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<h3>Activity (15 minutes)</h3>
<p>Time permitting, this activity gives students additional practice in reasoning about the solutions to inequalities without a context. Students can match the inequalities and solutions in a variety of ways—by testing different values, by solving a related equation and then testing values on either side of that solution, by reasoning about the parts of an inequality or its structure, or by graphing each side of an inequality as a linear function.</p>
<p>For all of the inequalities, once students find the boundary value by solving a related equation, the range of the solutions can be determined by analyzing the structure of the inequality. Monitor for students who do so and ask them to share their reasoning during class discussion later.</p>
<p>As students work, also make note of any common challenges or errors so they could be addressed.</p>
<br>
<h4>Student Activity</h4>
<p>Match each inequality to a graph that represents its solutions. Be prepared to show your reasoning.</p>
<ul>
<li>\(6x\;\leq\;3x\)</li>
<li>\(\frac14x>-\frac12\)</li>
<li>\(5x+4\geq7x\)</li>
<li>\(8x-2<-4(x-1)\)</li>
<li>\(\frac{4x-1}3>-1\)</li>
<li>\(\frac{12}5-\frac x5\leq x\)</li>
</ul>
<p>Graph A</p><img alt="Inequality graphed on a number line. Numbers from negative 10 to 10. At negative one half, open circle with line extending to the right. " src="https://k12.openstax.org/contents/raise/resources/2d621974e0882d1c29a952d999cd031e4b1acd5a">
<p>Graph B</p><img alt="Inequality graphed on a number line. Numbers from negative 10 to 10. At two, closed circle with line extending to the left. " src="https://k12.openstax.org/contents/raise/resources/5c1504f5073145ae656f9e928daa3f6a8b289e7c">
<p>Graph C</p><img alt="Inequality graphed on a number line. Numbers from negative 10 to 10. At two, closed circle with line extending to the right." src="https://k12.openstax.org/contents/raise/resources/e96d18ea04eb10c86b9797284fff1c676aa8c022">
<p>Graph D</p><img alt="Inequality graphed on a number line. Numbers from negative 10 to 10. At one half, open circle with line extending to the left. " src="https://k12.openstax.org/contents/raise/resources/e9b8072106740850d58157b4ee2d9e36e056e931">
<p>Graph E</p><img alt="Inequality graphed on a number line. Numbers from negative 10 to 10. At negative 2, open circle with line extending to the right." src="https://k12.openstax.org/contents/raise/resources/3afda547f9abbeb3d436ab4a16f63d661fc76af9">
<p>Graph F</p><img alt="Inequality graphed on a number line. Numbers from negative 10 to 10. At 0, closed circle with line extending to the left." src="https://k12.openstax.org/contents/raise/resources/099eae57745932eb4032ebbc413163b4adb69884">
<br><br>
<p>Use this inequality for questions 1 - 2:</p>
<blockquote><p>\(6x\le3x\)</p></blockquote>
<ol class="os-raise-noindent">
<li>
Match the graph to the equation.
<p><strong>Answer: </strong> Graph F.</p>
</li>
<li>
Explain or show your reasoning.
<p><strong>Answer: </strong> \(x\leq0\)</p>
</li>
<p>Use this inequality for questions 3 and 4:</p>
<blockquote><p>\(\frac14x\gt-\frac12\)</p></blockquote>
<li>
Match the graph to the equation.
<p><strong>Answer: </strong> Graph E.</p>
</li>
<li>
Explain or show your reasoning.
<p><strong>Answer: </strong> \(x>-2\)</p>
</li>
<p>Use this inequality for questions 5 - 6:</p>
<blockquote><p>\(5x+4\geq7x\)</p></blockquote>
<li>
Match the graph to the equation.
<p><strong>Answer: </strong> Graph B.</p>
</li>
<li>
Explain or show your reasoning.
<p><strong>Answer: </strong> \(x\leq2\)</p>
</li>
<p>Use this inequality for questions 7 and 8:</p>
<blockquote><p>\(8x-2\lt-4(x-1)\)</p></blockquote>
<li>
Match the graph to the equation.
<p><strong>Answer: </strong> Graph D.</p>
</li>
<li>
Explain or show your reasoning.
<p><strong>Answer: </strong> \(x\lt\frac12\)</p>
</li>
<p>Use this inequality for questions 9 and 10:</p>
<blockquote><p>\(\frac{4x-1}3\gt-1\)</p></blockquote>
<li>
Match the graph to the equation.
<p><strong>Answer: </strong> Graph A.</p>
</li>
<li>
Explain or show your reasoning.
<p><strong>Answer: </strong> \(x\gt-\frac12\)</p>
</li>
<p>Use this inequality for questions 11 and 12:</p>
<blockquote><p>\(\frac{12}5-\frac x5\leq x\)</p></blockquote>
<li>
Match the graph to the equation.
<p><strong>Answer: </strong> Graph C.</p>
</li>
<li>
Explain or show your reasoning.
<p><strong>Answer: </strong> \(x\geq2\)</p>
</li>
</ol>
<h4>Activity Synthesis</h4>
<p>Select students who used different strategies to share their thinking, especially those who made use of the structure of the inequalities. If no students found solution sets by thinking about the features of the inequalities, demonstrate the reasoning process with one or two examples. For instance:</p>
<p>\(6x\leq3x\): After finding \(x=0\) as the solution to \(6x=3x\), we can reason that for \(6x\) to be less than \(3x\), \(x\) must include negative numbers, so the solution must be \(x≤0\).</p>
<p>\(\frac{4x-1}{3}>-1\): After finding \(x=-\frac12\) as the solution to the related equation \(\frac{4x-1}{3}=-1\), we can reason that as \(x\) gets smaller, \(\frac{4x-1}{3}>-1\) is also going to get smaller. For that expression to be greater than \(-1\), \(x\) will have be to greater than \(-\frac12\).</p>
<p>If time permits, ask students to choose a different inequality and try reasoning this way about its solution.</p>