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<h4>Determine Whether an Ordered Pair Is a Solution of a System of Linear Inequalities</h4>
<p>The definition of a <span class="os-raise-ib-tooltip" data-store="glossary-tooltip" data-schema-version="1.0">system of linear inequalities</span> is very similar to the definition of a <span class="os-raise-ib-tooltip" data-store="glossary-tooltip" data-schema-version="1.0">system of linear equations</span>.<br></p>
<h5>System of Linear Inequalities<br></h5>
<p>Two or more linear inequalities grouped together form a system of linear inequalities.</p>
<p>A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.</p>
<p>\(\left\{\begin{array}{l}x+4y\geq10\\3x-2y<12\end{array}\right.\)</p>
<p>To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs \((x,y)\) that make both inequalities true.</p>
<h5>Solutions of A System of Linear Inequalities</h5>
<p>Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.</p>
<p>The <span class="os-raise-ib-tooltip" data-store="glossary-tooltip" data-schema-version="1.0">solution of a system of linear inequalities</span> is shown as a shaded region in the \((x, y)\) coordinate system that includes all the points whose ordered pairs make the inequalities true.<br></p>
<p>To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.<br></p>
<p><strong>Example</strong><br></p>
<p>Determine whether the ordered pair is a solution to this system: </p>
<p>\(\left\{\begin{array}{l}x+4y\geq10\\3x-2y<12\end{array}\right.\)</p>
<p>a.) \((−2,4)\) b.) \((3,1)\)<br></p>
<p>a.) Is the ordered pair \((−2,4)\) a solution?</p>
<p><strong><img alt="We substitute x equal to negative 2 and y equal to 4 into both inequalities. First inequality is x plus 4 times y greater than or equal to 10. So negative 2 plus 4 open parentheses 4 close parenthesis is greater than or equal to 10 or not. 14 is greater than or equal to 10 is true. Second inequality, 3 times x minus 2 times y is less than 12. Three open parentheses negative 2 close parentheses minus two open parentheses 4 close parentheses is less than 12 or not. Negative 14 is less than 12 is true." src="https://k12.openstax.org/contents/raise/resources/6aa8c92882cff834137c795fbb93eb4163315bad"></strong><br></p>
<p>The ordered pair \((−2,4)\) made both inequalities true. Therefore \((−2,4)\) is a solution to this system.</p>
<p>b.) Is the ordered pair \((3,1)\) a solution?</p>
<p><strong><img alt="We substitute x equal to three and y equal to one into both inequalities. First inequality is x plus four times y greater than or equal to ten. So three plus four open parentheses one close parenthesis is greater than or equal to ten or not. Seven greater than or equal to ten is false. Second inequality, three times x minus two times y is less than twelve. Three open parentheses three close parentheses minus two open parentheses one close parentheses is less than twelve or not. Seven less than 12 holds true." src="https://k12.openstax.org/contents/raise/resources/bb4c583f5689a1a325dd64a1c97f16b9d5bcd9e7"></strong><br></p>
<p>The ordered pair \((3,1)\) made one inequality true, but it made the other one false. Therefore, \((3,1)\) is not a solution to this system.</p>
<h4>Try It: Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities.</h4>
<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">
<p>Determine whether the ordered pair is a solution to the system: <br></p>
<p>\(\left\{\begin{array}{l}y>4x-2\\4x-y<20\end{array}\right.\)</p>
<p><span>a.) \((−2,1)\) b.) \((4,−1)\)</span></p>
</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">
<ol class="os-raise-noindent" start="1" type="a">
<li>Yes.</li>
</ol>
<p>Here is how to determine if each ordered pair is a solution:</p>
<p><strong>Step 1 - </strong>Choose an ordered pair.<br>
\((-2,1)\)<br><br>
<strong>Step 2 - </strong>Substitute into the inequality.<br>
\(y>4x−2\)<br>
\(1>4(-2)-2\)<br><br>
\(4x−y<20\)<br>
\(4(-2)-1<20\)<br><br>
<strong>Step 3 - </strong>Simplify.<br>
\(1>-10\)<br>
\(-9<20\)<br><br>
<strong>Step 4 - </strong> True or false statement?<br>
True<br>
True<br>
Since the ordered pair makes both inequalities true, it is a solution to the system of inequalities.</p>
<!-- END STEPS -->
<br>
<ol class="os-raise-noindent" start="2" type="a">
<li>No.</li>
</ol>
<p>Here is how to determine if each ordered pair is a solution:</p>
<p><strong>Step 1 - </strong>Choose an ordered pair.<br>
\((4, -1)\)<br><br>
<strong>Step 2 - </strong>Substitute into the inequality.<br>
\(y>4x−2\)<br>
\(-1>4(4)-2\)<br><br>
\(4x−y<20\)<br>
\(4(4)-1(-1)<20\)<br><br>
<strong>Step 3 - </strong>Simplify.<br>
\(-1>14\)<br>
\(17<20\)<br><br>
<strong>Step 4 - </strong> True or false statement?<br>
False<br>
True<br>
Since the ordered pair does not make a true statement in the first inequality, it is not a solution to the system of inequalities.</p>
<!-- END STEPS -->
</div>