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<h3>Activity (15 minutes)</h3>
<h4>Launch</h4>
<p>In preparation for the activity, review the properties of equality. When the same operation is performed with the same number to both sides of an equation, the equation remains true. This will be used as a base to build upon for operations performed on an inequality.</p>
<h4>Student Activity</h4>
<p>In this activity, we’re going to solve inequalities. Work with a partner to solve each inequality given. Then match it with the number line representing its solution.</p>
<p>As you complete the activity, discuss with your partner how you know each number line is the solution to its inequality. </p>
<p>When you think you’ve found the correct number line for an inequality, substitute a test point that is on the highlighted part of the number line into the inequality. Is it true?</p>
<p>Try a test point that is not highlighted. Is the inequality false? <br>
</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col"></th>
<th scope="col"></th>
</tr>
</thead>
<tbody>
<tr>
<td><ol class="os-raise-noindent">
<li>\(x\;–\;4\;\leq\;–6\;\) </li>
</ol></td>
<td><p>Graph A <img alt="A number line is shown. There is a closed circle on 2. The number line to the right of 2 is highlighted." height="36" src="https://k12.openstax.org/contents/raise/resources/382fc5944d156f96425a06cc472570c002f42690" width="300"></p></td>
</tr>
<tr>
<td><ol class="os-raise-noindent" start="2">
<li>\(x\;+\;6\;\geq\;8\;\)</li>
</ol></td>
<td><p>Graph B <img alt="A number line is shown. There is a closed circle on –2. The number line to the right of –2 is highlighted." height="36" src="https://k12.openstax.org/contents/raise/resources/94f73128258a93e934f991bef7309c37be97bea1" width="300"></p></td>
</tr>
<tr>
<td><ol class="os-raise-noindent" start="3">
<li>\(4x\;\leq\;8\) </li>
</ol></td>
<td><p>Graph C <img alt="A number line is shown. There is a closed circle on 2. The number line to the left of 2 is highlighted." height="36" src="https://k12.openstax.org/contents/raise/resources/7178cecf06bf2c3b25ec36162f73f1ea4cc50692" width="300"></p></td>
</tr>
<tr>
<td><ol class="os-raise-noindent" start="4">
<li>\(–\frac12x\;\leq\;1\;\)</li>
</ol></td>
<td><p>Graph D <img alt="A number line is shown. There is a closed circle on –2. The number line to the left of –2 is highlighted." height="36" src="https://k12.openstax.org/contents/raise/resources/6520962dc39f9e08095041073ca8d01671f0f75d" width="300"></p></td>
</tr>
</tbody>
</table>
<br>
<p>Use the table to choose the matching graph to the inequality.</p>
<ol class="os-raise-noindent">
<li>\(x\;–\;4\;\leq\;–6\;\)
</li>
</ol>
<p><strong>Answer: </strong>Graph C.</p>
<ol class="os-raise-noindent" start="2">
<li>\(x\;+\;6\;\geq\;8\;\) </li>
</ol>
<p><strong>Answer: </strong>Graph A.</p>
<ol class="os-raise-noindent" start="3">
<li>\(4x\;\leq\;8\)</li>
</ol>
<p><strong>Answer: </strong>Graph D.</p>
<ol class="os-raise-noindent" start="4">
<li>\(–\frac12x\;\leq\;1\;\)</li>
<p><strong>Answer: </strong>Graph B.</p>
</ol>
<br>
<p>For each of the inequalities above, match it to the property you used to solve.</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col"></th>
<th scope="col"></th>
</tr>
</thead>
<tbody>
<tr>
<td><ol class="os-raise-noindent" start="5">
<li>\(x\;–\;4\;\leq\;–6\;\) </li>
</ol></td>
<td><p>A. Multiplication Property of Inequality</p></td>
</tr>
<tr>
<td><ol class="os-raise-noindent" start="6">
<li>\(x\;+\;6\;\geq\;8\;\)</li>
</ol></td>
<td><p>B. Addition Property of Inequality</p></td>
</tr>
<tr>
<td><ol class="os-raise-noindent" start="7">
<li>\(4x\;\leq\;8\) </li>
</ol></td>
<td><p>C. Division Property of Inequality</p></td>
</tr>
<tr>
<td><ol class="os-raise-noindent" start="8">
<li>\(–\frac12x\;\leq\;1\;\)</li>
</ol></td>
<td><p>C. Subtraction Property of Inequality</p></td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent" start="5">
<li>\(x\;–\;4\;\leq\;–6\;\) </li></ol>
<p><strong>Answer: </strong>Addition Property of Inequality.</p>
<ol class="os-raise-noindent" start="6">
<li>\(x\;+\;6\;\geq\;8\;\)</li>
</ol>
<p><strong>Answer: </strong>Subtraction Property of Inequality.</p>
<ol class="os-raise-noindent" start="7">
<li>\(4x\;\leq\;8\)
</li>
</ol>
<p>Select <strong>two </strong>properties that could be used to solve the inequality</p>
<p><strong>Answer: </strong>Multiplication Property of Inequality and Division Property of Inequality.</p>
<ol class="os-raise-noindent" start="8">
<li>\(–\frac12x\;\leq\;1\;\)
</li>
</ol>
<p>Select <strong>two </strong>properties that could be used to solve the inequality</p>
<p><strong>Answer: </strong>Multiplication Property of Inequality and Division Property of Inequality.</p>
<ol class="os-raise-noindent" start="9">
<li>Is there anything unique about the method used to solve \(–\frac12x\leq1\)? Explain.</li>
</ol>
<p><strong>Answer: </strong>Yes. The inequality contains a less than or equal to symbol \((\leq)\). However, in the solution there is a greater than or equal to symbol \((\geq)\). When using the Multiplication and Division Properties of Inequality, you must reverse the symbol when you multiply or divide by a negative number.</p>
<h4>Anticipated Misconceptions</h4>
<p>Remind students that reversing the inequality symbol in a statement does not apply when adding or subtracting a negative number on both sides of an inequality. It only applies to multiplying or dividing with a negative number.</p>
<h4>Activity Synthesis</h4>
<p>Discuss how the Properties of Inequalities are similar to the Properties of Equality. Emphasize that performing the same action to each side of the inequality or equation maintains the truth of the statement. Mention that to solve the inequality, the goal is still to isolate the variable.</p>
<p>It should be remembered that the Multiplication and Division Properties of Inequality are unique because when multiplying or dividing by a negative number, the inequality symbol must be reversed to maintain the inequality’s truth.</p>
<br>
<h3>2.10.5: Self Check <br>
</h3>
<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>Solve the inequality. Which number line represents the solution? \(–13m \leq 5\)</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td><p><strong>Answers</strong></p></td>
<td><p><strong>Feedback</strong></p></td>
</tr>
<tr>
<td><br>
<p><img alt="There is a number line shown with a closed circle on 15. The number line to the left of 15 is highlighted." class="img-fluid atto_image_button_text-bottom" height="36" src="https://k12.openstax.org/contents/raise/resources/bce9b45c0860cae5f03cf88e7b8638bcdb744865" width="300"></p></td>
<td><p>Incorrect. Let’s try again a different way: This number line represents \(m\;\leq\;15\). However, substituting \(-16\) makes the inequality false. The answer is D, or \(m\;\geq\;-15\).</p></td>
</tr>
<tr>
<td><br>
<p><img alt="There is a number line shown with a closed circle on 15. The number line to the right of 15 is highlighted." class="img-fluid atto_image_button_text-bottom" height="36" src="https://k12.openstax.org/contents/raise/resources/8aac2fd9d1dbe2f99395c66587449e4a04eb629c" width="300"></p></td>
<td><p>Incorrect. Let’s try again a different way: This number line represents \(m\;\geq\;15\). However, substituting 14 makes the inequality true. The answer is D, or \(m\;\geq\;-15\).</p></td>
</tr>
<tr>
<td><br>
<p><img alt="There is a number line shown with a closed circle on -15. The number line to the left of -15 is highlighted." class="img-fluid atto_image_button_text-bottom" height="36" src="https://k12.openstax.org/contents/raise/resources/cf25238a45907e3f3e519345648129064e2cd2e5" width="300"></p></td>
<td><p>Incorrect. Let’s try again a different way: This number line represents \(m\;\leq\;-15\). However, substituting \(-16\) makes the inequality false. The answer is D, or \(m\;\geq\;-15\).</p></td>
</tr>
<tr>
<td><br>
<p><img alt="There is a number line shown with a closed circle on -15. The number line to the right of -15 is highlighted." class="img-fluid atto_image_button_text-bottom" height="36" src="https://k12.openstax.org/contents/raise/resources/23ef0bb0834b20bcfff8b36daafb2fe717ce9dfc" width="300"></p></td>
<td><p>That’s correct! Check yourself: The correct answer is \(m\;\geq\;-15\). Substituting \(-14\) makes the inequality true. Substituting \(-16\) makes the inequality false. </p></td>
</tr>
</tbody>
</table>
<br>
<!-- END SELF CHECK TABLE -->
<br>
<h3>2.10.5: Additional Resources</h3>
<br>
<p>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. <br>
</p>
<h3>Solving Linear Inequalities</h3>
<p >A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. So, when we solve linear equations, we are able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.</p>
<p >We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:</p>
<br>
<p ><img alt="Negative 4 is less than 2. Negative 4 minus 5 is less than 2 minus 5. Negative 9 is less than negative 3, which is true. Negative 4 is less than 2. Negative 4 plus 7 is less than 2 plus 7. 3 is less than 9, which is true." src="https://k12.openstax.org/contents/raise/resources/65d84793ccf7def97bb9e482b51b063196edf254"></p>
<br>
<p >Notice that the inequality sign stayed the same.</p>
<p >This leads us to the Addition and Subtraction Properties of Inequality.</p>
<h5>Addition and Subtraction Properties of Inequality</h5>
<hr>
<p>For any numbers \(a\), \(b\), and \(c\), if \(a<b\), then</p>
<p align="center">\(a+c<b+c\) \(a−c<b−c\)</p>
<p>For any numbers \(a\), \(b\), and \(c\), if \(a>b\), then </p>
<p align="center">\(a+c>b+c\) \(a−c>b−c\)</p>
<p>We can add or subtract
the same quantity from both sides of an inequality and still keep the
inequality.</p>
<br>
<p >What happens to an inequality when we divide or multiply both sides by a constant?</p>
<p >Let’s first multiply and divide both sides by a positive number.</p>
<p ><img alt="10 is less than 15. 10 times 5 is less than 15 times 5. 50 is less than 75 is true. 10 is less than 15. 10 divided by 5 is less than 15 divided by 5. 2 is less than 3 is true." src="https://k12.openstax.org/contents/raise/resources/a2e4b2e72fb097d63fa37dcb10ad973ab98dbeec"><br>
</p>
<p >The inequality signs stayed the same.</p>
<p >Does the inequality stay the same when we divide or multiply by a negative number?</p>
<p ><img alt="10 is less than 15 10 times negative 5 is blank 15 times negative 5? Negative 50 is blank negative 75. Negative 50 is greater than negative 75. 10 is less than 15. 10 divided by negative 5 is blank 15 divided by negative 5. Negative 2 is blank negative 3. Negative 2 is blank negative 3." src="https://k12.openstax.org/contents/raise/resources/0a933b3ce50677d122b976a29bcaac88bf759939"><br>
</p>
<p >Notice that when we filled in the inequality signs, the inequality signs reversed their direction.</p>
<p >When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.</p>
<p >This gives us the Multiplication and Division Properties of Inequality.</p>
<br>
<br>
<div class="os-raise-graybox">
<h5>Multiplication and Division Properties of Inequality</h5>
<hr>
<p>For any numbers \(a\), \(b\), and \(c\),</p>
<p>multiply or divide by a positive</p>
<p>if \(a< b\) and \(c > 0\), then \(ac < bc\) and \(\frac ac<\frac bc\)</p>
<p>if \(a> b\) and \(c > 0\), then \(ac > bc\) and \(\frac ac>\frac bc\)<br>
</p>
<p>Multiply or divide by a negative</p>
<p>if \(a< b\) and \(c < 0\), then \(ac > bc\) and \(\frac ac>\frac bc\)<br>
</p>
<p>if \(a> b\) and \(c < 0\), then \(ac < bc\) and \(\frac ac<\frac bc\)<br>
</p>
</div>
<br>
<p>When we divide or multiply an inequality by a:</p>
<ul>
<li>positive number, the inequality stays the same.</li>
<li>negative number, the inequality reverses.</li>
</ul>
<p>Sometimes when solving an inequality, as in the next example, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left.</p>
<p>\(x>a\) has the same meaning as \(a<x\).</p>
<p>Think about it as “If Xander is taller than Andy, then Andy is shorter than Xander.”</p>
<br>
<p >In Examples 1 and 2: solve the inequality, graph the solution on the number line, and write the solution in interval notation.</p>
<p ><strong>Example 1: </strong>\(x\;-\;\frac38\;\leq\;\frac34\)<br>
</p>
<p ><strong>Solution</strong></p>
<p><strong>Step 1 - </strong>Add \(\frac38\) to both sides of the inequality.<br>
</p>
<p>\(x-\frac38 +\frac38\leq\frac34+\frac38\)</p>
<p><strong>Step 2 - </strong>Simplify.<br>
</p>
<p>\(x\leq\frac98\)</p>
<p><strong>Step 3 - </strong>Graph the solution on the number line.<br>
</p>
<img alt="Graph of the solution, x is less than or equal to the fraction of 9 over 8." src="https://k12.openstax.org/contents/raise/resources/c83e52917f88c52d6badd18fc875c88fbf4621c8"> <br>
<br>
<p><strong>Step 4 - </strong>Write the solution in interval notation.</p>
<p>\((-\infty,\frac98]\)</p>
<br>
<p ><strong>Example 2:</strong> \(-15 < \frac35z\)<br>
</p>
<p ><strong>Solution</strong></p>
<p><strong>Step 1 - </strong>Multiply both sides of the inequality by \(\frac53\). Since \(\frac53\) is positive, the inequality
stays the same.<br>
</p>
<p>\((\frac53)(-15) \lt (\frac53)(\frac35z)\)</p>
<p><strong>Step 2 - </strong>Simplify.<br>
</p>
<p>\(-25\lt z\)</p>
<p><strong>Step 3 - </strong>Rewrite with the variable on the left.<br>
</p>
<p>\(z \gt -25\)</p>
<p><strong>Step 4 - </strong>Graph the solution on the number line.<br>
</p>
<img alt="Graph of a solution, z is greater than negative twenty five" src="https://k12.openstax.org/contents/raise/resources/4f24f66df266a47807d84a9fd9dd8e8d95c26cc2"> <br>
<br>
<p><strong>Step 5 - </strong>Write the solution in interval notation.</p>
<p>\((-25,\infty)\)</p>
<h4>Try It: Solving Linear Inequalities</h4>
<br>
<p>Solve the following linear inequalities:</p>
<!--Q1-->
<ol class="os-raise-noindent">
<li> \(9y\lt54\)</li>
</ol>
<p><strong>Answer: </strong></p>
<!--BEGIN STEPS-->
<p><strong>Step 1 - </strong>Divide both sides of the inequality by 9; since 9 is positive, the inequality stays the same.<br>
</p>
<p>\(\frac{9y}{9}\lt\frac{54}{9}\)</p>
<p><strong>Step 2 - </strong>Simplify.<br>
</p>
<p>\(y\lt6\)</p>
<p><strong>Step 3 - </strong>Graph the solution on the number line.<br>
</p>
<img alt="Graph of a solution, y is less than 6." src="https://k12.openstax.org/contents/raise/resources/de89c80e4ed83d804b7a227a69c3ba2a026b3076"> <br>
<br>
<p><strong>Step 4 - </strong>Write the solution in interval notation.</p>
<p>\((-\infty,6)\)</p>
<!-- END STEPS -->
<!--Q#2-->
<ol class="os-raise-noindent" start="2">
<li> \(-13m\geq65\)</li>
</ol>
<p><strong>Answer: </strong></p>
<!--BEGIN STEPS-->
<p><strong>Step 1 - </strong>Divide both sides of the inequality by \(-13\). Since \(-13\) is negative, the inequality reverses.<br>
</p>
<p>\(\frac{-13m}{-13}\color{red}\leq\frac{65}{-13}\)</p>
<p><strong>Step 2 - </strong>Simplify.<br>
</p>
<p>\(m\leq-5\)</p>
<p><strong>Step 3 - </strong>Graph the solution on the number line.<br>
</p>
<img alt="Graph of a solution m is less than or equal to negative five." src="https://k12.openstax.org/contents/raise/resources/e570c21627edfe973de18cd169aa0a19b5f82a7d"> <br>
<br>
<p><strong>Step 4 - </strong>Write the solution in interval notation.</p>
<p>\((-\infty,-5)\)</p>
<!-- END STEPS -->