Graphing Linear Inequalities

Now, we’re ready to put all this together to graph linear inequalities.

Example: How to Graph Linear Inequalities

Graph the linear inequality\(y\ge \frac{3}{4}x-2\).

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Identify and graph the boundary line. If the inequality is less than or equal to or greater than or equal to, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. The text in the second cell reads: “Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y equals three-fourths x minus 2. The inequality sign is greater than or equal to, so we draw a solid line. The third cell contains the graph of the line three-fourths x minus 2 on a coordinate plane.

In the third row of the table, the first cell says: “Step 3. Shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. In the second cell, the instructions say: The test point (0, 0) is a solution to y is greater than or equal to three-fourths x minus 2. So we shade in that side.” In the third cell is the graph of the line three-fourths x minus 2 on a coordinate plane with the region above the line shaded.

The steps we take to graph a linear inequality are summarized here.

Graph a linear inequality.

Identify and graph the boundary line.
- If the inequality is \(\le \text{or}\ge \), the boundary line is solid.
- If the inequality is < or >, the boundary line is dashed.
Test a point that is not on the boundary line. Is it a solution of the inequality?
Shade in one side of the boundary line.
- If the test point is a solution, shade in the side that includes the point.
- If the test point is not a solution, shade in the opposite side.

Example

Graph the linear inequality \(x-2y<5\).

Solution

First we graph the boundary line \(x-2y=5\). The inequality is \(<\) so we draw a dashed line.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right.

Then we test a point. We’ll use \(\left(0,0\right)\) again because it is easy to evaluate and it is not on the boundary line.

Is \(\left(0,0\right)\) a solution of \(x-2y<5\)?

The figure shows the inequality 0 minus 2 times 0 in parentheses is less than 5, with a question mark above the inequality symbol. The next line shows 0 minus 0 is less than 5, with a question mark above the inequality symbol. The third line shows 0 is less than 5.

The point \(\left(0,0\right)\) is a solution of \(x-2y<5\), so we shade in that side of the boundary line.

What if the boundary line goes through the origin? Then we won’t be able to use \(\left(0,0\right)\) as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

Example

Graph the linear inequality \(y\le -4x\).

Solution

First we graph the boundary line \(y=-4x\). It is in slope–intercept form, with \(m=-4\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=0\). The inequality is \(\le \) so we draw a solid line.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line s y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right.

Now, we need a test point. We can see that the point \(\left(1,0\right)\) is not on the boundary line.

Is \(\left(1,0\right)\) a solution of \(y\le -4x\)?

The figure shows 0 is less than or equal to negative 4 times 1 in parentheses, with a question mark above the inequality symbol. The next line shows 0 is not less than or equal to negative 4.

The point \(\left(1,0\right)\) is not a solution to \(y\le -4x\), so we shade in the opposite side of the boundary line. See the figure below.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right. The point (1, 0) is plotted, but not labeled. The region to the left of the line is shaded.

Some linear inequalities have only one variable. They may have an x but no y, or a y but no x. In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember?

\(\begin{array}{cccc}x=a\hfill & & & \text{vertical line}\hfill \\ y=b\hfill & & & \text{horizontal line}\hfill \end{array}\)

Example

Graph the linear inequality \(y>3\).

Solution

First we graph the boundary line \(y=3\). It is a horizontal line. The inequality is > so we draw a dashed line.

We test the point \(\left(0,0\right)\).

\(\require{cancel}\begin{array}{} y>3\hfill \\ 0\cancel{>}3\hfill \end{array}\)

\(\left(0,0\right)\) is not a solution to \(y>3\).

So we shade the side that does not include (0, 0).

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.

This lesson is part of:

Graphs and Equations

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