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<h4>Activity (10 minutes)</h4>
<p>Previously, students have learned that any point that is in the overlapping solution regions of the graphs of two inequalities is a solution to the system formed by those inequalities. In this activity, students take a closer look at whether points that are on the boundary lines are solutions to the system.</p>
<h4>Launch</h4>
<p>Arrange students in groups of 2. Display the system of inequalities and the graphs for all to see. </p>
<p>Give students a minute of quiet time to think about which region represents the solutions to each inequality and be prepared to explain how they know. Then, give students another minute to discuss their thinking with a partner. Follow with a class discussion.</p>
<p>Students are likely to identify the inequality that each graph represents by considering the equation of the boundary line. They may relate the solidly shaded region to \(x<y\) because the dashed line is the graph of \(x=y\). Or they may relate the hashed region to the solutions of \(y\leq-\;2x\;-\;6\) because the boundary line has a negative slope and it intersects the \(y\)-axis at \((0,-6)\).</p>
<p>Other students may test some coordinate pairs in each region and see if they make an inequality true. For example, they may say that all points above the graph of \(x=y\) have an \(x\)-value that is less than the \(y\)-value. </p>
<p>If these strategies for connecting the algebraic and graphical representations are not mentioned by students, bring them up.</p>
<p>Tell students that they will now think about whether certain points on the coordinate plane are solutions to the system.</p>
<h4>Student Activity</h4>
<p>Here are the graphs of the inequalities in this system:<br>
</p>
<p>\(\left\{\begin{array}{l}x<y\\y\leq-2x\;-\;6\end{array}\right.\)</p>
<p><img alt="A graph of two intersecting inequalities on a coordinate plane." src="https://k12.openstax.org/contents/raise/resources/d4029ba11276782f6550954c13749b37655b6b85"><br>
</p>
<p>Decide whether each point is a solution to the system. Be prepared to explain how you know.</p>
<ol class="os-raise-noindent">
<li>\((3,-5)\)
<p><strong>Answer: </strong>No. \((3,-5)\) is in the shaded region containing the solutions to one of the inequalities but not in the region where the shading overlaps.</p>
</li>
<li>\((0,5)\)
<p><strong>Answer: </strong>Yes. \((0,5)\) is in the region where the solutions to the two inequalities overlap.</p>
</li>
<li>\((-6,6)\)
<p><strong>Answer: </strong>Yes. \((-6,6)\) is on the solid boundary of \(y\geq-2x-6\) because \(0=-2(-3)-6\), and it is within the shaded region representing solutions to \(x\).</p>
</li>
<li>\((3,3)\)
<p><strong>Answer: </strong>No. While \((3,3)\) is clearly a solution to \(y\geq-2x-6\), it is on the dashed boundary of \(x < y\). The statement \(3<3\) is not true.</p>
</li>
<li>\((-2,-2)\)
<p><strong>Answer: </strong>No, because \(-2<-2\) is not true. Even though it is on the solid boundary of one, it is on the dashed boundary of the other.</p>
</li>
</ol>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<h5>Triangle as a Set of Solutions</h5>
<p>Find a system of inequalities with this triangle as its set of solutions.<br>
</p>
<p><img alt="Triangle on a coordinate grid. X and Y axis from negative 5 to 4. Vertices of the triangle are negative 2 comma 4, 2 comma 3, and 3 comma negative 3." src="https://k12.openstax.org/contents/raise/resources/58774892f20e984e35f56f0128ea5d45a7a93cb1"><br>
</p>
<p><strong>Answer: </strong>\(x+y\geq2\), \(x+4y\leq14\), \(4x+y\leq11\) or equivalent</p>
<h4> Video: Solving Systems of Inequalities</h4>
<p>Watch the following video to learn more </p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/a22e01b36d52e5c07062af988ec55db7cec27d65">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/0bd1399a3ba2b3f85aa262c8eec52b10c28a0167" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/a22e01b36d52e5c07062af988ec55db7cec27d65
</video></div>
</div>
<br>
<h4>Activity Synthesis</h4>
<p>Focus the discussion on the points on the boundary lines and how students determined if they are or are not solutions to the system.</p>
<p>Highlight explanations that state that a solution to a system of linear inequalities must be a solution to every inequality in the system. If a point on the boundary line is not included in the solution set of one inequality (so the graph is a dashed line), then it is also not included in the solution set of the system.</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: Access for Perception</p>
</div>
<div class="os-raise-extrasupport-body">
<p>
Prior to independent work, engage the whole class in developing a set of directions that displays criteria for checking if points are a solution. This can be written as a flow chart or as a list. Recommend students start with the step of plotting the point on their graph. Support them in articulating criteria that address evaluating shaded regions and boundary lines. Check for understanding by inviting students to rephrase directions in their own words. Consider keeping the display of directions visible throughout the activity. </p>
<p class="os-raise-text-italicize">Supports accessibility for: Language; Memory </p>
</div>
</div>
<br>
<h4>2.15.2 Self Check </h4>
<p> </p>
<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>
<!--SELF CHECK QUESTION GOES BEFORE THE Table -->
<p class="os-raise-text-bold">QUESTION:</p>
<p>Here is the graph of this system of inequalities. Which of the following ordered pairs falls within the solution set?</p>
<p><img alt="y is less than or equal to negative three-halves x plus thirty-four fourths and y is greater than two-sevenths x plus twenty-" class="img-fluid atto_image_button_text-bottom" height="299" src="https://k12.openstax.org/contents/raise/resources/f20e623c0948dffcb657171e551e097f14fe4e09" width="300"><br>
</p>
<p> \(6x\;+\;4y\;\leq\;34\)</p>
<p> \(-2x\;+\;7y\;>\;22\)</p>
<!--SELF CHECK table-->
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\((3, 4)\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This ordered pair falls on both boundary lines, but it only satisfies the first inequality. The answer is \((2, 4)\).</p>
</td>
</tr>
<tr>
<td>
<p>\((5, 1)\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This ordered pair falls on a boundary line and satisfies the first inequality, but it does not satisfy the second inequality. The answer is \((2, 4)\).</p>
</td>
</tr>
<tr>
<td>
<p>\((10, 6)\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This ordered pair falls on a dashed boundary line, and it does not satisfy either inequality. The answer is \((2, 4)\).</p>
</td>
</tr>
<tr>
<td>
<p>\((2, 4)\)</p>
</td>
<td>
<p>That’s correct! Check yourself: This ordered pair is a solution to both inequalities and falls within the solution region on the graph. This is the correct answer.</p>
</td>
</tr>
</tbody>
</table>
<br>
<!--END SELF CHECK INTRO BEFORE Tables -->
<strong><br>
</strong>
<h4>2.15.2: Additional Resources</h4>
<p><strong><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></strong></p>
<h4>Determine Solutions of Systems of Inequalities on Boundary Lines</h4>
<p>A solution to a system of inequalities only exists on a boundary line if the line is solid, not dashed.</p>
<p><strong>Example</strong></p>
<p>Solve the system by graphing:</p>
<p>\(\left\{\begin{array}{l}x-y>3\\y<-\frac15x+4.\end{array}\right.\)</p>
<p> </p>
<p><strong>Answer: </strong></p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col"></th>
<th scope="col"></th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(\left\{\begin{array}{l}x-y>3\\y<-\frac15x+4\end{array}\right.\)</p>
</td>
<td></td>
</tr>
<tr>
<td>
<p>Graph \(x − y > 3\), by graphing \(x − y = 3\) and testing a point.</p>
<br>
<p>The intercepts are \(x = 3\) and \(y = −3\), and the boundary line will be dashed.</p>
<br>
<p>Test \((0, 0)\), which makes the inequality false, so shade (red) the side that does not contain \((0, 0)\).</p>
</td>
<td>
<p><img alt="." class="img-fluid atto_image_button_text-bottom" height="303" src="https://k12.openstax.org/contents/raise/resources/05516f4f7a5b0aebfceb8022e98e45e52e709f9b" width="300"></p>
</td>
</tr>
<tr>
<td>
<p>Graph \(y<-\frac15x+4\) by graphing \(y=-\frac15x+4\) using the slope \(m=-\frac15\) and \(y\)-intercept \(b = 4\). The boundary line will be dashed.</p>
<br>
<p>Test \((0, 0)\), which makes the inequality true, so shade (blue) the side that contains \((0, 0)\).</p>
<br>
<p>Choose a test point in the solution and verify that it is a solution to both inequalities.</p>
</td>
<td>
<p><img alt="." class="img-fluid atto_image_button_text-bottom" height="303" src="https://k12.openstax.org/contents/raise/resources/06c3492d8a65550626639bb01d5f4c7d4ad3f240" width="300"></p>
</td>
</tr>
</tbody>
</table>
<br>
<p>The point of intersection of the two lines is not included because both boundary lines were dashed. The solution is the area shaded twice—which appears as the darkest shaded region.</p>
<p>Notice that the point \((0,-3)\) is NOT a solution because it falls on the red dashed line.</p>
<p>The point \((4, -2)\) IS a solution because it is included in the solutions to both inequalities.</p>
<h4>Try It: Determine Solutions of Systems of Inequalities on Boundary Lines</h4>
<p>Which of the following are solutions of the graph below?<br>
</p>
<ul>
<li>\((2,1)\)</li>
<li>\((2,3)\)</li>
<li>\((5,0)\)</li>
<li>\((3,-1)\)</li>
<li>\((-1,6)\)</li>
<li>\((1, 6)\)</li>
</ul>
<p><strong><img alt="y is less than or equal to 2x-1 and y is less than 5-x" class="atto_image_button_text-bottom" height="304" src="https://k12.openstax.org/contents/raise/resources/e5523716e1437ccb5db039a3424e1e7fc31aecd2" width="300"></strong><br>
</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
<h5>Answer</h5>
<p>Compare your answer:</p>
<p>Remember that when a coordinate is on a boundary line, it must be solid if it is a solution. The coordinate must also be in the overlapping shaded regions to be a solution to the entire system.</p>
<p>The answers are \((2, 1)\) and \((3, -1)\)</p>