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Graphs of Linear Inequalities

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 \(y \leq −1\).