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<title>Solving Absolute Value Equations and Inequalities</title>
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<div class="section" id="fwk-redden-ch02_s06" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">2.6</span> Solving Absolute Value Equations and Inequalities</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch02_s06_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch02_s06_o01" numeration="arabic">
<li>Review the definition of absolute value.</li>
<li>Solve absolute value equations.</li>
<li>Solve absolute value inequalities.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch02_s06_s01" version="5.0" lang="en">
<h2 class="title editable block">Absolute Value Equations</h2>
<p class="para block" id="fwk-redden-ch02_s06_s01_p01">Recall that the <span class="margin_term"><a class="glossterm">absolute value</a><span class="glossdef">The distance from the graph of a number <em class="emphasis">a</em> to zero on a number line, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1553" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow></mrow><mo>.</mo></math></span></span></span> of a real number <em class="emphasis">a</em>, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1554" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow></mrow></math></span>, is defined as the distance between zero (the origin) and the graph of that real number on the number line. For example, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1555" display="inline"><mrow><mrow><mo>|</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1556" display="inline"><mrow><mrow><mo>|</mo><mn>3</mn><mo>|</mo></mrow><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large block">
<img src="section_05/b4aaf730a9b49a73a73f1eee65bc4cc1.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s06_s01_p03">In addition, the absolute value of a real number can be defined algebraically as a piecewise function.</p>
<p class="para block" id="fwk-redden-ch02_s06_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1557" display="block"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>a</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>i</mi><mi>f</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>a</mi><mo>≥</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mi>a</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>i</mi><mi>f</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>a</mi><mo><</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch02_s06_s01_p05">Given this definition, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1558" display="inline"><mrow><mrow><mo>|</mo><mn>3</mn><mo>|</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1559" display="inline"><mrow><mrow><mo>|</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>=</mo><mo>−</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span> Therefore, the equation <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1560" display="inline"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span> has two solutions for <em class="emphasis">x</em>, namely {±3}. In general, given any algebraic expression <em class="emphasis">X</em> and any positive number <em class="emphasis">p</em>:</p>
<p class="para block" id="fwk-redden-ch02_s06_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1561" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mi>I</mi><mi>f</mi><mtext> </mtext><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mi>p</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>then</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>X</mi><mo>=</mo><mo>−</mo><mi>p</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>X</mi><mo>=</mo><mi>p</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s06_s01_p07">In other words, the <span class="margin_term"><a class="glossterm">argument of the absolute value</a><span class="glossdef">The number or expression inside the absolute value.</span></span> <em class="emphasis">X</em> can be either positive or negative <em class="emphasis">p</em>. Use this theorem to solve absolute value equations algebraically.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch02_s06_s01_p08">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1562" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p09">In this case, the argument of the absolute value is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1563" display="inline"><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span> and must be equal to 3 or −3.</p>
<div class="informalfigure large">
<img src="section_05/4a781f54e79090a87431680588379e8f.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s01_p11">Therefore, to solve this absolute value equation, set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1564" display="inline"><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span> equal to ±3 and solve each linear equation as usual.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p12"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1565" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>3</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mo>−</mo><mn>3</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>5</mn></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></mtd></mtr></mtable><mtext> </mtext></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p13">Answer: The solutions are −5 and 1.</p>
</div>
<p class="para block" id="fwk-redden-ch02_s06_s01_p14">To visualize these solutions, graph the functions on either side of the equal sign on the same set of coordinate axes. In this case, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1566" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow></math></span> is an absolute value function shifted two units horizontally to the left, and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1567" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span> is a constant function whose graph is a horizontal line. Determine the <em class="emphasis">x</em>-values where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1568" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large block">
<img src="section_05/8d0f6d91a1a35018cf3b4c0ce4b152da.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s06_s01_p16">From the graph we can see that both functions coincide where <em class="emphasis">x</em> = −5 and <em class="emphasis">x</em> = 1. The solutions correspond to the points of intersection.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch02_s06_s01_p17">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1569" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p18">Here the argument of the absolute value is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1570" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span> and can be equal to −4 or 4.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p19"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1571" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow><mo>|</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn></mrow></mtd><mtd><mrow><mtext>or</mtext></mrow></mtd><mtd><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>7</mn></mrow></mtd><mtd><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mtd><mtd><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p20">Check to see if these solutions satisfy the original equation.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p21"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1572" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></math></span></p></th>
<th align="center"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1573" display="inline"><mrow><mi> </mi><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1574" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mo>−</mo><mn>7</mn><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1575" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>1</mn><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mn>4</mn><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch02_s06_s01_p22">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1576" display="inline"><mrow><mo>−</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1577" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch02_s06_s01_p23">To apply the theorem, the absolute value must be isolated. The general steps for solving absolute value equations are outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch02_s06_s01_p24">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1578" display="inline"><mrow><mn>2</mn><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn><mo>=</mo><mn>9</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p25"><strong class="emphasis bold">Step 1</strong>: Isolate the absolute value to obtain the form <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1579" display="inline"><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mi>p</mi></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p26"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1580" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mi>A</mi><mi>d</mi><mi>d</mi><mtext> </mtext><mn>3</mn><mtext> </mtext><mi>t</mi><mi>o</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>12</mn></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mn>2</mn><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p27"><strong class="emphasis bold">Step 2</strong>: Set the argument of the absolute value equal to ±<em class="emphasis">p</em>. Here the argument is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1581" display="inline"><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1582" display="inline"><mrow><mi>p</mi><mo>=</mo><mn>6</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p28"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1583" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>=</mo><mo>−</mo><mn>6</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>6</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p29"><strong class="emphasis bold">Step 3</strong>: Solve each of the resulting linear equations.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p30"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1584" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>6</mn></mrow></mtd><mtd columnalign="right"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>5</mn><mi>x</mi></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>5</mn><mi>x</mi></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>7</mn><mn>5</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p31"><strong class="emphasis bold">Step 4</strong>: Verify the solutions in the original equation.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p32"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1585" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1586" display="inline"><mrow><mi> </mi><mi>x</mi><mo>=</mo><mfrac><mn>7</mn><mn>5</mn></mfrac></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1587" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mo>−</mo><mn>5</mn><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mo>−</mo><mn>6</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>12</mn><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1588" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>7</mn><mn>5</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mn>7</mn><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mn>6</mn><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>12</mn><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch02_s06_s01_p33">Answer: The solutions are −1 and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1589" display="inline"><mrow><mfrac><mn>7</mn><mn>5</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch02_s06_s01_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s06_s01_p34"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1590" display="inline"><mrow><mn>2</mn><mo>−</mo><mn>7</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>|</mo></mrow><mo>=</mo><mo>−</mo><mn>12</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p35">Answer: −6, −2</p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/G0EjbqreYmU" condition="http://img.youtube.com/vi/G0EjbqreYmU/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/G0EjbqreYmU" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para editable block" id="fwk-redden-ch02_s06_s01_p37">Not all absolute value equations will have two solutions.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch02_s06_s01_p38">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1591" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p39">Begin by isolating the absolute value.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p40"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1592" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>S</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mn>3</mn><mtext> </mtext><mi>o</mi><mi>n</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p41">Only zero has the absolute value of zero, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1593" display="inline"><mrow><mrow><mo>|</mo><mn>0</mn><mo>|</mo></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span> In other words, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1594" display="inline"><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> has one solution, namely <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1595" display="inline"><mrow><mi>X</mi><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span> Therefore, set the argument <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1596" display="inline"><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow></math></span> equal to zero and then solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p42"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1597" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>−</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>6</mn><mn>7</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p43">Geometrically, one solution corresponds to one point of intersection.</p>
<div class="informalfigure large">
<img src="section_05/8cfd0197857b87c9691270321c077d08.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s01_p45">Answer: The solution is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1598" display="inline"><mrow><mfrac><mn>6</mn><mn>7</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch02_s06_s01_n05">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch02_s06_s01_p46">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1599" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>5</mn><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p47">Begin by isolating the absolute value.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p48"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1600" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>5</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>S</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mn>5</mn><mtext> </mtext><mi>o</mi><mi>n</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn><mtext> </mtext></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p49">In this case, we can see that the isolated absolute value is equal to a negative number. Recall that the absolute value will always be positive. Therefore, we conclude that there is no solution. Geometrically, there is no point of intersection.</p>
<div class="informalfigure large">
<img src="section_05/f800ff00fb0c0579741140bd63f30a6c.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s01_p51">Answer: There is no solution, Ø.</p>
</div>
<p class="para block" id="fwk-redden-ch02_s06_s01_p52">If given an equation with two absolute values of the form <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1601" display="inline"><mrow><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow></mrow></math></span>, then <em class="emphasis">b</em> must be the same as <em class="emphasis">a</em> or opposite. For example, if <em class="emphasis">a</em> = 5, then <em class="emphasis">b</em> = ±5 and we have:</p>
<p class="para block" id="fwk-redden-ch02_s06_s01_p53"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1602" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mrow><mo>|</mo><mn>5</mn><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mrow><mo>|</mo><mn>5</mn><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mo>+</mo><mn>5</mn></mrow><mo>|</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s06_s01_p54">In general, given algebraic expressions <em class="emphasis">X</em> and <em class="emphasis">Y</em>:</p>
<p class="para block" id="fwk-redden-ch02_s06_s01_p55"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1603" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtext>If</mtext><mtext> </mtext><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>Y</mi><mo>|</mo></mrow><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>then</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>X</mi><mo>=</mo><mo>−</mo><mi>Y</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>X</mi><mo>=</mo><mi>Y</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s06_s01_p56">In other words, if two absolute value expressions are equal, then the arguments can be the same or opposite.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s01_n06">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch02_s06_s01_p57">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1604" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p58">Set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1605" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></math></span> equal to <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1606" display="inline"><mrow><mo>±</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span> and then solve each linear equation.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p59"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1607" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="right"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mo>|</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mo>|</mo></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo>|</mo></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="right"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>+</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi></mrow></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p60">To check, we substitute these values into the original equation.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p61"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1608" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><em class="emphasis">Check</em> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1609" display="inline"><mrow><mi> </mi><mi>x</mi><mo>=</mo><mn>3</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1610" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1611" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mn>1</mn><mo>|</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>|</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch02_s06_s01_p62">As an exercise, use a graphing utility to graph both <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1612" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1613" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow></mrow></math></span> on the same set of axes. Verify that the graphs intersect where <em class="emphasis">x</em> is equal to 1 and 3.</p>
<p class="para" id="fwk-redden-ch02_s06_s01_p63">Answer: The solutions are 1 and 3.</p>
</div>
<div class="callout block" id="fwk-redden-ch02_s06_s01_n06a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s06_s01_p64"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1614" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>10</mn></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s01_p65">Answer: −2, 6</p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/CskWmsQCBMU" condition="http://img.youtube.com/vi/CskWmsQCBMU/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/CskWmsQCBMU" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch02_s06_s02" version="5.0" lang="en">
<h2 class="title editable block">Absolute Value Inequalities</h2>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p01">We begin by examining the solutions to the following inequality:</p>
<p class="para block" id="fwk-redden-ch02_s06_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1615" display="block"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p03">The absolute value of a number represents the distance from the origin. Therefore, this equation describes all numbers whose distance from zero is less than or equal to 3. We can graph this solution set by shading all such numbers.</p>
<div class="informalfigure large block">
<img src="section_05/54d7fb08641f26cbbcabbdf76cd2ada7.png">
</div>
<p class="para block" id="fwk-redden-ch02_s06_s02_p05">Certainly we can see that there are infinitely many solutions to <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1616" display="inline"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math></span> bounded by −3 and 3. Express this solution set using set notation or interval notation as follows:</p>
<p class="para block" id="fwk-redden-ch02_s06_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1617" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mrow><mo>{</mo><mrow><mi>x</mi><mo>|</mo><mo>−</mo><mn>3</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>3</mn></mrow><mo>}</mo></mrow></mtd><mtd columnalign="left"><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></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo stretchy="false">[</mo><mo>−</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="left"><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></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p07">In this text, we will choose to express solutions in interval notation. In general, given any algebraic expression <em class="emphasis">X</em> and any positive number <em class="emphasis">p</em>:</p>
<p class="para block" id="fwk-redden-ch02_s06_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1618" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtext>If</mtext><mtext> </mtext><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≤</mo><mi>p</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>then</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mo>−</mo><mi>p</mi><mo>≤</mo><mi>X</mi><mo>≤</mo><mi>p</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p09">This theorem holds true for strict inequalities as well. In other words, we can convert any absolute value inequality involving “<em class="emphasis">less than</em>” into a compound inequality which can be solved as usual.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s02_n01">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch02_s06_s02_p10">Solve and graph the solution set: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1619" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p11">Bound the argument <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1620" display="inline"><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span> by −3 and 3 and solve.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p12"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1621" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>3</mn><mo><</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo><</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>3</mn><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle><mo><</mo><mi>x</mi><mo>+</mo><mn>2</mn><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle><mo><</mo><mn>3</mn><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mtd></mtr><mtr><mtd><mo>−</mo><mn>5</mn><mo><</mo><mi>x</mi><mo><</mo><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p13">Here we use open dots to indicate strict inequalities on the graph as follows.</p>
<div class="informalfigure large">
<img src="section_05/d198f40ad4498592e3fb836345d5d10d.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s02_p15">Answer: Using interval notation, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1622" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch02_s06_s02_p16">The solution to <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1623" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>3</mn></mrow></math></span> can be interpreted graphically if we let <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1624" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1625" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span> and then determine where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1626" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> by graphing both <em class="emphasis">f</em> and <em class="emphasis">g</em> on the same set of axes.</p>
<div class="informalfigure large block">
<img src="section_05/2f9d712a7228e5ab4d7b725b4b5dd02a.png">
</div>
<p class="para block" id="fwk-redden-ch02_s06_s02_p18">The solution consists of all <em class="emphasis">x</em>-values where the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1627" display="inline"><mi>f</mi></math></span> is below the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1628" display="inline"><mi>g</mi><mo>.</mo></math></span> In this case, we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1629" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>3</mn></mrow></math></span> where the <em class="emphasis">x</em>-values are between −5 and 1. To apply the theorem, we must first isolate the absolute value.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s02_n02">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch02_s06_s02_p19">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1630" display="inline"><mrow><mn>4</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>7</mn><mo>≤</mo><mn>5</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p20">Begin by isolating the absolute value.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p21"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1631" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>7</mn></mtd><mtd><mo>≤</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>≤</mo></mtd><mtd columnalign="left"><mn>12</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>≤</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p22">Next, apply the theorem and rewrite the absolute value inequality as a compound inequality.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p23"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1632" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>≤</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>3</mn><mo>≤</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>≤</mo><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p24">Solve.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p25"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1633" display="block"><mtable columnspacing="0.1em"><mtr><mtd><mo>−</mo><mn>3</mn><mo>≤</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo>≤</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>3</mn><mstyle color="#007fbf"><mo>−</mo><mn>3</mn></mstyle><mo>≤</mo><mi>x</mi><mo>+</mo><mn>3</mn><mstyle color="#007fbf"><mo>−</mo><mn>3</mn></mstyle><mo>≤</mo><mn>3</mn><mstyle color="#007fbf"><mo>−</mo><mn>3</mn></mstyle></mtd></mtr><mtr><mtd><mo>−</mo><mn>6</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>0</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p26">Shade the solutions on a number line and present the answer in interval notation. Here we use closed dots to indicate inclusive inequalities on the graph as follows:</p>
<div class="informalfigure large">
<img src="section_05/8abae6a57d4658e1e28b81d29a8ba467.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s02_p28">Answer: Using interval notation, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1634" display="inline"><mrow><mrow><mo>[</mo><mrow><mo>−</mo><mn>6</mn><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch02_s06_s02_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s06_s02_p29"><strong class="emphasis bold">Try this!</strong> Solve and graph the solution set: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1635" display="inline"><mrow><mn>3</mn><mo>+</mo><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>8</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p30">Answer: Interval notation: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1636" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
<div class="informalfigure large">
<img src="section_05/53fbc057eea3477b5709c94560299f26.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/sX6ppL2Fbq0" condition="http://img.youtube.com/vi/sX6ppL2Fbq0/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/sX6ppL2Fbq0" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p33">Next, we examine the solutions to an inequality that involves “<em class="emphasis">greater than</em>,” as in the following example:</p>
<p class="para block" id="fwk-redden-ch02_s06_s02_p34"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1637" display="block"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>≥</mo><mn>3</mn></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p35">This inequality describes all numbers whose distance from the origin is greater than or equal to 3. On a graph, we can shade all such numbers.</p>
<div class="informalfigure large block">
<img src="section_05/ee5443c82bae2d950182100b7ab9ff13.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p37">There are infinitely many solutions that can be expressed using set notation and interval notation as follows:</p>
<p class="para block" id="fwk-redden-ch02_s06_s02_p38"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1638" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mrow><mo>{</mo><mrow><mi>x</mi><mo>|</mo><mi>x</mi><mo>≤</mo><mo>−</mo><mn>3</mn><mtext> </mtext><mtext>or</mtext><mtext> </mtext><mi>x</mi><mo>≥</mo><mn>3</mn></mrow><mo>}</mo></mrow></mtd><mtd columnalign="left"><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></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><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>3</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><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></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p39">In general, given any algebraic expression <em class="emphasis">X</em> and any positive number <em class="emphasis">p</em>:</p>
<p class="para block" id="fwk-redden-ch02_s06_s02_p40"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1639" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtext>If</mtext><mtext> </mtext><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≥</mo><mi>p</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>then</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>X</mi><mo>≤</mo><mo>−</mo><mi>p</mi><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>X</mi><mo>≥</mo><mi>p</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p41">The theorem holds true for strict inequalities as well. In other words, we can convert any absolute value inequality involving “<em class="emphasis">greater than</em>” into a compound inequality that describes two intervals.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s02_n03">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch02_s06_s02_p42">Solve and graph the solution set: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1640" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>></mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solve</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p43">The argument <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1641" display="inline"><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span> must be less than −3 or greater than 3.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p44"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1642" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>></mo><mn>3</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi><mo>+</mo><mn>2</mn><mo><</mo><mo>−</mo><mn>3</mn><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mtext> </mtext><mi>x</mi><mo>+</mo><mn>2</mn><mo>></mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi><mo><</mo><mo>−</mo><mn>5</mn></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow><mi>x</mi><mo>></mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<div class="informalfigure large">
<img src="section_05/12b84b5b3331a000842bb1191680226e.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s02_p46">Answer: Using interval notation, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1643" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch02_s06_s02_p47">The solution to <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1644" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>></mo><mn>3</mn></mrow></math></span> can be interpreted graphically if we let <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1645" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1646" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span> and then determine where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1647" display="inline"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>></mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> by graphing both <em class="emphasis">f</em> and <em class="emphasis">g</em> on the same set of axes.</p>
<div class="informalfigure large block">
<img src="section_05/db019b8afdd81d3773f18c9a6e7658e4.png">
</div>
<p class="para block" id="fwk-redden-ch02_s06_s02_p49">The solution consists of all <em class="emphasis">x</em>-values where the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1648" display="inline"><mrow><mi>f</mi></mrow></math></span> is above the graph of <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1649" display="inline"><mrow><mi>g</mi></mrow><mo>.</mo></math></span> In this case, we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1650" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>></mo><mn>3</mn></mrow></math></span> where the <em class="emphasis">x</em>-values are less than −5 or are greater than 1. To apply the theorem we must first isolate the absolute value.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s02_n04">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch02_s06_s02_p50">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1651" display="inline"><mrow><mn>3</mn><mo>+</mo><mn>2</mn><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>≥</mo><mn>13</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p51">Begin by isolating the absolute value.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p52"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1652" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mo>+</mo><mn>2</mn><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mn>13</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p53">Next, apply the theorem and rewrite the absolute value inequality as a compound inequality.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p54"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1653" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>≥</mo><mn>5</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn><mo>≤</mo><mo>−</mo><mn>5</mn></mrow></mtd><mtd><mrow><mtext>or</mtext></mrow></mtd><mtd><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn><mo>≥</mo><mn>5</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p55">Solve.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p56"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1654" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow></mtd><mtd columnalign="right"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="right"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow></mtd><mtd columnalign="right"><mo>≥</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>4</mn><mi>x</mi></mrow></mtd><mtd columnalign="right"><mo>≤</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>4</mn><mi>x</mi></mrow></mtd><mtd columnalign="right"><mo>≥</mo></mtd><mtd columnalign="left"><mrow><mn>12</mn></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>4</mn><mi>x</mi></mrow></mtd><mtd columnalign="right"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>2</mn><mn>4</mn></mfrac></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="right"><mo>≥</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>4</mn><mi>x</mi></mrow></mtd><mtd columnalign="right"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p57">Shade the solutions on a number line and present the answer using interval notation.</p>
<div class="informalfigure large">
<img src="section_05/06fafde1bcd818b99ee494ab4c092fc0.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s02_p59">Answer: Using interval notation, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1655" 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>
<div class="callout block" id="fwk-redden-ch02_s06_s02_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s06_s02_p60"><strong class="emphasis bold">Try this!</strong> Solve and graph: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1656" display="inline"><mrow><mn>3</mn><mrow><mo>|</mo><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>2</mn><mo>></mo><mn>13</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p61">Answer: Using interval notation, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1657" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
<div class="informalfigure large">
<img src="section_05/8775b7be1a775881d4ce3e89ac626c6c.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/P6HjRz6W4F4" condition="http://img.youtube.com/vi/P6HjRz6W4F4/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/P6HjRz6W4F4" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p64">Up to this point, the solution sets of linear absolute value inequalities have consisted of a single bounded interval or two unbounded intervals. This is not always the case.</p>
<div class="callout block" id="fwk-redden-ch02_s06_s02_n05">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch02_s06_s02_p65">Solve and graph: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1658" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>5</mn><mo>></mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p66">Begin by isolating the absolute value.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p67"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1659" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>5</mn></mtd><mtd><mo>></mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>></mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p68">Notice that we have an absolute value greater than a negative number. For any real number <em class="emphasis">x</em> the absolute value of the argument will always be positive. Hence, any real number will solve this inequality.</p>
<div class="informalfigure large">
<img src="section_05/1ae423963434b72a27df0ffb19ce6b24.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s02_p70">Geometrically, we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1660" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>5</mn></mrow></math></span> is always greater than <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1661" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large">
<img src="section_05/4df21f5c6ca04b0cb00c064214bbf386.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s02_p72">Answer: All real numbers, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1662" display="inline"><mi>ℝ</mi><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch02_s06_s02_n06">
<h3 class="title">Example 12</h3>
<p class="para" id="fwk-redden-ch02_s06_s02_p73">Solve and graph: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1663" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>4</mn><mo>≤</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p74">Begin by isolating the absolute value.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p75"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1664" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>4</mn></mtd><mtd><mo>≤</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow></mtd><mtd><mo>≤</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn><mtext> </mtext></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s06_s02_p76">In this case, we can see that the isolated absolute value is to be less than or equal to a negative number. Again, the absolute value will always be positive; hence, we can conclude that there is no solution.</p>
<p class="para" id="fwk-redden-ch02_s06_s02_p77">Geometrically, we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1665" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>4</mn></mrow></math></span> is never less than <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1666" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large">
<img src="section_05/611732db35c3bdba71871ef237256092.png">
</div>
<p class="para" id="fwk-redden-ch02_s06_s02_p79">Answer: Ø</p>
</div>
<p class="para block" id="fwk-redden-ch02_s06_s02_p80">In summary, there are three cases for absolute value equations and inequalities. The relations =, <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1667" display="inline"><mo><</mo><mo>,</mo></math></span> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1668" display="inline"><mo>≤</mo><mo>,</mo></math></span> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1669" display="inline"><mo>></mo><mo>,</mo></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1670" display="inline"><mo>≥</mo></math></span> determine which theorem to apply.</p>
<p class="para block"> </p>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p81"><strong class="emphasis bold">Case 1</strong>: An absolute value equation:</p>
<p class="para block" id="fwk-redden-ch02_s06_s02_p82"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1671" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtext>If</mtext></mrow><mtext> </mtext><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mi>p</mi></mrow></mtd></mtr><mtr><mtd><mrow><mtext>then</mtext><mtext> </mtext></mrow><mrow><mi>X</mi><mo>=</mo><mo>−</mo><mi>p</mi><mtext> </mtext></mrow><mrow><mtext> </mtext><mtext>or</mtext><mtext> </mtext></mrow><mrow><mtext> </mtext><mi>X</mi><mo>=</mo><mi>p</mi></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"> <div class="informalfigure medium">
<img src="section_05/1e1718f40151f1b3c13b7f85115a285a.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p83"><strong class="emphasis bold">Case 2</strong>: An absolute value inequality involving “<em class="emphasis">less than</em>.”</p>
<p class="para block" id="fwk-redden-ch02_s06_s02_p84"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1672" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtext>If</mtext></mrow><mtext> </mtext><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≤</mo><mi>p</mi></mrow></mtd></mtr><mtr><mtd><mrow><mtext>then</mtext></mrow><mtext> </mtext><mrow><mo>−</mo><mi>p</mi><mo>≤</mo><mi>X</mi><mo>≤</mo><mi>p</mi></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"> <div class="informalfigure medium">
<img src="section_05/fdc16175aae956b7825143cc652dbb53.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s06_s02_p85"><strong class="emphasis bold">Case 3</strong>: An absolute value inequality involving “<em class="emphasis">greater than</em>.”</p>
<p class="para block" id="fwk-redden-ch02_s06_s02_p86"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1673" display="inline"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mtext>If</mtext></mrow><mtext> </mtext><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≥</mo><mi>p</mi></mrow></mtd></mtr><mtr><mtd><mrow><mtext>then</mtext></mrow><mtext> </mtext><mrow><mi>X</mi><mo>≤</mo><mo>−</mo><mi>p</mi></mrow><mtext> </mtext><mrow><mtext>or</mtext></mrow><mtext> </mtext><mrow><mi>X</mi><mo>≥</mo><mi>p</mi></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"> <div class="informalfigure medium">
<img src="section_05/0519655a76754718478f57cf009322ad.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<div class="key_takeaways block" id="fwk-redden-ch02_s06_s02_n07">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch02_s06_s02_l01" mark="bullet">
<li>To solve an absolute value equation, such as <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1674" display="inline"><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>=</mo><mi>p</mi></mrow></math></span>, replace it with the two equations <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1675" display="inline"><mrow><mi>X</mi><mo>=</mo><mo>−</mo><mi>p</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1676" display="inline"><mrow><mi>X</mi><mo>=</mo><mi>p</mi></mrow></math></span> and then solve each as usual. Absolute value equations can have up to two solutions.</li>
<li>To solve an absolute value inequality involving “less than,” such as <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1677" display="inline"><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≤</mo><mi>p</mi></mrow></math></span>, replace it with the compound inequality <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1678" display="inline"><mrow><mo>−</mo><mi>p</mi><mo>≤</mo><mi>X</mi><mo>≤</mo><mi>p</mi></mrow></math></span> and then solve as usual.</li>
<li>To solve an absolute value inequality involving “greater than,” such as <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1679" display="inline"><mrow><mrow><mo>|</mo><mi>X</mi><mo>|</mo></mrow><mo>≥</mo><mi>p</mi></mrow></math></span>, replace it with the compound inequality <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1680" display="inline"><mrow><mi>X</mi><mo>≤</mo><mo>−</mo><mi>p</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>X</mi><mo>≥</mo><mi>p</mi></mrow></math></span> and then solve as usual.</li>
<li>Remember to isolate the absolute value before applying these theorems.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch02_s06_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd01">
<h3 class="title">Part A: Absolute Value Equations Solve.</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd01_qd01">
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p01"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1681" display="inline"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>=</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p03"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1682" display="inline"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p05"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1683" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p07"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1684" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p09"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1685" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>12</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p11"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1686" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p13"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1687" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>|</mo></mrow><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p15"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1688" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>=</mo><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p17"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1689" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>13</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p19"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1690" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>16</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p21"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1692" display="inline"><mrow><mrow><mo>|</mo><mrow><mo>−</mo><mn>5</mn><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p23"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1694" display="inline"><mrow><mrow><mo>|</mo><mrow><mo>−</mo><mn>6</mn><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa13">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1696" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa14">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1698" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mfrac><mn>5</mn><mrow><mn>12</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p29"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1700" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>0.2</mn><mi>x</mi><mo>+</mo><mn>1.6</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>3.6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p31"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1701" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>0.3</mn><mi>x</mi><mo>−</mo><mn>1.2</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>2.7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p33"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1702" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p35"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1703" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mi>y</mi></mrow><mo>|</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p37"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1704" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p39"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1706" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>2</mn><mo>=</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p41"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1708" display="inline"><mrow><mn>9</mn><mo>+</mo><mrow><mo>|</mo><mrow><mn>7</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p43"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1710" display="inline"><mrow><mn>4</mn><mo>−</mo><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p45"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1712" display="inline"><mrow><mn>3</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>4</mn><mo>=</mo><mn>25</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p47"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1713" display="inline"><mrow><mn>2</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn><mo>=</mo><mn>17</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p49"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1714" display="inline"><mrow><mn>9</mn><mo>+</mo><mn>5</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1715" display="inline"><mrow><mn>11</mn><mo>+</mo><mn>6</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p53"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1716" display="inline"><mrow><mn>8</mn><mo>−</mo><mn>2</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p55"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1717" display="inline"><mrow><mn>12</mn><mo>−</mo><mn>5</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa29">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1718" display="block"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa30">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1719" display="block"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>|</mo></mrow><mo>+</mo><mn>1</mn><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p61"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1720" display="inline"><mrow><mo>−</mo><mn>2</mn><mrow><mo>|</mo><mrow><mn>7</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>4</mn><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p63"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1721" display="inline"><mrow><mo>−</mo><mn>3</mn><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>2</mn><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1722" display="inline"><mrow><mn>1.2</mn><mrow><mo>|</mo><mrow><mi>t</mi><mo>−</mo><mn>2.8</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>4.8</mn><mo>=</mo><mn>1.2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1723" display="inline"><mrow><mn>3.6</mn><mrow><mo>|</mo><mrow><mi>t</mi><mo>+</mo><mn>1.8</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>2.6</mn><mo>=</mo><mn>8.2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa35">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1724" display="block"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>|</mo><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>1</mn><mo>=</mo><mn>4</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa36">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1726" display="block"><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mrow><mo>|</mo><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>5</mn><mo>=</mo><mn>3</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1728" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1729" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>8</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>12</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1730" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><mi>y</mi><mo>+</mo><mn>8</mn></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1732" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>7</mn><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>5</mn><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p81"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1733" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>3</mn><mi>x</mi></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p83"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1735" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>7</mn><mi>x</mi></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa43">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1737" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>|</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa44">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1739" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>3</mn><mn>5</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow><mo>|</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p89"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1741" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>1.5</mn><mi>t</mi><mo>−</mo><mn>3.5</mn></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>2.5</mn><mi>t</mi><mo>+</mo><mn>0.5</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1742" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3.2</mn><mi>t</mi><mo>−</mo><mn>1.4</mn></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>1.8</mn><mi>t</mi><mo>+</mo><mn>2.8</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1743" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><mo>−</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1744" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd01_qd02" start="49">
<p class="para" id="fwk-redden-ch02_s06_qs01_p97"><strong class="emphasis bold">Assume all variables in the denominator are nonzero.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p98">Solve for <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1746" display="inline"><mrow><mi>p</mi><mrow><mo>|</mo><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mo>|</mo></mrow><mo>−</mo><mi>q</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p100">Solve for <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1748" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd02">
<h3 class="title">Part B: Absolute Value Inequalities</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd02_qd01" start="51">
<p class="para" id="fwk-redden-ch02_s06_qs01_p102"><strong class="emphasis bold">Solve and graph the solution set. In addition, give the solution set in interval notation.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1750" display="inline"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo><</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1752" display="inline"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1754" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>≤</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p109"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1756" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p111"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1758" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1759" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>|</mo></mrow><mo><</mo><mo>−</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p115"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1760" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>≤</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p117"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1762" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>27</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1764" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p121"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1766" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>10</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>25</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa61">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1768" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>|</mo></mrow><mo>≤</mo><mn>1</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa62">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1770" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>|</mo></mrow><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa63">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1772" display="inline"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>≥</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa64">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1774" display="inline"><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>></mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa65">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1776" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>></mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa66">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1778" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>≥</mo><mn>11</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa67">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1780" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa68">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1782" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>12</mn></mrow><mo>|</mo></mrow><mo>></mo><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa69">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1784" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo>≥</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa70">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1786" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>≥</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa71">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1788" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>></mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa72">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1790" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>≥</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa73">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1792" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mn>7</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>3</mn><mrow><mn>14</mn></mrow></mfrac></mrow><mo>|</mo></mrow><mo>></mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa74">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1794" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow><mo>|</mo></mrow><mo>></mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd02_qd02" start="75">
<p class="para" id="fwk-redden-ch02_s06_qs01_p151"><strong class="emphasis bold">Solve and graph the solution set.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p152"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1796" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>></mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p154"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1798" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>≤</mo><mn>21</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p156"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1800" display="inline"><mrow><mo>−</mo><mn>5</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow><mo>></mo><mo>−</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p158"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1802" display="inline"><mrow><mo>−</mo><mn>3</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>|</mo></mrow><mo>≤</mo><mo>−</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1804" display="inline"><mrow><mn>6</mn><mo>−</mo><mn>3</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow><mo><</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p162"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1806" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>|</mo></mrow><mo>≤</mo><mo>−</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p164"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1808" display="inline"><mrow><mn>6</mn><mo>−</mo><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo><</mo><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p166"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1810" display="inline"><mrow><mn>25</mn><mo>−</mo><mrow><mo>|</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>≥</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p168"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1812" display="inline"><mrow><mo>|</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>25</mn><mo>|</mo><mo>−</mo><mn>4</mn><mo>≥</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p170"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1814" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>−</mo><mn>8</mn><mo><</mo><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p172"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1816" display="inline"><mrow><mn>2</mn><mrow><mo>|</mo><mrow><mn>9</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>8</mn><mo>></mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p174"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1818" display="inline"><mrow><mn>3</mn><mrow><mo>|</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>4</mn><mo><</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p176"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1819" display="inline"><mrow><mn>5</mn><mrow><mo>|</mo><mrow><mn>4</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow><mo>|</mo></mrow><mo>−</mo><mn>10</mn><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p178"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1821" display="inline"><mrow><mn>6</mn><mrow><mo>|</mo><mrow><mn>1</mn><mo>−</mo><mn>4</mn><mi>x</mi></mrow><mo>|</mo></mrow><mo>−</mo><mn>24</mn><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p180"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1823" display="inline"><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>|</mo></mrow><mo>></mo><mo>−</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p182"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1825" display="inline"><mrow><mn>9</mn><mo>−</mo><mn>7</mn><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow><mo><</mo><mo>−</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p184"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1827" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn></mrow><mo>|</mo></mrow><mo>></mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p186"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1829" display="inline"><mrow><mrow><mo>|</mo><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>6</mn></mrow><mo>|</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa93">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1831" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mn>7</mn><mn>6</mn></mfrac></mrow><mo>|</mo></mrow><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>≤</mo><mo>−</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa94">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1833" display="block"><mrow><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>|</mo></mrow><mo>+</mo><mfrac><mn>3</mn><mrow><mn>20</mn></mrow></mfrac><mo>></mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1835" display="inline"><mrow><mn>12</mn><mo>+</mo><mn>4</mn><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>≤</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1837" display="inline"><mrow><mn>3</mn><mo>−</mo><mn>6</mn><mrow><mo>|</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>≥</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p196"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1839" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>3</mn><mo><</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p198"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1841" display="inline"><mrow><mn>2</mn><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn><mo>≤</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p200"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1843" display="inline"><mrow><mn>7</mn><mo>−</mo><mrow><mo>|</mo><mrow><mo>−</mo><mn>4</mn><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mn>4</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>></mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p202"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1845" display="inline"><mrow><mn>9</mn><mo>−</mo><mrow><mo>|</mo><mrow><mn>6</mn><mo>+</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>≥</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa101">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1847" display="block"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo><mrow><mo>|</mo><mrow><mn>2</mn><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi></mrow><mo>|</mo></mrow><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa102">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1849" display="block"><mrow><mfrac><mn>5</mn><mn>4</mn></mfrac><mo>−</mo><mrow><mo>|</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi></mrow><mo>|</mo></mrow><mo><</mo><mfrac><mn>3</mn><mn>8</mn></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd02_qd03" start="103">
<p class="para" id="fwk-redden-ch02_s06_qs01_p208"><strong class="emphasis bold">Assume all variables in the denominator are nonzero.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p209">Solve for <em class="emphasis">x</em> where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1851" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>p</mi><mo>></mo><mn>0</mn></mrow></math></span>: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1852" display="inline"><mrow><mi>p</mi><mrow><mo>|</mo><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mo>|</mo></mrow><mo>− </mo><mi>q</mi><mo>≤</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p211">Solve for <em class="emphasis">x</em> where <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1854" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>p</mi><mo>></mo><mn>0</mn></mrow></math></span>: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1855" display="inline"><mrow><mi>p</mi><mrow><mo>|</mo><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mo>|</mo></mrow><mo>−</mo><mi>q</mi><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd02_qd04" start="105">
<p class="para" id="fwk-redden-ch02_s06_qs01_p213"><strong class="emphasis bold">Given the graph of</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1857" display="inline"><mi>f</mi></math></span> <strong class="emphasis bold">and</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1858" display="inline"><mi>g</mi></math></span><strong class="emphasis bold">, determine the <em class="emphasis">x</em>-values where:</strong></p>
<ol class="orderedlist" id="fwk-redden-ch02_s06_qs01_o01" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1859" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1860" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>></mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1861" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span></li>
</ol>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa105">
<div class="question">
<div class="informalfigure large">
<img src="section_05/bc0d2fc192eab76856e471f5ac12e232.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa106">
<div class="question">
<div class="informalfigure large">
<img src="section_05/c30fedbf670eed3e2543ae2a040dd312.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa107">
<div class="question">
<div class="informalfigure large">
<img src="section_05/dc94f63ebaac262f5be8b9622190bb50.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa108">
<div class="question">
<div class="informalfigure large">
<img src="section_05/abe1275f6595df69b24710504d392f87.png">
</div>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s06_qs01_qd03_qd01" start="109">
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p218">Make three note cards, one for each of the three cases described in this section. On one side write the theorem, and on the other write a complete solution to a representative example. Share your strategy for identifying and solving absolute value equations and inequalities on the discussion board.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch02_s06_qs01_p219">Make your own examples of absolute value equations and inequalities that have no solution, at least one for each case described in this section. Illustrate your examples with a graph.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch02_s06_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch02_s06_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s06_qs01_p02_ans">−9, 9</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s06_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-ch02_s06_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s06_qs01_p06_ans">4, 10</p>