<p>solution with graph</p><a href="https://data.ulearngo.com/assets/1ad9d87a-4ba4-4739-9adc-f50d40b4c9e0"><img alt="" src="https://data.ulearngo.com/assets/1ad9d87a-4ba4-4739-9adc-f50d40b4c9e0"/></a>

<h2>Solving a System of Linear Inequalities By Graphing</h2>
<p>The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.</p>
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<h3>Example: How to Solve a System of Linear inequalities</h3>
<div class="ns-school-defn">
<p>Solve the system by graphing.</p>
<p>\(\begin{array}{c}y\ge 2x-1\hfill \\ y&lt;x+1\hfill \end{array}\)</p>
<h3 data-type="title">Solution</h3>
<div data-type="title"><img alt="This is a table with three columns and several rows. The first row says, “Step 1: Graph the first inequality. We will graph y is greater than or equal to 2x – 1.” There are two equations givens, y is greater than or equal to 2x – 1 and y is less than x + 1. The table then reads, “Graph the boundary line. We graph the line y = 2x – 1. It is a solid line because the inequality sign is greater than or equal to. Shade in the side of the boundary line where the inequality is true. We choose (0, 0) as a test point. It is a solution to y is greater than or equal to 2x – 1, so we shad in the left side of the boundary line.” There is a figure of a line graphed on an x y coordinate plane. The area to the left of the line is shaded." src="https://data.ulearngo.com/assets/4a9001fa-d712-4e5d-a4f2-1f3508831e43"/><img alt="The second row then says, “Step 2: On the same grid, graph the second inequality. We will graph y is less than x + 1 on the same grid. Grph the boundary line. We graph the lin y = x + 1. It is a dashed line because the inequality sign is less than. There is a graph which shows two lines graphed on an x y coordinate plane. The area to the left of one line is shaded. The area to the right of the second line is shaded. There is a small area where the shaded areas overlap. The table then says, “Shade in the side of that boundary line where the inequality is true. Again we use (0, 0) as a test point. It is a solution so we shade in that side of the line y = x + 1." src="https://data.ulearngo.com/assets/390a6aa4-0742-4447-8cc1-06265fac3445"/><img alt="The third row then says, “Step 3: The solution is the region where the shading overlaps. The poing where the boundary lines intersect is not a solution because it is not a solution to y is less than x + 1. The solution is all points in the purple shaded region.”" src="https://data.ulearngo.com/assets/c98e3404-5358-46fb-85ed-dfe870ba7875"/><img alt="The fourth row then says, “Step 4: Check by choosing a test point. We’ll use (-1, -1) as a test point. Is (-1, -1) a solution to y is greater than or equal to 2x – 1? -1 is greater than or equal to 2 times -1 – 1 or -1 is greater than or equal to -3 true.”" src="https://data.ulearngo.com/assets/43482f30-33e2-4966-878a-f3fea5220407"/></div>
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<div class="ns-school-defn2">
<header>
<h3 data-type="title">Solve a system of linear inequalities by graphing.</h3>
</header>
<ol type="1">
<li>Graph the first inequality.

<ul data-bullet-style="bullet" style="list-style-type: square;">
<li>Graph the boundary line.</li>
<li>Shade in the side of the boundary line where the inequality is true.</li>
</ul>
</li>
<li>On the same grid, graph the second inequality.

<ul data-bullet-style="bullet" style="list-style-type: square;">
<li>Graph the boundary line.</li>
<li>Shade in the side of that boundary line where the inequality is true.</li>
</ul>
</li>
<li>The solution is the region where the shading overlaps.</li>
<li>Check by choosing a test point.</li>
</ol>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
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<div data-type="example">
<div>
<p>Solve the system by graphing. \(\begin{array}{c}x-y&gt;3\hfill \\ y&lt;-\frac{1}{5}x+4\hfill \end{array}\)</p>
</div>
<div>
<h3>Solution</h3>
<table style="width: 784px;" summary="This figure has both words and a graph. The words say, “Graph x – y is greater than 3, by graphing x – y = 3 and testing a point. The intercepts are x = 3 and y = -3 and the boundary line will be dashed. Test (0, 0) which makes the inequality false so shade (red) the side that does not contain (0, 0).” The graph shows a line graphed on an x y-coordinate plane. The line is dashed and the area to the right of the line is shaded.">
<tbody>
<tr>
<td data-valign="top" style="width: 383px;">Graph <em>x</em> − <em>y</em> &gt; 3, by graphing <em>x</em> − <em>y</em> = 3 and testing a point.<br/>
The intercepts are <em>x</em> = 3 and <em>y</em> = −3 and the boundary line will be dashed.<br/>
Test (0, 0). It makes the inequality false. So, shade the side that does not contain (0, 0) red.</td>
<td data-valign="top" style="width: 401px;"><img alt="." src="https://data.ulearngo.com/assets/b4c82e21-9d69-428e-b214-732d5797aab2"/></td>
</tr>
</tbody>
</table>
<table summary="This figure has both words and a graph. The words say, “Graph y is less than (-1/5)x + 4 by graphing y = (-1/5)x + 4 using the slope m = (-1/5) and y-intercept b = 4. The boundary line will be dashed. Test (0, 0) which makes the inequality true, so shade (blue) the side that continas (0, 0). Choose a test point in the solution and verify that it is a solution to both inequalities.” The figure shows two lines graphed on an x y-coordinate plane. The figure shows on line that is dashed, and the area to the right is shaded. The other line is solid, and the area below it is shaded. There is an area where the two shaded areas overlap.">
<tbody>
<tr>
<td data-valign="top" style="width: 483px;">Graph \(y&lt;-\frac{1}{5}x+4\) by graphing \(y=-\frac{1}{5}x+4\) using the slope \(m=-\frac{1}{5}\) and <em style="font-family: inherit; font-size: inherit;">y</em><span style="font-family: inherit; font-size: inherit;">−intercept </span><em>b</em> = 4. The boundary line will be dashed.<br/>
Test (0, 0). It makes the inequality true, so shade the side that contains (0, 0) blue.<br/>
Choose a test point in the solution and verify that it is a solution to both inequalities.</td>
<td data-valign="top" style="width: 455px;"><img alt="." src="https://data.ulearngo.com/assets/81d04181-9b48-4354-861d-78a5ae8c5ed7"/></td>
</tr>
</tbody>
</table>
<p>The point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice which is the darker-shaded region.</p>
</div>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Solve the system by graphing. \(\begin{array}{c}x-2y&lt;5\hfill \\ y&gt;-4\hfill \end{array}\)</p>
</div>
<div>
<h3>Solution</h3>
<table summary="This figure has both words and a graph. The words say, “Graph x – 2y is less than 5, by graphing x – 2y = 5 and testing a point. The intercepts are x = 5 and y = -2.5 and the boundary line will be dashed. Test (0, 0) which makes the inequality true, so shade (red) the side that contains (0, 0).” The figure shows a dashed line graphed on an x y-coordinate plane. The area above the line is shaded.">
<tbody>
<tr>
<td data-valign="top" style="width: 516px;">Graph \(x-2y&lt;5\), by graphing \(x-2y=5\) and testing a point.<br/>
The intercepts are <em>x</em> = 5 and <em>y</em> = −2.5 and the boundary line will be dashed.<br/>
Test (0, 0). It makes the inequality true. So, shade the side that contains (0, 0) red.</td>
<td data-valign="top" style="width: 422px;"><img alt="." src="https://data.ulearngo.com/assets/312db6f3-70a9-4fbb-a8e6-7a211edbc0fe"/></td>
</tr>
</tbody>
</table>
<table summary="This figure has both words and a graph. The words say, “Graph y is greater than -4, by graphing y = -4 and recognizing that it is a horizontal line through y = -4. The boundary line will be dashed. Test (0, 0) which makes the inequality true so shade (blue) the side that contains (0, 0). This figure shows two line graphed in an x y-coordinate plane. One line is dashed, and the area above it is shaded. The second line is solid, and the area above it is shaded.">
<tbody>
<tr>
<td data-valign="top" style="width: 542px;">Graph <em>y</em> &gt; −4, by graphing <em>y</em> = −4 and recognizing that it is a horizontal line through <em style="font-family: inherit; font-size: inherit;">y</em><span style="font-family: inherit; font-size: inherit;"> = −4. The boundary line will be dashed.<br/>
</span>Test (0, 0). It makes the inequality true. So, shade (blue) the side that contains (0, 0) blue.</td>
<td data-valign="top" style="width: 395px;"><img alt="." src="https://data.ulearngo.com/assets/b85046c3-1810-4007-91d2-4a0b43a6457d"/></td>
</tr>
</tbody>
</table>
<p>The point (0, 0) is in the solution and we have already found it to be a solution of each inequality. The point of intersection of the two lines is not included as both boundary lines were dashed.</p>
<p>The solution is the area shaded twice which is the darker-shaded region.</p>
</div>
</div>
</div>
</div>
<p>Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in the example below.</p>
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<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Solve the system by graphing. \(\begin{array}{c}4x+3y\ge 12\hfill \\ y&lt;-\frac{4}{3}x+1\hfill \end{array}\)</p>
</div>
<div>
<h3>Solution</h3>
<table summary="This figure has both words and a graph. The first set of words says, “Graph 4x + 3y is greater than or equal to 12, by graphing 4x + 3 = 12 and testing a point. The intercepts are x = 3 and y = 4 and the boundary line will be solid. Test (0, 0) which makes the inequality false, s oshade (blue) the side that does not contain (0, 0).” The figure shows a line graphed on an x y-coordinate plane. The line is solid, and the area to the right of the line is shaded. The second set of words says, “Graph y is less than (-4/3)x + 1 by graphing y = (-4/3)x + 1 using the slope m = (-4/3) and the y-intercept b = 1. The boundarly line will be dashed. Test (0, 0) which makes the inequality true, so shade (red) the side that contains (0, 0). The figure shows two line graphed on an x y-coordinate plane. One line is solid, and the area to the right is shaded. The other line is dashed, and the area to the left is shaded. There is no area of overlap between the two shaded areas.">
<tbody>
<tr>
<td data-valign="top" style="width: 528px;">Graph \(4x+3y\ge 12\), by graphing \(4x+3y=12\) and testing a point.<br/>
The intercepts are <em>x</em> = 3 and <em>y</em> = 4 and the boundary line will be solid.<br/>
Test (0, 0). It makes the inequality false. So, shade the side that does not contain (0, 0) red.</td>
<td data-valign="top" style="width: 410px;"><img alt="." src="https://data.ulearngo.com/assets/822994c1-0a10-453f-9bda-cca24bb23d95"/></td>
</tr>
<tr>
<td data-valign="top" style="width: 528px;">Graph \(y&lt;-\frac{4}{3}x+1\) by graphing \(y=-\frac{4}{3}x+1\) using the slope \(m=\frac{4}{3}\) and the <em style="font-family: inherit; font-size: inherit;">y</em><span style="font-family: inherit; font-size: inherit;">-intercept </span><em style="font-family: inherit; font-size: inherit;">b</em><span style="font-family: inherit; font-size: inherit;"> = 1. The boundary line will be dashed.<br/>
</span>Test (0, 0). It makes the inequality true. So, shade the side that contains (0, 0) blue.</td>
<td data-valign="top" style="width: 410px;"><img alt="." src="https://data.ulearngo.com/assets/ef8b00ca-b466-4165-aa3f-213d3f97aec8"/></td>
</tr>
</tbody>
</table>
<p>There is no point in both shaded regions, so the system has no solution. This system has no solution.</p>
</div>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Solve the system by graphing. \(\begin{array}{c}y&gt;\frac{1}{2}x-4\hfill \\ x-2y&lt;-4\hfill \end{array}\)</p>
</div>
<div>
<h3>Solution</h3>
<table summary="This figure has both words and a graph. The first set of words says, “Graph y is greater than (1/2)x – 4 by graphing y = (1/2)x – 4 using the slope m = 1/2 and the intercept b = -4. The boundary line will be dashed. Tess (0, 0) which makes the inequality true, so shade (red) the side that contains (0, 0).” The graph shows a dashed line graphed on an x y-coordinate plane with the area above it shaded. The second set of words says, “Graph x -2 y is less than -4, by graphing x – 2y = -4 and testing a point. The intercepts are x = -4 and y = 2 and the boundary line will be dashed. Choose a test point in the solution and verify that it is a solution to both inequalities.” The graph shows to lines graphed. One is dashed, and the area above it is shaded. The other line is solid, and the area above it is shaded. There is an area where the shaded areas overlap.">
<tbody>
<tr>
<td data-valign="top" style="width: 501px;">Graph \(y&gt;\frac{1}{2}x-4\) by graphing \(y=\frac{1}{2}x-4\) using the slope \(m=\frac{1}{2}\) and the intercept <em>b</em> = −4. The boundary line will be dashed.<br/>
Test (0, 0). It makes the inequality true. So, shade the side that contains (0, 0) red.</td>
<td data-valign="top" style="width: 437px;"><img alt="." src="https://data.ulearngo.com/assets/dae4a128-30a7-4ed7-b3fe-8fc2031663a7"/></td>
</tr>
<tr>
<td data-valign="top" style="width: 501px;">Graph \(x-2y&lt;-4\) by graphing \(x-2y=-4\) and testing a point.<br/>
The intercepts are <em>x</em> = −4 and <em>y</em> = 2 and the boundary line will be dashed.<br/>
Choose a test point in the solution and verify that it is a solution to both inequalities.</td>
<td data-valign="top" style="width: 437px;"><img alt="." src="https://data.ulearngo.com/assets/d0b639c9-ae8c-40a2-8ea9-ca8deac86e26"/></td>
</tr>
</tbody>
</table>
<p>No point on the boundary lines is included in the solution as both lines are dashed.</p>
<p>The solution is the region that is shaded twice, which is also the solution to \(x-2y&lt;-4\).</p>
</div>
</div>
</div>
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Solving a System of Linear Inequalities By Graphing

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Solving a System of Linear Inequalities By Graphing

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