td>
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<td class="larger">
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<div class="center">(x+1)/(3−x) > 2</div></td>
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</tr>
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<tr>
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<td class="large">
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<div class="center"><</div></td>
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<td>
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<div class="center">less than</div></td>
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<td class="larger">
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<div class="center">x/(x+7) < −3</div></td>
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</tr>
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<tr>
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<td class="large">
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<div class="center">≥</div></td>
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<td>
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<div class="center">greater than or equal to</div></td>
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<td class="larger">
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<div class="center">(x−1)/(5−x) ≥<b> </b>0</div></td>
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</tr>
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<tr>
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<td class="large">
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<div class="center">≤</div></td>
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<td>
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<div class="center">less than or equal to </div></td>
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<td class="larger">
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<div class="center">(3−2x)/(x−1) ≤ 2</div></td>
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</tr>
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<tr height="6">
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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</tbody></table>
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<h2>Solving</h2>
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<p>Solving inequalities is very like <a href="equations-solving.html">solving equations</a> ... you do most of the same things.</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td align="center"><img src="images/inequality-rational-graph-function.svg" alt="Graph of Rational Inequality"></td>
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</tr>
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<tr>
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<td align="center"><span class="larger">When we solve <b>inequalities</b> <br>
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we try to find <b>interval(s)</b>, <br>
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</span>such as the ones marked "<0" or ">0"</td>
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</tr>
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</tbody></table>
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<p>These are the steps:</p>
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<div class="bigul">
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<ul>
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<li>find "points of interest":
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<ul>
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<li>the "=0" points (roots), and</li>
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<li>"vertical asymptotes" (where the function is undefined)</li>
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</ul></li>
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<li>in between the "points of interest", the function is either <b>greater than zero</b> (>0) or <b>less than zero</b> (<0)</li>
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<li>then pick a test value to find out which it is (>0 or <0)</li>
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</ul>
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<p>Here is an example:</p>
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</div>
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<div class="example">
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<h3>Example: <span class="intbl"><em>3x−10</em><strong>x−4</strong></span> > 2</h3>
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<p><b>First</b>, let us simplify!</p>
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<p class="larger">But You Cannot Multiply By (x−4)</p>
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<p>Because "x−4" could be positive or negative ... we don't know if we should change the direction of the inequality or not. This is all explained on <a href="inequality-solving.html">Solving Inequalities</a>.</p>
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<p>Instead, bring "2" to the left:</p>
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<p class="center larger"><span class="intbl"><em>3x−10</em><strong>x−4</strong></span> − 2 > 0</p>
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<p>Then multiply 2 by (x−4)/(x−4):</p>
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<p class="center larger"><span class="intbl"><em>3x−10</em><strong>x−4</strong></span> − 2<span class="intbl"><em>x−4</em><strong>x−4</strong></span> > 0</p>
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<p>Now we have a common denominator, let's bring it all together:</p>
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<p class="center larger"><span class="intbl"><em>3x−10 − 2(x−4)</em><strong>x−4</strong></span> > 0</p>
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<p>Simplify:</p>
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<p class="center larger"><span class="intbl"><em>x−2</em><strong>x−4</strong></span> > 0</p>
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<p> </p>
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<p><b>Second</b>, let us find "points of interest".</p>
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<p>At x=2 we have: <span class="larger">(0)/(x−4) > 0</span>, which is a "=0" point, or <b>root</b></p>
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<p>At x=4 we have: <span class="larger">(x−2)/(0) > 0</span>, which is <b>undefined</b></p>
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<p class="center larger"></p>
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<p class="center larger"></p>
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<p><b>Third</b>, do test points to see what it does in between:</p>
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<p><b>At x=0:</b></p>
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<ul>
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<li>x−2 = −2, which is <b>negative</b></li>
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<li>x−4 = −4, which is also <b>negative</b></li>
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<li>So (x−2)/(x−4) must be <b>positive</b></li>
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</ul>
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<p>We can do the same for <b>x=3</b> and <b>x=5</b>, and end up with these results:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr align="center">
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<th width="120"> </th>
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<th width="80">x=0</th>
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<th width="80">x=2</th>
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<th width="80">x=3</th>
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<th width="80">x=4</th>
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<th width="80">x=5</th>
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</tr>
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<tr align="center">
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<td> </td>
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<td>x−2 < 0</td>
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<td> </td>
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<td>x−2 > 0</td>
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<td> </td>
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<td>x−2 > 0</td>
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</tr>
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<tr align="center">
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<td> </td>
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<td>x−4 < 0 </td>
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<td> </td>
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<td>x−4 < 0 </td>
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<td> </td>
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<td>x−4 > 0</td>
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</tr>
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<tr align="center">
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<td>(x−2)/(x−4) is</td>
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<td>> 0</td>
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<td>0</td>
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<td>< 0</td>
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<td>undefined</td>
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<td>> 0</td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<p>That gives us a complete picture!</p>
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<p class="larger">And where is it > 0 ?</p>
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<ul>
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<li>Less than 2</li>
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<li>More than 4</li>
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</ul>
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<p>So our result is: </p>
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<p class="center"><span class="large">(−∞, 2) <b>U</b> (4, +∞)</span></p>
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</div>
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<p>We did all that without drawing a plot!</p>
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<p>But here is the <a href="../data/function-grapherd80d.html?func1=%28x-2%29/%28x-4%29&xmax=10&ymin=-6.17&ymax=7.17">plot of (x−2)/(x−4)</a> so you can see:</p>
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<p class="center"><img src="images/inequality-graph-xm2-xm4.gif" alt="Graph of Inequality" height="243" width="319"></p>
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<p> </p>
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<div class="questions">
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<script>getQ(581, 582, 9046, 1230, 9047, 2282, 9048, 1231, 9049, 2283);</script>
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</div>
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<div class="related">
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<a href="inequality.html">Inequalities</a>
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<a href="../equal-less-greater.html">Less Than or Greater Than</a>
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