1ce58775-5910-4d22-ad95-005021c7d142.html

<h3>Activity (10 minutes) </h3>
<p>In this activity, students continue to work with inequalities in two variables in context. They write an inequality that represents the constraints in a situation, graph its solutions, and interpret points in the solution region.</p>
<p>Earlier, students saw that some points in the solution region might satisfy an inequality (mathematically) but might not work in the given situation because other considerations were at play. Here, they see another reason that some values that satisfy an inequality might be unfeasible in the situation-namely, that the pair of values must be whole numbers.</p>
<p>As students work, look for those who plot discrete points to represent the solution region and those who shade a part of the plane.</p>
<h3>Launch</h3>
<p>Arrange (or keep) students in groups of 2. Give students a few minutes of quiet time to work on the first three questions, and then time to discuss their responses before they continue with the last two questions.</p>
<p>Emphasize that by analyzing the different combinations of necklaces and bracelets that can be made and sold, the vendor can prepare each week by making the right amount of each piece of jewelry. This information can also be used  to try to ensure that the vendor makes $100 in profit every week.</p>
<br>
<div class="os-raise-extrasupport">
  <div class="os-raise-extrasupport-header">
    <p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
    <p class="os-raise-extrasupport-name">Representation: Internalize Comprehension </p>
  </div>
  <div class="os-raise-extrasupport-body">
      <p>To support working memory, provide students with sticky notes or mini whiteboards. Some students may need extra time to explore possible combinations for multiples of 9 and 5. Encourage them to model these combinations by drawing pictures or listing multiples of each number using the whiteboard or sticky notes.</p>
      <p class="os-raise-text-italicize">Supports accessibility for: Memory; Organization</p>
</div>
</div>
<br>
<h4>Student Activity</h4>

<p>For questions 1–7, use the following scenario:</p>
<p>A vendor at the Saturday Market makes $9 profit on each necklace she sells and $5 profit on each bracelet. Find a combination of necklaces and bracelets that she could sell to make the profit listed in each question.</p>
<ol class="os-raise-noindent">
  <li>
    Exactly $100 profit.
  </li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here is a sample. 10 necklaces and 2 bracelets.</p>
<ol class="os-raise-noindent" start="2">
  <li>
    More than $100 profit.
  </li>
</ol>
<p><strong>Answer:</strong> Compare your answer: Your answer may vary, but here is a sample. 0 necklaces and 21 bracelets.</p>
<ol class="os-raise-noindent" start="3">
  <li>
  Write an equation whose solution is the combination of necklaces \((n)\) and bracelets \((b)\) she could sell and make exactly $100 profit. </li>
</ol>
<p><strong>Answer:</strong> Compare your answer: Your answer may vary, but here is a sample.</p>
<p>9n+5b=100, where n represents the number of necklaces sold and b represents the number of bracelets.</p>
<ol class="os-raise-noindent" start="4">
  <li>
  Write an inequality whose solutions are the combinations of necklaces \((n)\) and bracelets \((b)\) she could sell and make more than $100 profit. </li>
</ol>
<p><strong>Answer:</strong> Compare your answer: Your answer may vary, but here is a sample. \(9n+5b&gt;100\) where n represents the number of necklaces sold and b represents the number of bracelets.</p>
<ol class="os-raise-noindent" start="5">
  <li>Use the graphing tool or technology outside the course.Graph the linear inequality \(y&gt;\frac52x−4\) using the Desmos tool below.  Students were provided access to Desmos.
  </li>
</ol>
<p><strong>Answer:</strong> Compare your answer: Your answer may vary, but here is a sample.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/852888c4f18c108082752cb77bd5157597842ff7" alt="Graph of an inequality. Number of bracelets. Number of necklaces."/></p>
<ol class="os-raise-noindent" start="6">
  <li>
  Is \((3,18.6) \)a solution to the inequality? </li>
</ol>
<ul>
  <li>
    Yes
  </li>
  <li>
    No
  </li>
  <li>
    Both
  </li>
</ul>
<p><strong>Answer:</strong> Both</p>
<ol class="os-raise-noindent" start="7">
  <li>
    Explain your reasoning.
  </li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here is a sample.</p>
<p>The answer is both because:</p>
<ul>
  <li>
  Yes, it is a solution. The point \((3,18.6)\) is in the shaded region. </li>
  <li>
    No, it is not a solution. The vendor can't sell 18.6 bracelets.
  </li>
</ul>
<br>
<h3>Student Facing Extension</h3>

<h5>Are you ready for more?</h5>
<ol class="os-raise-noindent" >
  <li> Write an inequality using two variables, \(x\) and \(y\), where the solution would be represented by shading the entire coordinate plane. </li>
</ol>
<p><strong>Answer:</strong></p>
<p>Compare your answer:</p>
<p>Your answer may vary, but here is a sample.</p>
<p>\( x^2+y^2\geq 0 \)</p>
<ol class="os-raise-noindent"  start="2">
  <li> Write an inequality using two variables, \(x\) and \(y\), where the solution would be represented by not shading any of the coordinate plane. </li>
</ol>
<p><strong>Answer:</strong></p>
<p>Compare your answer:</p>
<p>Your answer may vary, but here is a sample.</p>
<p>\( x^2+y^2&lt; 0 \)</p>

<p>Students may first come up with good reasons why this can't be done (with linear inequalities). They may also respond with \( x &lt; \text{infinity} \) or something similar. Both of these tacks deserve praise, but tell students that this problem really is possible using real numbers. Students may ultimately need the hint that they will need to leave the realm of linear inequalities to answer these questions.</p>
<h3>Anticipated Misconceptions</h3>
<p>Some students may have trouble graphing the line that delineates the solution region from non-solution region because they are used to solving for the variable \( y \), but here the variables are \( b \) and \( n \). Ask these students to decide which variable to solve for based on the graph that has been set up. Ask them to notice which quantity is represented by the vertical axis.</p>
<h3>Activity Synthesis</h3>
<p>If some students plotted discrete points and some shaded the region, choose one of each and display these for all to see. Ask students why each one might be an appropriate representation of the solutions to the inequality.</p>
<p>Highlight that the discrete points represent the situation more accurately, because it is impossible to sell a fraction of a bracelet or 2.75 necklaces. It is, however, tedious to plot a bunch of points to show the solution region. It is much easier to shade the entire region, but with the understanding that, in this situation, only whole-number values make sense as solutions.</p>
<h4>2.12.3: Self Check </h4>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>An artist is designing a
  sculpture to be displayed hanging in the lobby of a museum. She is using \( x \) wooden rods and \( y \) metal pipes.</p>
<ul>
  <li>Each wooden rod, \( x \), weighs 4 pounds.</li>
  <li>Each metal pipe, \( y \), weighs 7 pounds. </li>
  <li>The total weight of these materials cannot be more than
    510 pounds.</li>
</ul>
<p>Which of these ordered pairs represents an amount of materials that meets the artist’s constraints?</p>
<table class="os-raise-textheavytable">
  <thead>
    <tr>
      <th scope="col">Answers</th>
      <th scope="col">Feedback</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>\( (60, 50) \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: <br>
        The weight must be less or equal to 510 pounds. <br>
        Does this ordered pair meet the weight constraint? <br>
        The correct answer is \( (70, 20) \).</td>
    </tr>
    <tr>
      <td>\( (20, 70) \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: <br>
        Notice \( x \) represents the weight of the wooden rod and \( y \) represents the weight of the metal pipe. <br>
        The total must be less than or equal to 510 pounds. <br>
        Does this ordered pair meet the weight constraint? <br>
        The correct answer is \( (70, 20) \).</td>
    </tr>
    <tr>
      <td>\( (50, 60) \)</td>
      <td>Incorrect. Let&rsquo;s try again a different way: <br>
        The weight must be less than or equal to 510 pounds. <br>
        Does this ordered pair meet the maximum weight constraint? <br>
        The correct answer is \( (70, 20) \).</td>
    </tr>
    <tr>
      <td>\( (70, 20) \)</td>
      <td>That&rsquo;s correct! <br>
        Check yourself: The weight of 70 wooden rods, 70(4), plus the weight of 20 metal pipes, 20(7), equals 420 pounds, which is less than 510 pounds.  <br></td>
    </tr>
  </tbody>
</table>
<br>
<h3>2.12.3: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on their experience with the self check. Students will not automatically have access to this content, so you may wish to share it with those who could benefit from it.</em></p>
<h4>Graph Linear Inequalities in Two Variables</h4>
<p>Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.</p>
<p><strong>Example </strong><br>
  How to Graph a Linear Equation in Two Variables</p>
<p>Graph the linear inequality \( y &ge; \frac{3}{4} x &minus; 2 \).</p>
<p><b>Step 1 - </b> Identify and graph the boundary line. </p>
<p>  If the inequality is \(\leq\) or \(\geq\)
  boundary line is solid. </p>
<p>If the inequality is &lt; or &gt;, the boundary
line is dashed. </p>
<p>Replace the
inequality sign with an equal sign to find the boundary line</p>
<p>\(y=\frac34x-2.\)</p>
<p>The
  inequality sign is \(\geq\), so we draw a solid line.</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="205" role="presentation" src="https://k12.openstax.org/contents/raise/resources/73a02b5f0522899713366bc7f2696871112ac313" width="200"></p>
<p><b>Step 2 -</b> Test a point that is not on the boundary line. Is it a solution of the
inequality?</p>
<p>We&rsquo;ll test
\((0,0)\).</p>
<p>Is it a solution
  of the inequality?</p>
<p>At \((0, 0)\),
is \(y \geq \frac34x-2.\)?</p>
<p>\(0\overset?\geq\frac34(0)-2\)</p>
<p>\(0\geq-2\)</p>
<p>So, \((90,0)\)
  is a solution.</p>
<p><b>Step 3 -</b> 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 to \(y\geq\frac34x-2.\)
So we shade in that side.</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="202" role="presentation" src="https://k12.openstax.org/contents/raise/resources/3fb1b437b5f627f97dd1a57c0ef8f9697fa730b6" width="200"></p>
<p>All points in
  the shaded region and on the boundary line represent the solutions to \(y\geq\frac34x-2.\)</p>
<br>
<h4>Try It: Graph Linear Inequalities in Two Variables</h4>
<ol class="os-raise-noindent">
<li>Graph the linear inequality \(y\;&gt;\;\frac52x\;-\;4\). Use the graphing tool or technology outside the course. Graph the inequality. Students were provided access to Desmos.</li> 
</ol>
<p><strong>Answer: </strong></p>
<p>All points in the shaded region represent the solutions to \( y &gt; \frac{5}{2}x &minus; 4 \).</p>
<img alt class="img-fluid atto_image_button_text-bottom" height="308" role="presentation" src="https://k12.openstax.org/contents/raise/resources/3a2c4243b36188d45c7f1556d44666cb5fbeef8f" width="300"><br>