Graph Linear Inequalities

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

Example 4.72

How to Graph Linear Inequalities

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

Solution

Step 1. Identify and graph the boundary line.

  • If the inequality is \(\leq\) or \(\geq\), the boundary line is solid.
  • If the inequality is \( < \) or \(>\), the boundary line is dashed.

Replace the inequality sign with an equal sign to find the boundary line.

Graph the boundary line \(y=\frac{3}{4} x-2\)
The inequality sign is \(\geq\). so we draw a solid line.

 

Graph the boundary line y=3/4x−2
Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality? We'll test \((0,0)\).
Is it a solution of the inequality?		 \begin{array}{l}
\text { At }(0,0) \text { , is } \begin{aligned}
y & \geq \frac{3}{4} x-2 ? \\
& 0 \geq \frac{3}{4}(0)-2 \\
0 & \geq-2
\end{aligned}\\
\text { So, }(0,0) \text { is a solution. }
\end{array}
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.
 The test point \((0,0)\), is a solution to \(y \geq \frac{3}{4} x-2\). So
we shade in that side.		 Shade in one side of the boundary line.
Try It 4.143

Graph the linear inequality \(y \geq \frac{5}{2} x-4\).

Try It 4.144

Graph the linear inequality \(y < \frac{2}{3} x-5\).

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

HOW TO

Graph a linear inequality.
  1. Step 1. Identify and graph the boundary line.
    • If the inequality is \(\leq\) or \(\geq\), the boundary line is solid.
    • If the inequality is \( < \) or \(>\), the boundary line is dashed.
  2. Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  3. 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.

Example 4.73

Graph the linear inequality \(y≤−4x\).

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

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

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

\(\begin{align} \begin{array}{r} 0-2(0) \stackrel{?}{ < }5 \\ 0-0\stackrel{?}{ < }5 \\ 0 < 5 \end{array} \end{align}\)

The point \((0, 0)\) is a solution of \(x−2y < 5\), so we shade in that side of the boundary 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
Try It 4.145

Graph the linear inequality \( 2x−3y≤6\).

Try It 4.146

Graph the linear inequality \(2x−y > 3\).

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

Example 4.74

Graph the linear inequality \(y≤−4x\).

Solution

First we graph the boundary line \(y=-4 x .\) It is in slope-intercept form, with \(m=-4\) and \(b=0\). The inequality is \(\leq\) 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

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

Is \((1,0)\) a solution of \(y \leq-4 x ?\)

\(0\stackrel{?}{x}-4(1)\)

\(0 \nleq-4\)

The point \((1,0)\) is not a solution to \(y≤−4x\), so we shade in the opposite side of the boundary line. See Figure 4.35.


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 i

Figure 4.35

Try It 4.147

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

Try It 4.148

Graph the linear inequality \(y \geq −2x\).

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?

\(\begin{aligned} x&=a \quad \quad \text{vertical line} \\ y&=b \quad \quad \text{horizontal line}\end{aligned} \)

Example 4.75

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 \((0,0)\).

\( \begin{array}{l} y>3\\ 0 \ngtr 3 \end{array} \)

\((0,0)\) 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 a

Try It 4.149

Graph the linear inequality \(y < 5\).

Try It 4.150

Graph the linear inequality \(y \leq −1\).

Back to '5.9: Graphing Linear Inequality of Two Variables on the Coordinate Plane'