a188eee7-ce2b-40e7-a9fe-6d1571061656.html

<h4 class="os-raise-text-bold os-raise-text-italicize">Connecting Representations of Inequalities </h4>
<p>Tables can be used to help you to write and solve inequalities in two variables.</p>
<p><strong>Part 1:</strong> Writing Inequalities in Two Variables Given A Set of Data in a Table</p>
<p>Write a linear inequality in two variables based on the given data:</p>
<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">\(x\)</th>
      <th scope="col">\(y\)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>-4</td>
      <td>-3</td>
    </tr>
    <tr>
      <td>-2</td>
      <td>-2</td>
    </tr>
    <tr>
      <td>0</td>
      <td>-1</td>
    </tr>
    <tr>
      <td>2</td>
      <td>0</td>
    </tr>
  </tbody>
</table>
<br>
<p>Remember that for an inequality the points could be on the line, above the the line, or below the line.  The type of inequality will differ based on these conditions.</p>
<table class="os-raise-horizontaltable">
  <thead>
  </thead>
  <tbody>
    <tr>
      <th scope="row">On the line</th>
      <td> \(\leq\) or \(\geq\) </td>
    </tr>
    <tr>
      <th scope="row">Above the line</th>
      <td>\(\gt\)</td>
    </tr>
    <tr>
      <th scope="row">Below the line</th>
      <td>\(\lt\)</td>
    </tr>
  </tbody>
</table>
<br>
<p>If you are given a set of possible points like the table above, there could be multiple lines that the data points could represent.  Let&rsquo;s figure out a few possibilities for this given set.</p>
<p><strong>Step 1 -</strong> Graph the points on a coordinate plane.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/b68cb27bb14fbe6ed09f6a4b71d795de41b1ebf7"/></p>
<p><strong>Step 2 -</strong> Determine possible lines.</p>
<p>Remember that the dots can be on the line, but they all have to be either above or below to create an inequality.  If the line has points on both sides, then the line isn&rsquo;t a possibility.</p>
<p>Let&rsquo;s try a few possibilities. We can start with the equation \(y=x\) to see if the points are above the line or below the line.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/60b97dc16d33f7b3edfcf5efbb96256bfadd96a7"/></p>
<p>The equation \(y=x\) cannot be a possibility because there are points above and below the line.</p>
<p><strong>Step 3 -</strong> Adjust the slope or \(y\)-intercept</p>
<p>We can change the slope and the y-intercept to see how we can shift the line to be able to capture all the points. Let&rsquo;s move the line to below the \((3, -5)\) and \((-4, -6)\).  We can leave the slope as 1 and change the y-intercept to -7.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/e710802ff3c9d7b912f10604c9c314a8719c3a60" width="300"/></p>
<p>There is still one point below the line.  We could decrease the slope of the line to \(m=13\). </p>
<p><img src="https://k12.openstax.org/contents/raise/resources/cfcb49dc42e98e04d4fd1a580dfed336c09536d0" width="300"/></p>
<p>Now all of the points are above the line.  This means that the value of \(y\) is always greater than the \(x\) value.</p>
<p>So, one of the possible solutions could be \(y&gt; 13x-7\).</p>
<br>
<p class="os-raise-text-bold"><strong>Part 2:</strong> Using Tables to Solve Inequalities in Two Variables</p>
<p>Graph the linear inequality \(y &lt; \frac{1}{2}x − 1\) using a table.</p>
<p><strong>Step 1</strong> - Create a table of values using the related equation \(y = \frac{1}{2}x − 1\) to identify points on the boundary line of the inequality.</p>
<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">\(x\)</th>
      <th scope="col">\(y\)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>−4</td>
      <td>−3</td>
    </tr>
    <tr>
      <td>−2</td>
      <td>−2</td>
    </tr>
    <tr>
      <td>0</td>
      <td>-1</td>
    </tr>
    <tr>
      <td>2</td>
      <td>0</td>
    </tr>
  </tbody>
</table>
<br>
<p><strong>Step 2</strong> - Plot the points and sketch the boundary line.</p>
<p>If the inequality is \(\leq\) or \(\geq\), the boundary line is solid.</p>
<p>If the inequality is \(&lt;\) or \(&gt;\), the boundary line is dashed.</p>
<p>The inequality sign is \(&lt;\), so we draw a dashed line.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/58a574ba91345f694a3db3762d2bc5ee16c86792" width="300"/></p>
<p><strong>Step 3</strong> - Test a point that is not on the boundary line. Is it a solution of the inequality?</p>
<p>Test \((0, 0)\) as it is not on the boundary line of the inequality.</p>
<p>\(y \overset?&lt; \frac{1}{2} x − 1\)</p>
<p>\((0) &lt; \frac{1}{2}(0) − 1\)</p>
<p>\(0 &lt; − 1\) is False.</p>
<p>So, \((0, 0)\) is not a solution.</p>
<p><strong>Step 4</strong> - Shade in one side of the boundary line.</p>
<p>If the test point is a solution, shade in the side that includes the point.</p>
<p>If the test point is not a solution, shade in the opposite side.</p>
<p>The test point \((0, 0)\) is not a solution, so shade in the side of the boundary line that does not include \((0, 0)\).</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/35b11285cf8c33fdf425e2dca7b9b7bf7c9afeca" width="300"/></p>
<p>All points in the shaded region and not on the boundary line represent the solutions to \(y &lt; \frac{1}{2} x − 1\).</p>
<br>
<h4>Try It: Graph Linear Inequalities in Two Variables</h4>
<p><br>
  <!--Q#-->
</p>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1a" data-schema-version="1.0">
  <div class="os-raise-ib-cta-content">
    <ol class="os-raise-noindent">
  <li>Is the inequality  \(y&lt; 3x+6\) a solution for the table of values? Explain your reasoning. </li>
</ol>
<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">\(x\)</th>
      <th scope="col">\(y\)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>-4</td>
      <td>-3</td>
    </tr>
    <tr>
      <td>-2</td>
      <td>-2</td>
    </tr>
    <tr>
      <td>0</td>
      <td>-1</td>
    </tr>
    <tr>
      <td>2</td>
      <td>0</td>
    </tr>
  </tbody>
</table>
    </div>
  <div class="os-raise-ib-cta-prompt">
    <p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
  </div>
</div>
<div class="os-raise-ib-content" data-wait-for-event="Reveal1a" data-schema-version="1.0">
  <p>Compare your answer:</p>
    <p>No, the point \((-4, -6)\) sits on the line so it can&rsquo;t be greater than or less than.</p>
</div>
<!--Interaction End -->
<br>
<br>
<!--Q#-->
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
  <div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
  <li>Use a table to graph the inequality \(y  = 2x + 3\).</li>
</ol>
<p>&nbsp;</p>

  </div>
  <div class="os-raise-ib-cta-prompt">
    <p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
  </div>
</div>
<div class="os-raise-ib-content" data-wait-for-event="Reveal1" data-schema-version="1.0">
<p>Compare your answer:</p>
<p>Here&rsquo;s how to solve an inequality using a table:</p>
<p><strong>Step 1</strong> - Create a table of values using the related equation \(y = 2x + 3\) to identify points on the boundary line of the inequality.</p>
<table class="os-raise-skinnytable">
  <thead>
    <tr>
      <th scope="col">x</th>
      <th scope="col">y</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>−4</td>
      <td>−5</td>
    </tr>
    <tr>
      <td>−2</td>
      <td>−1</td>
    </tr>
    <tr>
      <td>0</td>
      <td>3</td>
    </tr>
    <tr>
      <td>2</td>
      <td>7</td>
    </tr>
  </tbody>
</table>
<br>

<p><strong>Step 2</strong> - Plot the points and sketch the boundary line.</p>
<p>If the inequality is \(\leq\) or \(\geq\), the boundary line is solid.</p>
<p>If the inequality is \(&lt;\) or \(&gt;\), the boundary line is dashed.</p>
<p>The inequality sign is , so we draw a solid line.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/341d1376d80f11af33d4dedb611828816f5d0fb9" width="300"/></p>
<p><strong>Step 3</strong> - Test a point that is not on the boundary line. Is it a solution of the inequality?</p>
<p>Test \((0, 0)\) as it is not on the boundary line of the inequality.</p>
<p>\(y  \leq 2x + 3\)</p>
<p>\((0)  \stackrel {?}{\leq} 2(0) + 3\)</p>
<p>\(0  \leq 3\) is True.</p>
<p>So, (0, 0) is a solution.</p>
<p><strong>Step 4</strong> - Shade in one side of the boundary line.</p>
<p>If the test point is a solution, shade in the side that includes the point.</p>
<p>If the test point is not a solution, shade in the opposite side.</p>
<p>The test point \((0, 0)\) is a solution, so shade in the side of the boundary line that includes \((0, 0)\).</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/37fe97b84f8a51f20da54c56d324e29532ed6e40" width="300"/></p>
<p>All points in the shaded region and not on the boundary line represent the solutions to \(y  \leq 2x + 3\).</p>


</div>
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