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<title>Solving Inequalities with Two Variables</title>
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<div class="section" id="fwk-redden-ch02_s07" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">2.7</span> Solving Inequalities with Two Variables</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch02_s07_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch02_s07_o01" numeration="arabic">
<li>Identify and check solutions to inequalities with two variables.</li>
<li>Graph solution sets of linear inequalities with two variables.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch02_s07_s01" version="5.0" lang="en">
<h2 class="title editable block">Solutions to Inequalities with Two Variables</h2>
<p class="para editable block" id="fwk-redden-ch02_s07_s01_p01">We know that a linear equation with two variables has infinitely many ordered pair solutions that form a line when graphed. A <span class="margin_term"><a class="glossterm">linear inequality with two variables</a><span class="glossdef">An inequality relating linear expressions with two variables. The solution set is a region defining half of the plane.</span></span>, on the other hand, has a solution set consisting of a region that defines half of the plane.</p>
<p class="para block" id="fwk-redden-ch02_s07_s01_p02"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">Linear Equation</p></th>
<th align="center"><p class="para">Linear Inequality</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1868" display="inline"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1869" display="inline"><mrow><mi>y</mi><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"> <div class="informalfigure medium">
<img src="section_05/c0ba389d132dac23daf68ad7350690f2.png">
</div>
</td>
<td align="left"> <div class="informalfigure medium">
<img src="section_05/a6b9541aed8ef1677c915bd3e20e30e6.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s07_s01_p03">For the inequality, the line defines the boundary of the region that is shaded. This indicates that any ordered pair in the shaded region, including the boundary line, will satisfy the inequality. To see that this is the case, choose a few <span class="margin_term"><a class="glossterm">test points</a><span class="glossdef">A point not on the boundary of the linear inequality used as a means to determine in which half-plane the solutions lie.</span></span> and substitute them into the inequality.</p>
<p class="para block" id="fwk-redden-ch02_s07_s01_p04"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><strong class="emphasis bold">Test point</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1870" display="inline"><mrow><mi>y</mi><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">(0, 0)</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1871" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>0</mn><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>≤</mo><mn>3</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">(2, 1)</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1872" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>1</mn><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>≤</mo><mn>3</mn><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>≤</mo><mn>6</mn><mi> </mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">(−2, −1)</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1873" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mo>−</mo><mn>1</mn><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>1</mn><mo>≤</mo><mo>−</mo><mn>3</mn><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>1</mn><mo>≤</mo><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s07_s01_p05">Also, we can see that ordered pairs outside the shaded region do not solve the linear inequality.</p>
<p class="para block" id="fwk-redden-ch02_s07_s01_p06"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><strong class="emphasis bold">Test point</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1874" display="inline"><mrow><mi>y</mi><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">(−2, 3)</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1875" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>3</mn><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn><mo>≤</mo><mo>−</mo><mn>3</mn><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn><mo>≤</mo><mn>0</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#ff0000"><mtext>✗</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch02_s07_s01_p07">The graph of the solution set to a linear inequality is always a region. However, the boundary may not always be included in that set. In the previous example, the line was part of the solution set because of the “or equal to” part of the inclusive inequality <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1876" display="inline"><mo>≤</mo><mo>.</mo></math></span> If given a strict inequality <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1877" display="inline"><mo><</mo></math></span>, we would then use a dashed line to indicate that those points are not included in the solution set.</p>
<p class="para block" id="fwk-redden-ch02_s07_s01_p08"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">Non-Inclusive Boundary</p></th>
<th align="center"><p class="para">Inclusive Boundary</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1878" display="inline"><mrow><mi>y</mi><mo><</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1879" display="inline"><mrow><mi>y</mi><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"> <div class="informalfigure medium">
<img src="section_05/32164708ce230b1f6cb8f1ed75ececdb.png">
</div>
</td>
<td align="left"> <div class="informalfigure medium">
<img src="section_05/a6b9541aed8ef1677c915bd3e20e30e6.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s07_s01_p09">Consider the point (0, 3) on the boundary; this ordered pair satisfies the linear equation. It is the “or equal to” part of the inclusive inequality that makes the ordered pair part of the solution set.</p>
<p class="para block" id="fwk-redden-ch02_s07_s01_p10"></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_m1880" display="inline"><mrow><mi>y</mi><mo><</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1881" display="inline"><mrow><mi>y</mi><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1882" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>3</mn><mo><</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn><mo><</mo><mn>0</mn><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn><mo><</mo><mn>3</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mtext> </mtext><mstyle color="#ff0000"><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_m1883" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>3</mn><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn><mo>≤</mo><mn>0</mn><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn><mo>≤</mo><mn>3</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s07_s01_p11">So far we have seen examples of inequalities that were “less than.” Now consider the following graphs with the same boundary:</p>
<p class="para block" id="fwk-redden-ch02_s07_s01_p12"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">Greater Than (Above)</p></th>
<th align="center"><p class="para">Less Than (Below)</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1884" display="inline"><mrow><mi>y</mi><mo>≥</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
<td align="left"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1885" display="inline"><mrow><mi>y</mi><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="left"> <div class="informalfigure medium">
<img src="section_05/2a39ff5c1efb787eb4a1d8c4e4816031.png">
</div>
</td>
<td align="left"> <div class="informalfigure medium">
<img src="section_05/a6b9541aed8ef1677c915bd3e20e30e6.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s07_s01_p13">Given the graphs above, what might we expect if we use the origin (0, 0) as a test point?</p>
<p class="para block" id="fwk-redden-ch02_s07_s01_p14"></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_m1886" display="inline"><mrow><mi>y</mi><mo>≥</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1887" display="inline"><mrow><mi>y</mi><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1888" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>0</mn><mo>≥</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>≥</mo><mn>0</mn><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>≥</mo><mn>3</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#ff0000"><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_m1889" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>0</mn><mo>≤</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>≤</mo><mn>0</mn><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>≤</mo><mn>3</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<div class="callout block" id="fwk-redden-ch02_s07_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch02_s07_s01_p15">Determine whether or not <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1890" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span> is a solution to <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1891" display="inline"><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo><</mo><mn>10</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s07_s01_p16">Substitute the <em class="emphasis">x</em>- and <em class="emphasis">y</em>-values into the equation and see if a true statement is obtained.</p>
<p class="para" id="fwk-redden-ch02_s07_s01_p17"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1892" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo><</mo></mtd><mtd columnalign="left"><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mtd><mtd><mo><</mo></mtd><mtd columnalign="left"><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mo>−</mo><mn>1</mn></mtd><mtd><mo><</mo></mtd><mtd columnalign="left"><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn></mtd><mtd><mo><</mo></mtd><mtd columnalign="left"><mn>10</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch02_s07_s01_p18">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1893" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span> is a solution.</p>
</div>
<p class="para block" id="fwk-redden-ch02_s07_s01_p19">These ideas and techniques extend to nonlinear inequalities with two variables. For example, all of the solutions to <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1894" display="inline"><mrow><mi>y</mi><mo>></mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> are shaded in the graph below.</p>
<div class="informalfigure large block">
<img src="section_05/d729e8f8fa4d185c5fa93825c8fae657.png">
</div>
<p class="para editable block" id="fwk-redden-ch02_s07_s01_p21">The boundary of the region is a parabola, shown as a dashed curve on the graph, and is not part of the solution set. However, from the graph we expect the ordered pair (−1,4) to be a solution. Furthermore, we expect that ordered pairs that are not in the shaded region, such as (−3, 2), will not satisfy the inequality.</p>
<p class="para block" id="fwk-redden-ch02_s07_s01_p22"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Check</em> (−1,4)</p></th>
<th align="center"><p class="para"><em class="emphasis">Check</em> (−3, 2)</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1895" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mi>y</mi><mo>></mo><msup><mi>x</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mn>4</mn><mo>></mo><msup><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mn>4</mn><mo>></mo><mn>1</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><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_m1896" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mi>y</mi><mo>></mo><msup><mi>x</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mn>2</mn><mo>></mo><msup><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mn>2</mn><mo>></mo><mn>9</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#ff0000"><mtext>✗</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s07_s01_p23">Following are graphs of solutions sets of inequalities with inclusive parabolic boundaries.</p>
<p class="para block" id="fwk-redden-ch02_s07_s01_p24"></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_m1897" display="inline"><mrow><mi>y</mi><mo>≤</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1898" display="inline"><mrow><mi>y</mi><mo>≥</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"> <div class="informalfigure medium">
<img src="section_05/c2f435fe1aac6d09820b94447436bc78.png">
</div>
</td>
<td align="center"> <div class="informalfigure medium">
<img src="section_05/ea928f1ceec1321852c0e45b4fd99236.png">
</div>
</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch02_s07_s01_p25">You are encouraged to test points in and out of each solution set that is graphed above.</p>
<p class="para block"> </p>
<div class="callout block" id="fwk-redden-ch02_s07_s01_n01a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s07_s01_p26"><strong class="emphasis bold">Try this!</strong> Is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1899" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span> a solution to <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1900" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo><</mo><mn>0</mn></mrow></math></span>?</p>
<p class="para" id="fwk-redden-ch02_s07_s01_p27">Answer: No</p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/IMTudaflcik" condition="http://img.youtube.com/vi/IMTudaflcik/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/IMTudaflcik" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch02_s07_s02" version="5.0" lang="en">
<h2 class="title editable block">Graphing Solutions to Inequalities with Two Variables</h2>
<p class="para editable block" id="fwk-redden-ch02_s07_s02_p01">Solutions to linear inequalities are a shaded half-plane, bounded by a solid line or a dashed line. This boundary is either included in the solution or not, depending on the given inequality. If we are given a strict inequality, we use a dashed line to indicate that the boundary is not included. If we are given an inclusive inequality, we use a solid line to indicate that it is included. The steps for graphing the solution set for an inequality with two variables are shown in the following example.</p>
<div class="callout block" id="fwk-redden-ch02_s07_s02_n01">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch02_s07_s02_p02">Graph the solution set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1901" display="inline"><mrow><mi>y</mi><mo>></mo><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<ul class="itemizedlist" id="fwk-redden-ch02_s07_s02_l01" mark="none">
<li>
<p class="para"><strong class="emphasis bold">Step 1:</strong> Graph the boundary. Because of the strict inequality, we will graph the boundary <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1902" display="inline"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span> using a dashed line. We can see that the slope is <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1903" display="inline"><mrow><mi>m</mi><mo>=</mo><mo>−</mo><mn>3</mn><mo>=</mo><mfrac><mrow><mo>−</mo><mn>3</mn></mrow><mn>1</mn></mfrac><mo>=</mo><mfrac><mrow><mi>r</mi><mi>i</mi><mi>s</mi><mi>e</mi></mrow><mrow><mi>r</mi><mi>u</mi><mi>n</mi></mrow></mfrac></mrow></math></span> and the <em class="emphasis">y</em>-intercept is (0, 1).</p>
<div class="informalfigure large">
<img src="section_05/82284673731d10796e50613e26fc9742.png">
</div>
</li>
<li>
<p class="para"><strong class="emphasis bold">Step 2:</strong> Test a point that is <em class="emphasis bolditalic">not</em> on the boundary. A common test point is the origin, (0, 0). The test point helps us determine which half of the plane to shade.</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p03"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><strong class="emphasis bold">Test point</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1904" display="inline"><mrow><mi>y</mi><mo>></mo><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">(0, 0)</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1905" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>0</mn><mo>></mo><mo>−</mo><mn>3</mn><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>></mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mtext>✗</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
</li>
<li>
<strong class="emphasis bold">Step 3:</strong> Shade the region containing the solutions. Since the test point (0, 0) was not a solution, it does not lie in the region containing all the ordered pair solutions. Therefore, shade the half of the plane that does not contain this test point. In this case, shade above the boundary line.</li>
</ul>
<p class="para" id="fwk-redden-ch02_s07_s02_p04">Answer: </p>
<div class="informalfigure large">
<img src="section_05/8e4e5aab40ea431ec175f16e5f75b13d.png">
</div>
</div>
<p class="para block" id="fwk-redden-ch02_s07_s02_p05">Consider the problem of shading above or below the boundary line when the inequality is in slope-intercept form. If <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1906" display="inline"><mrow><mi>y</mi><mo>></mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></math></span>, then shade above the line. If <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1907" display="inline"><mrow><mi>y</mi><mo><</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></math></span>, then shade below the line. Shade with caution; sometimes the boundary is given in standard form, in which case these rules do not apply.</p>
<div class="callout block" id="fwk-redden-ch02_s07_s02_n02">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch02_s07_s02_p06">Graph the solution set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1908" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>≥</mo><mo>−</mo><mn>10</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p07">Here the boundary is defined by the line <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1909" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>10</mn></mrow><mo>.</mo></math></span> Since the inequality is inclusive, we graph the boundary using a solid line. In this case, graph the boundary line using intercepts.</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p08"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para">To find the <em class="emphasis">x</em>-intercept, set <em class="emphasis">y</em> = 0.</p></th>
<th align="center"><p class="para">To find the <em class="emphasis">y</em>-intercept, set <em class="emphasis">x</em> = 0.</p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1910" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>10</mn></mrow></math></span></p>
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1911" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>5</mn></mtd></mtr></mtable></math></span></p>
</td>
<td align="center">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1912" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>10</mn></mrow></math></span></p>
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1913" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></math></span></p>
</td>
</tr>
<tr>
<td align="center"><p class="para"><em class="emphasis">x</em>-intercept: (−5, 0)</p></td>
<td align="center"><p class="para"><em class="emphasis">y</em>-intercept: (0, 2)</p></td>
</tr>
</tbody>
</table>
</div>
<div class="informalfigure large">
<img src="section_05/008c62b07e5818001cad1191ad888028.png">
</div>
<p class="para" id="fwk-redden-ch02_s07_s02_p10">Next, test a point; this helps decide which region to shade.</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p11"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><strong class="emphasis bold">Test point</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1914" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>≥</mo><mo>−</mo><mn>10</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">(0, 0)</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1915" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>−</mo><mn>5</mn><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch02_s07_s02_p12">Since the test point is in the solution set, shade the half of the plane that contains it.</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p13">Answer: </p>
<div class="informalfigure large">
<img src="section_05/87613d604fcf319bcb23a31a3ceae2ba.png">
</div>
</div>
<p class="para block" id="fwk-redden-ch02_s07_s02_p14">In this example, notice that the solution set consists of all the ordered pairs below the boundary line. This may seem counterintuitive because the original inequality involved “greater than” <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1916" display="inline"><mo>≥</mo><mo>.</mo></math></span> This illustrates that it is a best practice to actually test a point. Solve for <em class="emphasis">y</em> and you see that the shading is correct.</p>
<p class="para block" id="fwk-redden-ch02_s07_s02_p15"><span class="informalequation"><math xml:id="fwk-redden-ch02_m1917" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mstyle color="#007fbf"><mo>−</mo><mn>2</mn><mi>x</mi></mstyle></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>10</mn><mstyle color="#007fbf"><mo>−</mo><mn>2</mn><mi>x</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>≥</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mo>−</mo><mn>5</mn><mi>y</mi></mrow><mrow><mstyle color="#007fbf"><mo>−</mo><mn>5</mn></mstyle></mrow></mfrac></mtd><mtd><mstyle color="#007f3f"><mo>≤</mo></mstyle></mtd><mtd columnalign="left"><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>10</mn></mrow><mrow><mstyle color="#007fbf"><mo>−</mo><mn>5</mn></mstyle></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>R</mi><mi>e</mi><mi>v</mi><mi>e</mi><mi>r</mi><mi>s</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>i</mi><mi>n</mi><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>t</mi><mi>y</mi></mstyle><mo>.</mo></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>≤</mo></mtd><mtd columnalign="left"><mfrac><mn>2</mn><mn>5</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch02_s07_s02_p16">In slope-intercept form, you can see that the region below the boundary line should be shaded. An alternate approach is to first express the boundary in slope-intercept form, graph it, and then shade the appropriate region.</p>
<div class="callout block" id="fwk-redden-ch02_s07_s02_n03">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch02_s07_s02_p17">Graph the solution set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1918" display="inline"><mrow><mi>y</mi><mo><</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p18">First, graph the boundary line <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1919" display="inline"><mrow><mi>y</mi><mo>=</mo><mn>2</mn></mrow></math></span> with a dashed line because of the strict inequality. Next, test a point.</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p19"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><strong class="emphasis bold">Test point</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1920" display="inline"><mrow><mi>y</mi><mo><</mo><mn>2</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">(0, 0)</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1921" display="inline"><mrow><mn>0</mn><mo><</mo><mn>2</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch02_s07_s02_p20">In this case, shade the region that contains the test point.</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p21">Answer: </p>
<div class="informalfigure large">
<img src="section_05/1e02c94b53d1c81c022ac740fe3d39e6.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch02_s07_s02_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s07_s02_p22"><strong class="emphasis bold">Try this!</strong> Graph the solution set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1922" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo><</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s07_s02_p23">Answer:</p>
<div class="informalfigure large">
<img src="section_05/f1ea4eef0408ae6f20dc13ef3b76a347.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/53Glfim7gGc" condition="http://img.youtube.com/vi/53Glfim7gGc/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/53Glfim7gGc" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para editable block" id="fwk-redden-ch02_s07_s02_p25">The steps are the same for nonlinear inequalities with two variables. Graph the boundary first and then test a point to determine which region contains the solutions.</p>
<div class="callout block" id="fwk-redden-ch02_s07_s02_n04">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch02_s07_s02_p26">Graph the solution set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1923" display="inline"><mrow><mi>y</mi><mo><</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p27">The boundary is a basic parabola shifted 2 units to the left and 1 unit down. Begin by drawing a dashed parabolic boundary because of the strict inequality.</p>
<div class="informalfigure large">
<img src="section_05/bab33695ff768523151d53025c4d8240.png">
</div>
<p class="para" id="fwk-redden-ch02_s07_s02_p29">Next, test a point.</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p30"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><strong class="emphasis bold">Test point</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1924" display="inline"><mrow><mi>y</mi><mo><</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">(0, 0)</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1925" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>0</mn><mo><</mo><msup><mrow><mo>(</mo><mrow><mn>0</mn><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo><</mo><mn>4</mn><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo><</mo><mn>3</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><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_s07_s02_p31">In this case, shade the region that contains the test point <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1926" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s07_s02_p32">Answer: </p>
<div class="informalfigure large">
<img src="section_05/a98dc80c5e9cc5e75b35a7d89077f6f5.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch02_s07_s02_n05">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch02_s07_s02_p33">Graph the solution set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1927" display="inline"><mrow><mi>y</mi><mo>≥</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p34">The boundary is a basic parabola shifted 3 units up. It is graphed using a solid curve because of the inclusive inequality.</p>
<div class="informalfigure large">
<img src="section_05/90af2dbcef6cf415315b67bd5e8d4501.png">
</div>
<p class="para" id="fwk-redden-ch02_s07_s02_p36">Next, test a point.</p>
<p class="para" id="fwk-redden-ch02_s07_s02_p37"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><strong class="emphasis bold">Test point</strong></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1928" display="inline"><mrow><mi>y</mi><mo>≥</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">(0, 0)</p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1929" display="inline"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mn>0</mn><mo>≥</mo><msup><mn>0</mn><mn>2</mn></msup><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>≥</mo><mn>3</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mtext>✗</mtext></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch02_s07_s02_p38">In this case, shade the region that does not contain the test point <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1930" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s07_s02_p39">Answer: </p>
<div class="informalfigure large">
<img src="section_05/5eea9b3abe173e98fabd5ac780b0d129.png">
</div>
</div>
<div class="callout block" id="fwk-redden-ch02_s07_s02_n05a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch02_s07_s02_p40"><strong class="emphasis bold">Try this!</strong> Graph the solution set <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1931" display="inline"><mrow><mi>y</mi><mo><</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch02_s07_s02_p41">Answer: </p>
<div class="informalfigure large">
<img src="section_05/bb08ba367ef8a8087bde192b416f6e05.png">
</div>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/BVbX_unvcOs" condition="http://img.youtube.com/vi/BVbX_unvcOs/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/BVbX_unvcOs" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<div class="key_takeaways editable block" id="fwk-redden-ch02_s07_s02_n06">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch02_s07_s02_l02" mark="bullet">
<li>Linear inequalities with two variables have infinitely many ordered pair solutions, which can be graphed by shading in the appropriate half of a rectangular coordinate plane.</li>
<li>To graph the solution set of an inequality with two variables, first graph the boundary with a dashed or solid line depending on the inequality. If given a strict inequality, use a dashed line for the boundary. If given an inclusive inequality, use a solid line. Next, choose a test point not on the boundary. If the test point solves the inequality, then shade the region that contains it; otherwise, shade the opposite side.</li>
<li>Check your answer by testing points in and out of the shading region to verify that they solve the inequality or not.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch02_s07_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s07_qs01_qd01">
<h3 class="title">Part A: Solutions to Inequalities with Two Variables</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s07_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch02_s07_qs01_p01"><strong class="emphasis bold">Is the ordered pair a solution to the given inequality?</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa01">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1932" display="inline"><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>></mo><mo>−</mo><mn>2</mn></mrow><mtext>; </mtext><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa02">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1933" display="inline"><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo><</mo><mo>−</mo><mn>8</mn></mrow><mtext>; </mtext><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mo>,</mo><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa03">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1934" display="block"><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>15</mn><mi>y</mi><mo>≥</mo><mo>−</mo><mn>1</mn></mrow><mtext>; </mtext><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><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_s07_s02_qs01_qa04">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1935" display="block"><mrow><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>≥</mo><mn>2</mn></mrow><mtext>; </mtext><mrow><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa05">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1936" display="block"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>y</mi><mo><</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mtext>; </mtext><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa06">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1937" display="block"><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>y</mi><mo>></mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mtext>; </mtext><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1938" display="inline"><mrow><mi>y</mi><mo>≤</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1939" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1940" display="inline"><mrow><mi>y</mi><mo>≥</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1941" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1942" display="inline"><mrow><mi>y</mi><mo>≥</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1943" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1944" display="inline"><mrow><mi>y</mi><mo>≤</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>3</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1945" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1946" display="inline"><mrow><mi>y</mi><mo>></mo><mn>3</mn><mo>−</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1947" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn><mo>,</mo><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1948" display="inline"><mrow><mi>y</mi><mo><</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>−</mo><mn>8</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1949" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>,</mo><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p26"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1950" display="inline"><mrow><mi>y</mi><mo>></mo><mrow><mo>|</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>3</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1951" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1952" display="inline"><mrow><mi>y</mi><mo><</mo><mrow><mo>|</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch02_m1953" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s07_qs01_qd02">
<h3 class="title">Part B: Graphing Solutions to Inequalities with Two Variables.</h3>
<ol class="qandadiv" id="fwk-redden-ch02_s07_qs01_qd02_qd01" start="15">
<p class="para" id="fwk-redden-ch02_s07_qs01_p30"><strong class="emphasis bold">Graph the solution set.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p31"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1954" display="inline"><mrow><mi>y</mi><mo><</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p33"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1955" display="inline"><mrow><mi>y</mi><mo>></mo><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p35"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1956" display="inline"><mrow><mi>y</mi><mo>≥</mo><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p37"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1957" display="inline"><mrow><mi>y</mi><mo>≤</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p39"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1958" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>≤</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p41"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1959" display="inline"><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>≤</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p43"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1960" display="inline"><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>></mo><mn>30</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p45"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1961" display="inline"><mrow><mn>8</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo><</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p47"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1962" display="inline"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo><</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p49"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1963" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1964" display="inline"><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p53"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1965" display="inline"><mrow><mi>x</mi><mo>−</mo><mi>y</mi><mo>≥</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p55"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1966" display="inline"><mrow><mi>y</mi><mo>≤</mo><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p57"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1967" display="inline"><mrow><mi>y</mi><mo>></mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p59"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1968" display="inline"><mrow><mi>x</mi><mo><</mo><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p61"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1969" display="inline"><mrow><mi>x</mi><mo>≥</mo><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa31">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1970" display="block"><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac><mi>y</mi><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa32">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1971" display="block"><mrow><mfrac><mn>3</mn><mn>8</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>≥</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa33">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1972" display="block"><mrow><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>y</mi><mo><</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa34">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1973" display="block"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>9</mn></mfrac><mi>y</mi><mo>></mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa35">
<div class="question">
<p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1974" display="inline"><mrow><mn>5</mn><mi>x</mi><mo>≤</mo><mo>−</mo><mn>4</mn><mi>y</mi><mo>−</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1975" display="inline"><mrow><mo>−</mo><mn>4</mn><mi>x</mi><mo>≤</mo><mn>12</mn><mo>−</mo><mn>3</mn><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1976" display="inline"><mrow><mn>4</mn><mi>y</mi><mo>+</mo><mn>2</mn><mo><</mo><mn>3</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1977" display="inline"><mrow><mn>8</mn><mi>x</mi><mo><</mo><mn>9</mn><mo>−</mo><mn>6</mn><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1978" display="inline"><mrow><mn>5</mn><mo>≥</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>15</mn><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa40">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch02_m1979" display="block"><mrow><mn>2</mn><mi>x</mi><mo>≥</mo><mn>6</mn><mo>−</mo><mn>9</mn><mi>y</mi></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p83">Write an inequality that describes all points in the upper half-plane above the <em class="emphasis">x</em>-axis.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p85">Write an inequality that describes all points in the lower half-plane below the <em class="emphasis">x</em>-axis.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p87">Write an inequality that describes all points in the half-plane left of the <em class="emphasis">y</em>-axis.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p89">Write an inequality that describes all points in the half-plane right of the <em class="emphasis">y</em>-axis.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p91">Write an inequality that describes all ordered pairs whose <em class="emphasis">y</em>-coordinate is at least <em class="emphasis">k</em> units.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p93">Write an inequality that describes all ordered pairs whose <em class="emphasis">x</em>-coordinate is at most <em class="emphasis">k</em> units.</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch02_s07_qs01_qd02_qd02" start="47">
<p class="para" id="fwk-redden-ch02_s07_qs01_p95"><strong class="emphasis bold">Graph the solution set.</strong></p>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p96"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1986" display="inline"><mrow><mi>y</mi><mo>≤</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p98"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1987" display="inline"><mrow><mi>y</mi><mo>></mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p100"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1988" display="inline"><mrow><mi>y</mi><mo>≤</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p102"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1989" display="inline"><mrow><mi>y</mi><mo>≥</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p104"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1990" display="inline"><mrow><mi>y</mi><mo>></mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p106"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1991" display="inline"><mrow><mi>y</mi><mo>></mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p108"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1992" display="inline"><mrow><mi>y</mi><mo>≤</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1993" display="inline"><mrow><mi>y</mi><mo>≤</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p112"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1994" display="inline"><mrow><mi>y</mi><mo><</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p114"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1995" display="inline"><mrow><mi>y</mi><mo>></mo><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1996" display="inline"><mrow><mi>y</mi><mo>≥</mo><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_s07_s02_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1997" display="inline"><mrow><mi>y</mi><mo><</mo><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_s07_s02_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1998" display="inline"><mrow><mi>y</mi><mo><</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m1999" display="inline"><mrow><mi>y</mi><mo>≤</mo><mrow><mo>|</mo><mrow><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_s07_s02_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2000" display="inline"><mrow><mi>y</mi><mo>></mo><mo>−</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>|</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p126"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2001" display="inline"><mrow><mi>y</mi><mo>≤</mo><mo>−</mo><mrow><mo>|</mo><mrow><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_s07_s02_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2002" display="inline"><mrow><mi>y</mi><mo>≥</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p130"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2003" display="inline"><mrow><mi>y</mi><mo>≥</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p132"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2004" display="inline"><mrow><mi>y</mi><mo><</mo><mo>−</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>|</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2005" display="inline"><mrow><mi>y</mi><mo>></mo><mo>−</mo><mrow><mo>|</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>|</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p136"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2006" display="inline"><mrow><mi>y</mi><mo>></mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p138"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2007" display="inline"><mrow><mi>y</mi><mo>≤</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p140"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2008" display="inline"><mrow><mi>y</mi><mo>≤</mo><msqrt><mi>x</mi></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p142"><span class="inlineequation"><math xml:id="fwk-redden-ch02_m2009" display="inline"><mrow><mi>y</mi><mo>></mo><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p144">A rectangular pen is to be constructed with at most 200 feet of fencing. Write a linear inequality in terms of the length <em class="emphasis">l</em> and the width <em class="emphasis">w</em>. Sketch the graph of all possible solutions to this problem.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch02_s07_qs01_p146">A company sells one product for $8 and another for $12. How many of each product must be sold so that revenues are at least $2,400? Let <em class="emphasis">x</em> represent the number of products sold at $8 and let <em class="emphasis">y</em> represent the number of products sold at $12. Write a linear inequality in terms of <em class="emphasis">x</em> and <em class="emphasis">y</em> and sketch the graph of all possible solutions.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch02_s07_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s07_qs01_p03_ans">No</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_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_s07_s02_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s07_qs01_p07_ans">Yes</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s07_qs01_p11_ans">Yes</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s07_qs01_p15_ans">No</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s07_qs01_p19_ans">No</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s07_qs01_p23_ans">No</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch02_s07_qs01_p27_ans">Yes</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
<ol class="qandadiv" start="15">
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa15_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/3b58bc925724eb2e654aad3552cae177.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa17_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/41979c37e8432038bb301e1da496cca9.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa19_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/54495e3682309827df22ba552a9f43f5.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa21_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/8b08957be424586a3b83cb93bc7573f5.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa23_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/074ba8ea7ffb486e143f347576de7ee9.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa24_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa25_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/807140900d7e77f51c38090c8234954d.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa26_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa27_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/2d3e09b2a3300febc1c26a19936b9299.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa28_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa29_ans">
<div class="answer">
<div class="informalfigure large">
<img src="section_05/8f88ccde13297ab57abbc1a711f4e56e.png">
</div>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch02_s07_s02_qs01_qa30_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>