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<h4>Graph Linear Inequalities in Two Variables</h4>
<p>Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.</p>
<p><strong>Example </strong><br>
How to Graph a Linear Equation in Two Variables</p>
<p>Graph the linear inequality \( y ≥ \frac{3}{4} x − 2 \).</p>
<p><b>Step 1 - </b> Identify and graph the boundary line. </p>
<p> If the inequality is \(\leq\) or \(\geq\)
boundary line is solid. </p>
<p>If the inequality is < or >, the boundary
line is dashed. </p>
<p>Replace the
inequality sign with an equal sign to find the boundary line</p>
<p>\(y=\frac34x-2.\)</p>
<p>The
inequality sign is \(\geq\), so we draw a solid line.</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="205" role="presentation" src="https://k12.openstax.org/contents/raise/resources/73a02b5f0522899713366bc7f2696871112ac313" width="200"></p>
<p><b>Step 2 -</b> Test a point that is not on the boundary line. Is it a solution of the
inequality?</p>
<p>We’ll test
\((0,0)\).</p>
<p>Is it a solution
of the inequality?</p>
<p>At \((0, 0)\),
is \(y \geq \frac34x-2.\)?</p>
<p>\(0\overset?\geq\frac34(0)-2\)</p>
<p>\(0\geq-2\)</p>
<p>So, \((90,0)\)
is a solution.</p>
<p><b>Step 3 -</b> Shade in one side of the boundary line. </p>
<p>If the test point is a solution, shade in the side
that includes the point. </p>
<p> If the test point is not a solution, shade in
the opposite side. </p>
<p>The test
point \((0,0)\) is a solution to \(y\geq\frac34x-2.\)
So we shade in that side.</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="202" role="presentation" src="https://k12.openstax.org/contents/raise/resources/3fb1b437b5f627f97dd1a57c0ef8f9697fa730b6" width="200"></p>
<p>All points in
the shaded region and on the boundary line represent the solutions to \(y\geq\frac34x-2.\)</p>
<br>
<h4>Try It: Graph Linear Inequalities in Two Variables</h4>
<!--BEGIN ADD DESMOS INTERATIVE-->
<!--START QUESTION #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">
<!-- INSERT ANY VALID HTML HERE -->
<ol class="os-raise-noindent">
<!-- INSERT ANY VALID HTML HERE -->
<!--CHANGE THE TEXT BEFORE Use the graphing tool or technology outside the course. TO FIT THE PROBLEM-->
<li>Graph the linear inequality \(y\;>\;\frac52x\;-\;4\). Use the graphing tool or technology outside the course. Graph the inequality using the Desmos tool below. </li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<!-- INSERT ANY VALID HTML HERE -->
<p>When you have finished graphing, 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>
<!--REPLACE IMAGE BELOW WITH CORRECT IMAGE-->
<p>All points in the shaded region represent the solutions to \(y\;>\;\frac52x\;-\;4\)</p>
<img alt class="img-fluid atto_image_button_text-bottom" height="234" role="presentation" src=" https://k12.openstax.org/contents/raise/resources/3a2c4243b36188d45c7f1556d44666cb5fbeef8f" width="228">
<br>
<br>
</div>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<!--END ADD DESMOS INTERATIVE-->