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<h4>Graphing Linear Inequalities</h4>
<p><br>
What number would make the inequality \(x > 3\) true? Are you thinking, “\(x\) could be 4”? That’s correct, but \(x\) could be 6, too, or 37, or even 3.001. Any number greater than 3 is a solution to the inequality \(x > 3\). We show all the solutions to the inequality \(x > 3\) on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open circle at 3. <br>
</p>
<p>We can also represent inequalities using interval notation. There is no upper end to the solution to this inequality. In interval notation, we express \( x > 3 \) as \((3, ∞)\). The symbol ∞ is read as “infinity.” It is not an actual number.</p>
<p>The figure below shows both the number line and the interval notation.</p>
<p> \(x > 3\)<br>
<img alt="A number line is shown. There is an open circle on 3. The number line to the right of 3 is highlighted." class="img-fluid atto_image_button_text-bottom" height="48" src="https://k12.openstax.org/contents/raise/resources/1f11f9f52563f8687b96d97d1f8d166c69c5b4d7" width="400"><br>
\((3, ∞)\)<br>
</p>
<p>The inequality \(x≤1\) means all numbers less than or equal to 1. Here we need to show that 1 is a solution, too. We do that by putting a closed circle at \(x = 1\). We then shade in all the numbers to the left of 1, to show that all numbers less than 1 are solutions. See the figure below.</p>
<p> \( x ≤ 1 \)<br>
<img alt="A number line is shown. There is a closed circle on 1. The number line to the left of 1 is highlighted." class="img-fluid atto_image_button_text-bottom" height="49" src="https://k12.openstax.org/contents/raise/resources/c89f24114c86a88a457ff1ee55c0119e4aea0a91" width="400"><br>
\((-∞,1]\)<br>
</p>
<p id="docs-internal-guid-17ab7c97-7fff-0e49-3be6-d15506d843a8">There is no lower end to those numbers. We write \(x ≤ 1 \) in interval notation as \( (−∞,1] \). The symbol −\( ∞ \) is read as “negative infinity.” The figure below shows both the number line and interval notation.</p>
<h4>Inequalities, Number Lines, and Interval Notation</h4>
<img alt="Four different number lines are shown.The > symbol corresponds to an open circle and the number line to the right being highlighted. Its set notation is (a, ∞). The < symbol corresponds to a closed circle and the number line to the right being highlighted. Its set notation is [a, ∞). The ≥ symbol corresponds to a closed circle and the number line to the right being highlighted. Its set notation is (–∞, a). The ≤ symbol corresponds to a closed circle and the number line to the left being highlighted. Its set notation is (–∞, a]." class="img-fluid atto_image_button_text-bottom" height="89" role="presentation" src="https://k12.openstax.org/contents/raise/resources/ae108513af5043e10799df8d1eed5b40269dbece" width="400"> <br>
<br>
<h4>Try It: Graphing Linear Inequalities</h4>
<br>
<p>For questions 1 - 2 use the following inequality:</p>
<blockquote>
<p>\( x > -3 \)</p>
</blockquote>
<!-- BEGIN CLICK TO REVEAL NO TEXTBOX -->
<br>
<!--Q#1-->
<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>Write the inequality on a number line.</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-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<img alt="A number line is shown. There is an open circle on
–3. The number line to the right of –3 is highlighted.
" class="img-fluid atto_image_button_text-bottom" height="49" role="presentation" src="https://k12.openstax.org/contents/raise/resources/10effd40cd28ca12937e70f7415e5166edad0307" width="400">
</div>
<!--Interaction End -->
<br>
<!-- END CLICK TO REVEAL NO TEXTBOX -->
<!--Text Entry Interaction Start -->
<!--Q #2-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="1bd47f94-dd9b-484a-8901-cd9dbb83e9c8" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>Write the inequality in interval notation.</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer: \((-3, ∞)\)</p>
</div>
</div>
<!--Interaction End -->
<p>For questions 3 and 4 use the following inequality:</p>
<blockquote>
<p>\( x ≤ 5 \)</p>
</blockquote>
<!-- BEGIN CLICK TO REVEAL NO TEXTBOX -->
<br>
<!--Q#3-->
<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="3">
<li>Write the inequality on a number line.</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-schema-version="1.0" data-wait-for-event="Reveal2">
<p>Compare your answer:</p>
<img alt=" A number line is shown. There is a closed circle
on 5. The number line to the left of 5 is highlighted.
" class="atto_image_button_text-bottom" height="49" role="presentation" src="https://k12.openstax.org/contents/raise/resources/b549d66de4f02bdbf801752a89ee06481e8d9689" width="400">
</div>
<!--Interaction End -->
<br>
<!-- END CLICK TO REVEAL NO TEXTBOX -->
<!--Text Entry Interaction Start -->
<!--Q #2-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="378fe726-dd95-40f3-8d18-99e7d7de0a68" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li>Write the inequality in interval notation.</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer: \((∞, 5]\)</p>
</div>
</div>
<!--Interaction End -->