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edc2f00a-d965-45a4-a33f-9dd4e8174564.html
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<h4>Reasoning about Solution Sets to Linear Inequalities</h4>
<p>You can derive a lot of information about the solution set to an inequality by studying the inequality itself. <br></p>
<p>Let’s look at the inequality shown.</p>
<p>\(12x>-3(x−1)\)</p>
<p>The left side of the inequality contains \(12x\). The right side of the inequality contains \(-3x\).</p>
<p>We can think to ourselves: “What values of \(x\) would make \(12x\) greater than \(-3x\)?”</p>
<p>If \(x\) is a large positive number, then \(12x\) will still be positive. If \(x\) is a large positive number, then \(-3x\) will be a negative number. This makes the inequality true, so the solution set must include large positive numbers.</p><br>
<p>Let’s solve the related equation.</p>
<p>\(\begin{array}{rcl}12x&>&-3(x-1)\\12x\;&=&\;–3(x\;–\;1)\\12x\;&=&\;–3x\;+3\\15x\;&=&\;3\\x\;&=&\;\frac{3}{15}\end{array}\)</p>
<p>Since we know that the solution set must include large positive numbers, then \(x > \frac{3}{15}\) must be the solution to the inequality.</p>
<h4>Try It: Reasoning about Solution Sets to Linear Inequalities</h4>
<br>
<p>For questions 1 - 2, solve the related equation for each inequality. Use what you know about the inequality to determine the solution set.</p>
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<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="6cb1aa90-d0f6-4181-8f42-6ced1b3378bb" data-fire-event="eventShow" data-schema-version="1.0">
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<li>\(-5y\geq30\)</li>
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<p>Enter your answer here:</p>
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<p>Compare your answer:</p>
<p>The solution \(y = -6\). Since any positive number substituted for \(y\) makes \(-5y\) a negative value, then the solution set cannot include positive numbers. This means the solution is \(y\leq-6\).</p>
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<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="b85f6455-a628-4c86-b23e-ae985278b83c" data-fire-event="eventShow" data-schema-version="1.0">
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<ol class="os-raise-noindent" start="2">
<li>\(4p<-4\)</li>
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<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
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<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>The solution \(p = -1\). Since any positive number substituted for \(p\) makes \(4p\) a positive value, the solution set cannot include positive numbers. This means the solution is \(p < -1\).</p>
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