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<p > </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>
<div class="os-raise-graybox">
<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>
</div>
<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>
<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>
<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.<br?</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.<br?</p>
<p>\((-25,\infty)\)</p>
<h4>Try It: Solving Linear Inequalities</h4>
<p>Solve the following linear inequalities:</p>
<!-- BEGIN TEXT REVEAL -->
<br>
<!--Q#-->
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<li> \(9y\lt54\)</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-wait-for-event="Reveal1" data-schema-version="1.0">
<p>Compare your answer:</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.<br?</p>
<p>\((-\infty,6)\)</p>
<!-- END STEPS -->
</div>
<!--Interaction End -->
<br>
<!-- BEGIN TEXT REVEAL -->
<br>
<!--Q#-->
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li> \(-13m\geq65\)</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-wait-for-event="Reveal2" data-schema-version="1.0">
<p>Compare your answer:</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.<br?</p>
<p>\((-\infty,-5)\)</p>
<!-- END STEPS -->
</div>
<!--Interaction End -->
<br>
<!-- END TEXT REVEAL -->