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<h3>Activity (15 minutes)</h3>
<p>This activity is an extension of the expectations in the TEKS. This activity allows students to visualize an inequality in one variable in another way: by graphing the expressions on each side and comparing the values of the expressions at different values of the variable. Doing so allows them to see the values at which the inequality is satisfied.<br></p>
<p>This activity works best when each student has access to devices that can run the Desmos applet because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the synthesis. (Students can still graph the equations in the activity using the graphing technology available in the classroom.)<br></p>
<h3>Launch</h3>
<p>Consider projecting for all to see the graph of \( y=-3x+7 \). Ask students:</p>
<ul>
<li>From the graph, can you tell what \(x\)-value gives a \(y\)-value of 4? In other words, what \(x\)-value makes the value of \( -3x+7 \) exactly 4? (1)</li>
<li>Which \(x\)-values give a \(y\)-value that is greater than 4? (\(x\)-values less than 1)</li>
<li>Which \(x\)-values give a \(y\)-value that is less than 10? (\(x\)-values greater than \(-1\))</li>
<li>Which \(x\)-values give a \(y\)-value that is greater than \(-5\)? (\(x\)-values less than 4)</li>
</ul>
<p>If students have individual access to Desmos or another tool with a slider function, consider demonstrating how moving the slider for \( x \) in the applet could help them see the answers to these questions more clearly. Otherwise, consider showing the slider during discussion (after students have analyzed the graphs and estimated the values visually).</p>
<h4>Student Activity</h4>
<p>Consider the inequality \({-} \frac12 x + 6 < 4x−3 \). Let’s look at another way to find its solutions.</p>
<ol class="os-raise-noindent">
<li>Use the graphing tool or technology outside the course. Graph \(y=-\frac12 x + 6 \)and \(y=4x−3\) on the same coordinate plane. Students were given access to Desmos. </li>
<p><Strong>Answer: </Strong></p>
<img alt="Graph of 2 intersecting lines." src="https://k12.openstax.org/contents/raise/resources/3de7e6fae22f182297a1b8cbf6f6733973a8005f">
</li>
</ol>
<p>Use your graphs to answer the following questions:</p>
<ol class="os-raise-noindent" start="2">
<li>Find the values of \( -\frac {1}{2}x+6 \) and \( 4x-3 \) when \( x \) is 1.
<p><strong>Answer: </strong>When \( x \) is 1, the value of \( - \frac{1}{2} x+6 \) is \( 5 \frac{1}{2} \),and the value of \( 4x-3 \) is 1.</p>
</li>
<li>What value of \( x \) makes \( -\frac {1}{2}x+6 \) and \( 4x-3 \) equal?
<p><strong>Answer: </strong>When \( x \) is equal to 2, the value of \( - \frac{1}{2} x+6 \) is equal to that of \( 4x-2 \).</p>
</li>
<li>For what values of \( x \) is \( - \frac12 x + 6 \) less than \( 4x−3 \)?
<p><strong>Answer: </strong>When \( x \) is greater than 2, the value of \( - \frac{1}{2} x+6 \) is less than that of \(4x-3\).</p>
</li>
<li>For what values of \( x \) is \( - \frac12 x + 6 \) greater than \( 4x−3 \)?
<p><strong>Answer: </strong> When \( x \) is less than 2, the value of \( - \frac{1}{2} x+6 \) is greater than that of \( 4x-3 \).</p>
</li>
<li>What is the solution to the inequality \( - \frac12 x+ 6 < 4x−3 \)? Be prepared to explain how you know.
<p><strong>Answer: </strong>\( x >2 \)</p>
</li>
</ol>
<h3>Anticipated Misconceptions</h3>
<p>Some students may need help parsing the phrase for what values of \(x\) is the \(y\)-value . . . Ask them: When \( x \) is 0, what is the \( y \)-value in \( y= -\frac{1}{2}x+6 \)? What about in \( y=4x−3 \)? Which \(y\)-value is greater? Compare the \(y\)-values with another value of \( x \). Then, ask students, Is there a value of \( x \) that would make the two \(y\)-values equal? What is that \(x\)-value?<br></p>
<h3>Activity Synthesis</h3>
<p>Focus the discussion on how this way of solving an inequality in one variable is like and unlike the strategy of solving a related equation, which students used in an earlier activity. Discuss questions such as:</p>
<ul>
<li>Previously, we saw that we could solve an inequality like this by first solving a related equation: \(-\frac{1}{2}x+6<4x-3\). Is the method of graphing similar to that process in any way? (It is similar in that we can find the value of \(x\) that makes the two expressions equal.)</li>
<li>How is the graphing method different? (Instead of comparing the values of the expressions by calculation, we can graph \(y=\) [expression on the left] and \(y=\) [expression on the right] and see where the two graphs intersect. The intersection tells us the \(x\)-value that produces the same \(y\)-value.)</li>
<li>Previously, to find the solutions to \(-\frac{1}{2}x+6<4x-3\), we would test \(x\)-values that are greater and less than the solution to \( -\frac{1}{2}x+6=4x-3\) and see which one would make the inequality true. How is the graphing method similar, and how is it different? (It is similar in that we are still comparing the values of the two expressions. It is different in that the graphs allow us to compare visually and see which graph has a greater \(y\)-value on either side of their intersection.)</li>
</ul>