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329 lines
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329 lines
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HTML
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<!-- Mirrored from www.mathsisfun.com/algebra/inequality-quadratic-solving.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:02:34 GMT -->
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<title>Solving Quadratic Inequalities</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Solving Quadratic Inequalities</h1>
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<p align="center">... and more ...</p>
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<h2>Quadratic</h2>
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<p>A <a href="quadratic-equation.html">Quadratic Equation</a> (in Standard Form) looks like:</p>
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<p align="center"><img src="images/quadratic-equation.svg" alt="Quadratic Equation" />
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<br /> A Quadratic<b> Equation</b> in Standard Form
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<br /> (<b>a</b>, <b>b</b>, and <b>c</b> can have any value, except that <b>a</b> can't be 0.) </p>
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<p> </p>
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<p>The above is an <b>equation</b> (=) but sometimes we need to solve <a href="inequality.html">inequalities</a> like these:</p>
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<table border="0" align="center">
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<tr>
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<th>
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<div align="center"><b>Symbol</b></div>
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</th>
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<th> </th>
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<th>
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<div align="center"><b>Words</b></div>
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</th>
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<th> </th>
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<th>
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<div align="center"><b>Example</b></div>
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</th>
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</tr>
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<tr height="6">
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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<tr>
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<td class="large">
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<div align="center">></div>
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</td>
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<td> </td>
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<td>
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<div align="center">greater than</div>
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</td>
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<td> </td>
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<td class="larger">
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<div align="center">x<sup>2</sup> + 3x > 2</div>
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</td>
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</tr>
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<tr>
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<td class="large">
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<div align="center"><</div>
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</td>
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<td> </td>
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<td>
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<div align="center">less than</div>
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</td>
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<td> </td>
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<td class="larger">
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<div align="center">7x<sup>2</sup> < 28</div>
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</td>
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</tr>
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<tr>
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<td class="large">
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<div align="center">≥</div>
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</td>
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<td> </td>
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<td>
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<div align="center">greater than or equal to</div>
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</td>
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<td> </td>
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<td class="larger">
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<div align="center">5 ≥ x<sup>2</sup> − x </div>
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</td>
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</tr>
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<tr>
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<td class="large">
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<div align="center">≤</div>
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</td>
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<td> </td>
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<td>
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<div align="center">less than or equal to </div>
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</td>
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<td> </td>
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<td class="larger">
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<div align="center">2y<sup>2</sup> + 1 ≤ 7y</div>
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</td>
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</tr>
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<tr height="6">
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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<td></td>
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</tr>
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</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> ... we do most of the same things. </p>
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<table border="0" align="center">
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<tr>
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<td align="center"><span class="larger">When solving <b>equations</b> we try to find<b> points</b>, </span>
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<br /> such as the ones marked "=0"</td>
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</tr>
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<tr>
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<td align="center"><img src="images/inequality-graph-function.svg" alt="Graph of Inequality" /></td>
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</tr>
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<tr>
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<td align="center"><span class="larger">But when we solve <b>inequalities</b>
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we try to find <b>interval(s)</b>,</span>
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<br />
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such as the ones marked ">0" or "<0"</td>
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</tr>
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</table>
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<p>So this is what we do:</p>
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<div class="bigul">
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<ul>
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<li>find the "=0" <b>points</b> </li>
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<li>in between the "=0" points, are <b>intervals</b> that are either
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<ul>
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<li><b>greater than zero</b> (>0), or</li>
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<li><b>less than zero</b> (<0)</li>
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</ul>
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</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: x<sup>2</sup> − x − 6 < 0</h3>
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<p><b>x<sup>2</sup> − x − 6</b> has these simple <a href="factoring-quadratics.html">factors</a> (because I wanted to make it easy!):</p>
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<p align="center" class="larger">(x+2)(x−3) < 0</p>
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<p> </p>
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<p><b>Firstly</b>, let us find where it is <b>equal to</b> zero:</p>
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<p align="center" class="larger">(x+2)(x−3) <span class="hilite">=</span> 0</p>
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<p align="center">It is equal to zero when <span class="large">x = −2</span> or <span class="large">x = +3</span>
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<br> <i>because when x = −2, then (x+2) is zero<br>
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or
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when x = +3, then (x−3) is zero</i></p>
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<p> </p>
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<p>So <b>between</b> −2 and +3, the function will either be </p>
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<ul>
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<li>always <b>greater</b> than zero, or</li>
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<li>always <b>less</b> than zero </li>
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</ul>
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<p>We don't know which ... yet!</p>
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<p align="center"><b>Let's pick a value in-between and test it:</b></p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="larger"><b>At x=0:</b></span></span><span class="right"><span class="larger">x<sup>2</sup> − x − 6 </span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger">= 0 − 0 − 6 </span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger">= <b>−6</b></span></span></div>
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</div>
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<p>So between −2 and +3, the function is <b>less</b> than zero.</p>
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<p>And that is the region we want, so...</p>
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<p align="center"><span class="large">x<sup>2</sup> − x − 6 < 0 </span>in the interval<span class="large"> (−2, 3)</span></p>
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<p> </p>
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<p>Note: <b>x<sup>2</sup> − x − 6 <span class="hilite">></span> 0</b> on the interval <b>(−∞,−2)</b> and <b>(3, +∞)</b></p>
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</div>
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<table border="0" align="center">
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<tr>
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<td valign="top">
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<p> </p>
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<p>And here is the plot of <b>x<sup>2</sup> − x − 6:</b></p>
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<ul>
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<li>The <b>equation</b> equals zero <b>at</b> −2 and 3</li>
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<li>The <b>inequality</b> "<0" is true <b>between</b> −2 and 3. </li>
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</ul>
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</td>
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<td> </td>
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<td><img src="images/inequality-graph-x2mxm6.gif" alt="x^2-x-6" width="164" height="223" /></td>
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</tr>
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</table>
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<p>Also try the <a href="../data/inequality-grapher.html">Inequality Grapher</a>.</p>
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<h2>What If It Doesn't Go Through Zero?</h2>
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<table border="0" align="center">
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<tr>
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<td><img src="images/inequality-graph-x2mxp1.gif" alt="x^2-x-1" width="139" height="122" /></td>
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<td>
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<p>Here is the plot of <b>x<sup>2</sup> − x + 1</b></p>
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<p align="center" class="larger">There are no "=0" points!</p>
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<p>But that makes things easier!</p>
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</td>
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</tr>
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<tr>
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<td colspan="2">
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<p>Because the line does not cross through y=0, it must be either:</p>
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<ul>
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<li><b>always > 0</b>, or </li>
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<li><b>always < 0</b></li>
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</ul>
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<p>So all we have to do is <b>test one value</b> (say x=0) to see if it is above or below.</p>
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</td>
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</tr>
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</table>
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<h2>A "Real World" Example</h2>
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<div class="example">
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<h3>A stuntman will jump off a 20 m building. <br>
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<br>
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A high-speed camera is ready to film him between 15 m and 10 m above the ground.</h3>
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<p><b>When should the camera film him? </b></p>
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<p>We can use this formula for distance and time: </p>
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<p align="center"><span class="large">d = 20 − 5t<sup>2</sup></span> </p>
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<ul>
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<li>d = distance above ground (m), and </li>
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<li>t = time from jump (seconds)</li>
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</ul>
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<p><i>(Note: if you are curious about the formula, it is simplified from </i><b>d = d<sub>0</sub> + v<sub>0</sub>t + ½a<sub>0</sub>t<sup>2</sup></b><i> , where </i><b>d<sub>0</sub>=20</b><i>, </i><b>
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v<sub>0</sub>=0</b><i>, and </i><b>a<sub>0</sub>=−9.81</b><i>,
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the
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<a href="../measure/metric-acceleration.html">acceleration</a> due to gravity.)</i></p>
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<p>OK, let's go.</p>
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<p> </p>
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<h3><b>First</b>, let us sketch the question: </h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/inequality-jump-sketch.gif" alt="Jump Sketch" width="237" height="156" /></p>
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<p>The distance we want is from <b>10 m</b> to <b>15 m</b>:</p>
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<p align="center" class="larger">10 < d < 15</p>
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<p>And we know the formula for <span class="larger">d</span>:</p>
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<p align="center" class="larger">10 < 20 − 5t<sup>2</sup> < 15</p>
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<p> </p>
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<h3>Now let's solve it!</h3>
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<p>First, let's subtract 20 from both sides:</p>
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<p align="center" class="larger">−10 < −5t<sup>2</sup> <−5</p>
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<p> </p>
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<p>Now multiply both sides by −(1/5). <i>But because we are multiplying by a negative number, the inequalities will change direction ... read <a href="inequality-solving.html">Solving Inequalities</a> to see why.</i></p>
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<p align="center" class="larger">2 <span class="hilite">></span> t<sup>2</sup> <span class="hilite">></span> 1</p>
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<p> </p>
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<p>To be neat, the smaller number should be on the left, and the larger on the right. So let's swap them over (and make sure the inequalities still point correctly): </p>
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<p align="center" class="larger">1 < t<sup>2</sup> < 2</p>
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<p> </p>
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<p>Lastly, we can safely take square roots, since all values are greater then zero:</p>
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<p align="center" class="larger">√1 < t < √2</p>
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<p>We can tell the film crew:</p>
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<p align="center" class="large">"Film from 1.0 to 1.4 seconds after jumping"</p>
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</div>
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<h2>Higher Than Quadratic</h2>
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<p>The same ideas can help us solve more complicated inequalities:</p>
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<div class="example">
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<h3>Example: x<sup>3 </sup>+ 4 ≥ 3x<sup>2</sup> + x</h3>
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<p>First, let's put it in standard form:</p>
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<p align="center" class="larger">x<sup>3</sup> − 3x<sup>2 </sup>− x + 4 ≥ 0</p>
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<p>This is a <b>cubic equation</b> (the highest exponent is a cube, i.e. <span class="larger">x<sup>3</sup></span>), and is hard to solve, so let us <a href="../data/function-grapher670b.html?func1=x^3-3x^2-x+4&xmin=-10&xmax=10&ymin=-6.17&ymax=7.17">graph it</a> instead:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/inequality-graph-x3m3x2mxp4.gif" alt="Graph of Inequality" width="217" height="219" /></p>
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<p>The zero points are <b>approximately</b>:</p>
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<ul>
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<li>−1.1</li>
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<li>1.3</li>
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<li>2.9</li>
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</ul>
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<p>And from the graph we can see the intervals where it is greater than (or equal to) zero:</p>
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<ul>
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<li>From −1.1 to 1.3, and</li>
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<li>From 2.9 on</li>
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</ul>
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<p>In <a href="../sets/intervals.html">interval notation</a> we can write: </p>
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<p align="center"><i>Approximately:</i><span class="large"> [−1.1, 1.3] <b>U</b> [2.9, +∞)</span></p>
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