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<title>Solving Quadratic Inequalities</title>
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<div class="section" id="fwk-redden-ch06_s05" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">6.5</span> Solving Quadratic Inequalities</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch06_s05_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch06_s05_o01" numeration="arabic">
<li>Check solutions to quadratic inequalities with one variable.</li>
<li>Understand the geometric relationship between solutions to quadratic inequalities and their graphs.</li>
<li>Solve quadratic inequalities.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch06_s05_s01" version="5.0" lang="en">
<h2 class="title editable block">Solutions to Quadratic Inequalities</h2>
<p class="para editable block" id="fwk-redden-ch06_s05_s01_p01">A <span class="margin_term"><a class="glossterm">quadratic inequality</a><span class="glossdef">A mathematical statement that relates a quadratic expression as either less than or greater than another.</span></span> is a mathematical statement that relates a quadratic expression as either less than or greater than another. Some examples of quadratic inequalities solved in this section follow.</p>
<p class="para block" id="fwk-redden-ch06_s05_s01_p02">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1138" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>11</mn><mo>≤</mo><mn>0</mn></mrow></math></span></td>
<td align="center"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1139" display="block"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>></mo><mn>0</mn></mrow></math></span></td>
<td align="center"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1140" display="block"><mrow><mn>9</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></math></span></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch06_s05_s01_p03">A solution to a quadratic inequality is a real number that will produce a true statement when substituted for the variable.</p>
<div class="callout block" id="fwk-redden-ch06_s05_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch06_s05_s01_p04">Are −3, −2, and −1 solutions to <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1141" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>6</mn><mo>≤</mo><mn>0</mn></mrow></math></span>?</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s05_s01_p05">Substitute the given value in for <em class="emphasis">x</em> and simplify.</p>
<p class="para" id="fwk-redden-ch06_s05_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1142" display="block"><mrow><mtable columnalign="left" columnlines="none none solid none none none none solid none none none none none" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>6</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="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>6</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="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mo>≤</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>3</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>3</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>6</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="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>6</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="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>6</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>9</mn><mo>+</mo><mn>3</mn><mo>−</mo><mn>6</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="right"><mrow><mn>4</mn><mo>+</mo><mn>2</mn><mo>−</mo><mn>6</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="right"><mrow><mn>1</mn><mo>+</mo><mn>1</mn><mo>−</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mo>≤</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>6</mn></mtd><mtd columnalign="left"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s01_p07">Answer: −2 and −1 are solutions and −3 is not.</p>
</div>
<p class="para block" id="fwk-redden-ch06_s05_s01_p08">Quadratic inequalities can have infinitely many solutions, one solution, or no solution. If there are infinitely many solutions, graph the solution set on a number line and/or express the solution using interval notation. Graphing the function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1143" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>6</mn></mrow></math></span> found in the previous example we have</p>
<div class="informalfigure large block">
<img src="section_09/a30fca8d168ce536e5ffe8782af1a050.png">
</div>
<p class="para editable block" id="fwk-redden-ch06_s05_s01_p10">The result of evaluating for any <em class="emphasis">x</em>-value will be negative, zero, or positive.</p>
<p class="para block" id="fwk-redden-ch06_s05_s01_p11"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1144" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>P</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>></mo></mstyle></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>0</mn></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>f</mtext><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</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><mstyle color="#007fbf"><mi>Z</mi><mi>e</mi><mi>r</mi><mi>o</mi></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>=</mo></mstyle></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>0</mn></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>f</mtext><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>N</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo><</mo></mstyle></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>0</mn></mstyle></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s05_s01_p12">The values in the domain of a function that separate regions that produce positive or negative results are called <span class="margin_term"><a class="glossterm">critical numbers</a><span class="glossdef">The values in the domain of a function that separate regions that produce positive or negative results.</span></span>. In the case of a quadratic function, the critical numbers are the roots, sometimes called the zeros. For example, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1145" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>6</mn><mo>=</mo><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>3</mn></mrow><mo>)</mo></mrow></mrow></math></span> has roots −2 and 3. These values bound the regions where the function is positive (above the <em class="emphasis">x</em>-axis) or negative (below the <em class="emphasis">x</em>-axis).</p>
<div class="informalfigure large block">
<img src="section_09/49a51b102bc7bcdb764372ec98f905cc.png">
</div>
<p class="para block" id="fwk-redden-ch06_s05_s01_p14">Therefore <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1146" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>6</mn><mo>≤</mo><mn>0</mn></mrow></math></span> has solutions where <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1147" display="inline"><mrow><mo>−</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>3</mn></mrow></math></span>, using interval notation <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1148" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>]</mo></mrow></mrow><mo>.</mo></math></span> Furthermore, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1149" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>6</mn><mo>≥</mo><mn>0</mn></mrow></math></span> has solutions where <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1150" display="inline"><mrow><mi>x</mi><mo>≤</mo><mo>−</mo><mn>2</mn></mrow></math></span> or <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1151" display="inline"><mrow><mi>x</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, using interval notation <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1152" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>]</mo></mrow><mo>∪</mo><mrow><mo>[</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<div class="callout block" id="fwk-redden-ch06_s05_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch06_s05_s01_p15">Given the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1153" display="inline"><mi>f</mi></math></span> determine the solutions to <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1154" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span>:</p>
<div class="informalfigure large">
<img src="section_09/cd3b1eec65c7220ad5735a581e29bd5e.png">
</div>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s05_s01_p17">From the graph we can see that the roots are −4 and 2. The graph of the function lies above the <em class="emphasis">x</em>-axis (<span class="inlineequation"><math xml:id="fwk-redden-ch06_m1155" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span>) in between these roots.</p>
<div class="informalfigure large">
<img src="section_09/3a8b1549706b08b21b5cfa087977afc9.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s01_p19">Because of the strict inequality, the solution set is shaded with an open dot on each of the boundaries. This indicates that these critical numbers are not actually included in the solution set. This solution set can be expressed two ways,</p>
<p class="para" id="fwk-redden-ch06_s05_s01_p20"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1156" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="center"><mrow><mrow><mo>{</mo><mrow><mi>x</mi><mo>|</mo><mo>−</mo><mn>4</mn><mo><</mo><mi>x</mi><mo><</mo><mn>2</mn></mrow><mo>}</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>S</mi><mi>e</mi><mi>t</mi><mtext> </mtext><mi>N</mi><mi>o</mi><mi>t</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="center"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mtext> </mtext><mi>N</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s01_p21">In this textbook, we will continue to present answers in interval notation.</p>
<p class="para" id="fwk-redden-ch06_s05_s01_p22">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1157" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch06_s05_s01_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s05_s01_p23"><strong class="emphasis bold">Try this!</strong> Given the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1158" display="inline"><mi>f</mi></math></span> determine the solutions to <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1159" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><mn>0</mn></mrow></math></span>:</p>
<div class="informalfigure large">
<img src="section_09/1ea58324d9c9b824a79b719ff7ce5d17.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s01_p25">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1160" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/kkBaoM8buBo" condition="http://img.youtube.com/vi/kkBaoM8buBo/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/kkBaoM8buBo" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch06_s05_s02" version="5.0" lang="en">
<h2 class="title editable block">Solving Quadratic Inequalities</h2>
<p class="para editable block" id="fwk-redden-ch06_s05_s02_p01">Next we outline a technique used to solve quadratic inequalities without graphing the parabola. To do this we make use of a <span class="margin_term"><a class="glossterm">sign chart</a><span class="glossdef">A model of a function using a number line and signs (+ or −) to indicate regions in the domain where the function is positive or negative.</span></span> which models a function using a number line that represents the <em class="emphasis">x</em>-axis and signs (+ or −) to indicate where the function is positive or negative. For example,</p>
<div class="informalfigure large block">
<img src="section_09/8f2989ef066ce6883861208396130683.png">
</div>
<p class="para editable block" id="fwk-redden-ch06_s05_s02_p03">The plus signs indicate that the function is positive on the region. The negative signs indicate that the function is negative on the region. The boundaries are the critical numbers, −2 and 3 in this case. Sign charts are useful when a detailed picture of the graph is not needed and are used extensively in higher level mathematics. The steps for solving a quadratic inequality with one variable are outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch06_s05_s02_n01">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch06_s05_s02_p04">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1161" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>7</mn><mo>≥</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p05">It is important to note that this quadratic inequality is in standard form, with zero on one side of the inequality.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p06"><strong class="emphasis bold">Step 1</strong>: Determine the critical numbers. For a quadratic inequality in standard form, the critical numbers are the roots. Therefore, set the function equal to zero and solve.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p07"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1162" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p08">The critical numbers are −1 and 7.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p09"><strong class="emphasis bold">Step 2</strong>: Create a sign chart. Since the critical numbers bound the regions where the function is positive or negative, we need only test a single value in each region. In this case the critical numbers partition the number line into three regions and we choose test values <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1163" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1164" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1165" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>10</mn></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large">
<img src="section_09/915e05bcffaaab7874eda54ada0751af.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p11">Test values may vary. In fact, we need only determine the sign (+ or −) of the result when evaluating <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1166" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>7</mn><mo>=</mo><mo>−</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Here we evaluate using the factored form.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p12"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1167" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>3</mn></mstyle></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>3</mn></mstyle><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>3</mn></mstyle><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>N</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mstyle color="#007f3f"><mn>0</mn></mstyle><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>0</mn></mstyle><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>0</mn></mstyle><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>+</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>P</mi><mi>o</mi><mi>s</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>10</mn></mstyle></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>10</mn></mstyle><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>10</mn></mstyle><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>11</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>N</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p13">Since the result of evaluating for −3 was negative, we place negative signs above the first region. The result of evaluating for 0 was positive, so we place positive signs above the middle region. Finally, the result of evaluating for 10 was negative, so we place negative signs above the last region, and the sign chart is complete.</p>
<div class="informalfigure large">
<img src="section_09/6fb23054f7edfb70117b8b3df7880587.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p15"><strong class="emphasis bold">Step 3</strong>: Use the sign chart to answer the question. In this case, we are asked to determine where <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1168" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn></mrow></math></span>, or where the function is positive or zero. From the sign chart we see this occurs when <em class="emphasis">x</em>-values are inclusively between −1 and 7.</p>
<div class="informalfigure large">
<img src="section_09/539d2e7400fed2d8137b1919416ad81c.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p17">Using interval notation, the shaded region is expressed as <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1169" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>7</mn></mrow><mo>]</mo></mrow></mrow><mo>.</mo></math></span> The graph is not required; however, for the sake of completeness it is provided below.</p>
<div class="informalfigure large">
<img src="section_09/ca4d179ae46f208616be623bb29df57d.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p19">Indeed the function is greater than or equal to zero, above or on the <em class="emphasis">x</em>-axis, for <em class="emphasis">x</em>-values in the specified interval.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p20">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1170" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>7</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch06_s05_s02_n02">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch06_s05_s02_p21">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1171" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>></mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p22">Begin by finding the critical numbers, in this case, the roots of <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1172" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p23"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1173" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>1</mn></mtd><mtd><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p24">The critical numbers are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1174" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> and 3. Because of the strict inequality > we will use open dots.</p>
<div class="informalfigure large">
<img src="section_09/7e9d03426a5bb9e54fe75ad9123b3018.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p26">Next choose a test value in each region and determine the sign after evaluating <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1175" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>=</mo><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Here we choose test values −1, 2, and 5.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p27"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1176" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>+</mo></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mstyle color="#007f3f"><mn>2</mn></mstyle><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>2</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>2</mn></mstyle><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mo>+</mo><mo>)</mo></mrow><mrow><mo>(</mo><mo>−</mo><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mstyle color="#007f3f"><mn>5</mn></mstyle><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>[</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>5</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>5</mn></mstyle><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mo>+</mo><mo>)</mo></mrow><mrow><mo>(</mo><mo>+</mo><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>+</mo></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p28">And we can complete the sign chart.</p>
<div class="informalfigure large">
<img src="section_09/b032f46c76e66584153dfe8c5f0c7f5a.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p30">The question asks us to find the <em class="emphasis">x</em>-values that produce positive results (greater than zero). Therefore, shade in the regions with a + over them. This is the solution set.</p>
<div class="informalfigure large">
<img src="section_09/6a7b9408ba1c281e883447cd3b5e8023.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p32">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1177" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s05_s02_p33">Sometimes the quadratic function does not factor. In this case we can make use of the quadratic formula.</p>
<div class="callout block" id="fwk-redden-ch06_s05_s02_n03">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch06_s05_s02_p34">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1178" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>11</mn><mo>≤</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p35">Find the critical numbers.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p36"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1179" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>11</mn><mo>=</mo><mn>0</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p37">Identify <em class="emphasis">a</em>, <em class="emphasis">b</em>, and <em class="emphasis">c</em> for use in the quadratic formula. Here <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1180" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1181" display="inline"><mrow><mi>b</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1182" display="inline"><mrow><mi>c</mi><mo>=</mo><mo>−</mo><mn>11</mn></mrow><mo>.</mo></math></span> Substitute the appropriate values into the quadratic formula and then simplify.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p38"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1183" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mi>b</mi><mo>±</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></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><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>±</mo><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>11</mn></mstyle></mrow><mo>)</mo></mrow></mrow></msqrt></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow></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><mrow><mn>2</mn><mo>±</mo><msqrt><mrow><mn>48</mn></mrow></msqrt></mrow><mn>2</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><mrow><mn>2</mn><mo>±</mo><mn>4</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>±</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p39">Therefore the critical numbers are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1184" display="inline"><mrow><mn>1</mn><mo>−</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>≈</mo><mo>−</mo><mn>2.5</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1185" display="inline"><mrow><mn>1</mn><mo>+</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>≈</mo><mn>4.5</mn></mrow><mo>.</mo></math></span> Use a closed dot on the number to indicate that these values will be included in the solution set.</p>
<div class="informalfigure large">
<img src="section_09/73f014a5f1f4df1a3d101b98a87d3ed9.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p41">Here we will use test values −5, 0, and 7.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p42"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1186" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>5</mn></mstyle></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>5</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>5</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>11</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>25</mn><mo>+</mo><mn>10</mn><mo>−</mo><mn>11</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>+</mo></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mstyle color="#007f3f"><mn>0</mn></mstyle><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mstyle color="#007f3f"><mn>0</mn></mstyle><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>0</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>11</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mo>+</mo><mn>0</mn><mo>−</mo><mn>11</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mstyle color="#007f3f"><mn>7</mn></mstyle><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mstyle color="#007f3f"><mn>7</mn></mstyle><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>7</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>11</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>49</mn><mo>−</mo><mn>14</mn><mo>−</mo><mn>11</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>+</mo></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p43">After completing the sign chart shade in the values where the function is negative as indicated by the question (<span class="inlineequation"><math xml:id="fwk-redden-ch06_m1187" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≤</mo><mn>0</mn></mrow></math></span>).</p>
<div class="informalfigure large">
<img src="section_09/4f5ed76e0a0d29888d8c470642c41dfa.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p45">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1188" display="inline"><mrow><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>,</mo><mn>1</mn><mo>+</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch06_s05_s02_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s05_s02_p46"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1189" display="inline"><mrow><mn>9</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p47">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1190" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/7gMJ8gUvASw" condition="http://img.youtube.com/vi/7gMJ8gUvASw/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/7gMJ8gUvASw" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para editable block" id="fwk-redden-ch06_s05_s02_p49">It may be the case that there are no critical numbers.</p>
<div class="callout block" id="fwk-redden-ch06_s05_s02_n04">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch06_s05_s02_p50">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1191" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>></mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p51">To find the critical numbers solve,</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p52"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1192" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p53">Substitute <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1193" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1194" display="inline"><mrow><mi>b</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1195" display="inline"><mrow><mi>c</mi><mo>=</mo><mn>3</mn></mrow></math></span> into the quadratic formula and then simplify.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p54"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1196" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mi>b</mi><mo>±</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></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><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>±</mo><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow><mrow><mo>(</mo><mstyle color="#007f3f"><mn>3</mn></mstyle><mo>)</mo></mrow></mrow></msqrt></mrow><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow></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><mrow><mn>2</mn><mo>±</mo><msqrt><mrow><mo>−</mo><mn>8</mn></mrow></msqrt></mrow><mn>2</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><mrow><mn>2</mn><mo>±</mo><mn>2</mn><mi>i</mi><msqrt><mn>2</mn></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>+</mo><mi>i</mi><msqrt><mn>2</mn></msqrt></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p55">Because the solutions are not real, we conclude there are no real roots; hence there are no critical numbers. When this is the case, the graph has no <em class="emphasis">x</em>-intercepts and is completely above or below the <em class="emphasis">x</em>-axis. We can test any value to create a sign chart. Here we choose <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1197" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p56"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1198" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mo>+</mo></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p57">Because the test value produced a positive result the sign chart looks as follows:</p>
<div class="informalfigure large">
<img src="section_09/5d9c4984c0c6d540edcef79225c19764.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p59">We are looking for the values where <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1199" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span>; the sign chart implies that any real number for <em class="emphasis">x</em> will satisfy this condition.</p>
<div class="informalfigure large">
<img src="section_09/f5b2ce61993fc861ec77863eba8c9c37.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p61">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1200" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s05_s02_p62">The function in the previous example is graphed below.</p>
<div class="informalfigure large block">
<img src="section_09/002dbb2426b3523d8983c2ffd4293a5a.png">
</div>
<p class="para block" id="fwk-redden-ch06_s05_s02_p64">We can see that it has no <em class="emphasis">x</em>-intercepts and is always above the <em class="emphasis">x</em>-axis (positive). If the question was to solve <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1201" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo><</mo><mn>0</mn></mrow></math></span>, then the answer would have been no solution. The function is never negative.</p>
<div class="callout block" id="fwk-redden-ch06_s05_s02_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s05_s02_p65"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1202" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>4</mn><mo>≤</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p66">Answer: One solution, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1203" display="inline"><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>.</mo></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/E7VcOYVV_Ds" condition="http://img.youtube.com/vi/E7VcOYVV_Ds/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/E7VcOYVV_Ds" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<div class="callout block" id="fwk-redden-ch06_s05_s02_n05">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch06_s05_s02_p68">Find the domain: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1204" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></msqrt></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p69">Recall that the argument of a square root function must be nonnegative. Therefore, the domain consists of all real numbers for <em class="emphasis">x</em> such that <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1205" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></math></span> is greater than or equal to zero.</p>
<p class="para" id="fwk-redden-ch06_s05_s02_p70"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1206" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo>≥</mo><mn>0</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p71">It should be clear that <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1207" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span> has two solutions <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1208" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>±</mo><mn>2</mn></mrow></math></span>; these are the critical values. Choose test values in each interval and evaluate <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1209" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p72"><span class="informalequation"><math xml:id="fwk-redden-ch06_m1210" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>9</mn><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>+</mo></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>f</mi><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>9</mn><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>+</mo></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s05_s02_p73">Shade in the <em class="emphasis">x</em>-values that produce positive results.</p>
<div class="informalfigure large">
<img src="section_09/8d2a68272d08c7ea0a09124dc643a7f6.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_s02_p75">Answer: Domain: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1211" display="inline"><mrow><mrow><mo>(</mo> <mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>2</mn></mrow> <mo>]</mo></mrow><mo>∪</mo><mrow><mo>[</mo> <mrow><mn>2</mn><mo>,</mo><mi>∞</mi></mrow> <mo>)</mo></mrow></mrow></math></span></p>
</div>
<div class="key_takeaways editable block" id="fwk-redden-ch06_s05_s02_n06">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch06_s05_s02_l01" mark="bullet">
<li>Quadratic inequalities can have infinitely many solutions, one solution or no solution.</li>
<li>We can solve quadratic inequalities graphically by first rewriting the inequality in standard form, with zero on one side. Graph the quadratic function and determine where it is above or below the <em class="emphasis">x</em>-axis. If the inequality involves “less than,” then determine the <em class="emphasis">x</em>-values where the function is below the <em class="emphasis">x</em>-axis. If the inequality involves “greater than,” then determine the <em class="emphasis">x</em>-values where the function is above the <em class="emphasis">x</em>-axis.</li>
<li>We can streamline the process of solving quadratic inequalities by making use of a sign chart. A sign chart gives us a visual reference that indicates where the function is above the <em class="emphasis">x</em>-axis using positive signs or below the <em class="emphasis">x</em>-axis using negative signs. Shade in the appropriate <em class="emphasis">x</em>-values depending on the original inequality.</li>
<li>To make a sign chart, use the function and test values in each region bounded by the roots. We are only concerned if the function is positive or negative and thus a complete calculation is not necessary.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch06_s05_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s05_qs01_qd01">
<h3 class="title">Part A: Solutions to Quadratic Inequalities</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s05_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch06_s05_qs01_p01"><strong class="emphasis bold">Determine whether or not the given value is a solution.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1212" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo><</mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1213" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1214" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>></mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1215" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1216" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>9</mn><mo>≤</mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1217" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1218" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>4</mn><mo><</mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1219" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1220" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>≥</mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1221" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1222" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>≤</mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1223" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1224" display="inline"><mrow><mn>2</mn><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1225" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1226" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1227" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1228" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>9</mn><mo><</mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1229" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1230" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>6</mn><mo>≥</mo><mn>0</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1231" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s05_qs01_qd01_qd02" start="11">
<p class="para" id="fwk-redden-ch06_s05_qs01_p22"><strong class="emphasis bold">Given the graph of <em class="emphasis">f</em> determine the solution set.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p23"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1232" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≤</mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/61dbc9fb935db3dcd56af132b074816f.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p25"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1234" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/4d4b622fed9ab041b48ba48f6f5a356f.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p27"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1236" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/5e53d237df176b67470002d44f78da66.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p29"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1238" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≤</mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/d63a68a2b51c5f9fb944e5d38546e7a4.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p31"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1240" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/40866592c28bd420f3115c7b791973e7.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p33"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1242" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/af4427394ed2204619910de3c8395be9.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p35"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1244" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/e6f4611f5f2fd7fe2d271d247b4f8b56.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p37"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1246" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/513a57cc3615140a25f42ca4231c08df.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p39"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1248" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/4b717b91a3a4ed52c9aef0d259c4a655.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p41"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1250" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><mn>0</mn></mrow></math></span>; </p>
<div class="informalfigure large">
<img src="section_09/c77963b766fafce1a4ab94a09ede8bf5.png">
</div>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s05_qs01_qd01_qd03" start="21">
<p class="para" id="fwk-redden-ch06_s05_qs01_p43"><strong class="emphasis bold">Use the transformations to graph the following and then determine the solution set.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1252" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p46"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1254" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1256" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p50"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1258" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p52"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1260" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1262" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1264" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1266" display="inline"><mrow><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1268" display="inline"><mrow><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p62"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1270" display="inline"><mrow><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s05_qs01_qd02">
<h3 class="title">Part B: Solving Quadratic Inequalities</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s05_qs01_qd02_qd01" start="31">
<p class="para" id="fwk-redden-ch06_s05_qs01_p64"><strong class="emphasis bold">Use a sign chart to solve and graph the solution set. Present answers using interval notation.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1272" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>12</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1274" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>16</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1276" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>24</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1278" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>15</mn><mi>x</mi><mo>+</mo><mn>54</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1280" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>23</mn><mi>x</mi><mo>−</mo><mn>24</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1282" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>20</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1284" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>−</mo><mn>6</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1286" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>17</mn><mi>x</mi><mo>−</mo><mn>6</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p81"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1288" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>18</mn><mi>x</mi><mo>−</mo><mn>5</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p83"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1290" display="inline"><mrow><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>17</mn><mi>x</mi><mo>+</mo><mn>6</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p85"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1292" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>25</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p87"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1294" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>40</mn><mi>x</mi><mo>+</mo><mn>25</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p89"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1296" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1298" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>4</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1300" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>30</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1302" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>27</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1304" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1306" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>81</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1308" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1310" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1312" display="inline"><mrow><mn>25</mn><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1314" display="inline"><mrow><mn>1</mn><mo>−</mo><mn>49</mn><msup><mi>x</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p109"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1316" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p111"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1318" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>75</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1320" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p115"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1322" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p117"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1323" display="inline"><mrow><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1325" display="inline"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p121"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1327" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p123"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1328" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p125"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1330" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>9</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p127"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1332" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>4</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p129"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1334" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p131"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1336" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p133"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1337" display="inline"><mrow><mn>2</mn><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p135"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1339" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p137"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1341" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>9</mn><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p139"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1343" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>6</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p141"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1344" display="inline"><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p143"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1346" display="inline"><mrow><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s05_qs01_qd02_qd02" start="71">
<p class="para" id="fwk-redden-ch06_s05_qs01_p145"><strong class="emphasis bold">Find the domain of the function.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p146"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1348" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p148"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1350" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p150"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1352" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>2</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p152"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1354" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p154"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1356" display="inline"><mrow><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>16</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p156"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1358" display="inline"><mrow><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p158"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1360" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1362" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>9</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p162">A robotics manufacturing company has determined that its weekly profit in thousands of dollars is modeled by <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1364" display="inline"><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>30</mn><mi>n</mi><mo>−</mo><mn>200</mn></mrow></math></span> where <em class="emphasis">n</em> represents the number of units it produces and sells. How many units must the company produce and sell to maintain profitability. (Hint: Profitability occurs when profit is greater than zero.)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p164">The height in feet of a projectile shot straight into the air is given by <span class="inlineequation"><math xml:id="fwk-redden-ch06_m1365" display="inline"><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>400</mn><mi>t</mi></mrow></math></span> where <em class="emphasis">t</em> represents the time in seconds after it is fired. In what time intervals is the projectile under 1,000 feet? Round to the nearest tenth of a second.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s05_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s05_qs01_qd03_qd01" start="81">
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p166">Does the sign chart for any given quadratic function always alternate? Explain and illustrate your answer with some examples.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p167">Research and discuss other methods for solving a quadratic inequality.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch06_s05_qs01_p168">Explain the difference between a quadratic equation and a quadratic inequality. How can we identify and solve each? What is the geometric interpretation of each?</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch06_s05_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p03_ans">No</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_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-ch06_s05_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p07_ans">Yes</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_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-ch06_s05_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p11_ans">No</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_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-ch06_s05_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p15_ans">Yes</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_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-ch06_s05_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p19_ans">Yes</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_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-ch06_s05_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p24_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1233" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mn>2</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_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-ch06_s05_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p28_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1237" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>3</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa14_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-ch06_s05_qs01_qa15_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p32_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1241" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa16_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-ch06_s05_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p36_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1245" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>8</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa18_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-ch06_s05_qs01_qa19_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p40_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1249" display="inline"><mrow><mrow><mo>{</mo><mrow><mo>−</mo><mn>10</mn></mrow><mo>}</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa20_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-ch06_s05_qs01_qa21_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_09/360201a1639d837951419cb31d16ecae.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_qs01_p45_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1253" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa22_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-ch06_s05_qs01_qa23_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_09/2c8e579416618bc09a22816aa15029fe.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_qs01_p49_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1257" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa24_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-ch06_s05_qs01_qa25_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_09/3e5e7957122a85bcd178d4f0803e6899.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_qs01_p53_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1261" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa26_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-ch06_s05_qs01_qa27_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_09/1d4dd18cb17c91b732dd84dcf0ead918.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_qs01_p57_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1265" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa28_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-ch06_s05_qs01_qa29_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_09/91cc511b49796998368b01809fad4efa.png">
</div>
<p class="para" id="fwk-redden-ch06_s05_qs01_p61_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1269" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa30_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>
</ol>
<ol class="qandadiv" start="31">
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa31_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p66_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1273" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>4</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa32_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-ch06_s05_qs01_qa33_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p70_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1277" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>6</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa34_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-ch06_s05_qs01_qa35_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p74_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1281" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>24</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa36_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-ch06_s05_qs01_qa37_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m1285" display="block"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>]</mo></mrow><mo>∪</mo><mrow><mo>[</mo><mrow><mn>6</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa38_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-ch06_s05_qs01_qa39_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m1289" display="block"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa40_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-ch06_s05_qs01_qa41_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m1293" display="block"><mrow><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa42_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-ch06_s05_qs01_qa43_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m1297" display="block"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa44_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-ch06_s05_qs01_qa45_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p94_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1301" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>6</mn><mo>,</mo><mn>5</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa46_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-ch06_s05_qs01_qa47_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p98_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1305" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>8</mn><mo>,</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa48_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-ch06_s05_qs01_qa49_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m1309" display="block"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>]</mo></mrow><mo>∪</mo><mrow><mo>[</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa50_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-ch06_s05_qs01_qa51_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m1313" display="block"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow><mo>]</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s05_qs01_qa52_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-ch06_s05_qs01_qa53_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch06_s05_qs01_p110_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m1317" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>