4.7 Graphs of Linear Inequalities

Learning Objectives

By the end of this section, you will be able to:

  • Verify solutions to an inequality in two variables
  • Recognize the relation between the solutions of an inequality and its graph
  • Graph linear inequalities

Verify Solutions to an Inequality in Two Variables

We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. Inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business would make a profit.

Linear Inequality

A linear inequality is an inequality that can be written in one of the following forms:

where AandB are not both zero.

Do you remember that an inequality with one variable had many solutions? The solution to the inequality x>3 is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See Figure 4.30.

Similarly, inequalities in two variables have many solutions. Any ordered pair (x,y) that makes the inequality true when we substitute in the values is a solution of the inequality.

Solution of a Linear Inequality

An ordered pair (x,y) is a solution of a linear inequality if the inequality is true when we substitute the values of x and y.

Recognize the Relation Between the Solutions of an Inequality and its Graph

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in Figure 4.30 again. The point x=3 separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See Figure 4.31.

The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity. An arrow above the number line extends from 3 and points to the left. It is labeled “numbers less than 3.” An arrow above the number line extends from 3 and points to the right. It is labeled “numbers greater than 3.”
Figure 4.31

The solution to x>3 is the shaded part of the number line to the right of x=3.

Similarly, the line y=x+4 separates the plane into two regions. On one side of the line are points with y<x+4. On the other side of the line are the points with y>x+4. We call the line y=x+4 a boundary line.

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not a is included in the solution:

The figure shows two number lines. The number line on the left is labeled x is less than a. The number line shows a parenthesis at a and an arrow that points to the left. The number line on the right is labeled x is less than or equal to a. The number line shows a bracket at a and an arrow that points to the left.

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to indicate whether or not it the line is included in the solution. This is summarized in Table 4.48

Now, let’s take a look at what we found in Example 4.69. We’ll start by graphing the line y=x+4, and then we’ll plot the five points we tested. See Figure 4.32.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15).
Figure 4.32

In Example 4.69 we found that some of the points were solutions to the inequality y>x+4 and some were not.

Which of the points we plotted are solutions to the inequality y>x+4? The points (1,6) and (−8,12) are solutions to the inequality y>x+4. Notice that they are both on the same side of the boundary line y=x+4.

The two points (0,0) and (−5,−15) are on the other side of the boundary line y=x+4, and they are not solutions to the inequality y>x+4. For those two points, y<x+4.

What about the point (2,6)? Because 6=2+4, the point is a solution to the equation y=x+4. So the point (2,6) is on the boundary line.

Let’s take another point on the left side of the boundary line and test whether or not it is a solution to the inequality y>x+4. The point (0,10) clearly looks to be to the left of the boundary line, doesn’t it? Is it a solution to the inequality?

Any point you choose on the left side of the boundary line is a solution to the inequality y>x+4. All points on the left are solutions.

Similarly, all points on the right side of the boundary line, the side with (0,0) and (−5,−15), are not solutions to y>x+4. See Figure 4.33.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15). To the upper left of the line is the inequality y is greater than x plus 4. To the right of the line is the inequality y is less than x plus 4.
Figure 4.33

The graph of the inequality y>x+4 is shown in Figure 4.34 below. The line y=x+4 divides the plane into two regions. The shaded side shows the solutions to the inequality y>x+4.

The points on the boundary line, those where y=x+4, are not solutions to the inequality y>x+4, so the line itself is not part of the solution. We show that by making the line dashed, not solid.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as a dashed arrow extending from the bottom left toward the upper right. The coordinate plane to the upper left of the line is shaded.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as a dashed arrow extending from the bottom left toward the upper right. The coordinate plane to the upper left of the line is shaded.
Figure 4.34 The graph of the inequality y>x+4.

Graph Linear Inequalities

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

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

How To

Graph a linear inequality.

  • Step 1. Identify and graph the boundary line.
    • If the inequality is ≤or≥, the boundary line is solid.
    • If the inequality is < or >, the boundary line is dashed.
  • Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  • 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.

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.

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?

Section 4.7 Exercises

Practice Makes Perfect

Verify Solutions to an Inequality in Two Variables

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

504.Determine whether each ordered pair is a solution to the inequality y>x−1:ⓐ (0,1)
ⓑ (−4,−1)
ⓒ (4,2)
ⓓ (3,0)
ⓔ (−2,−3)

505.Determine whether each ordered pair is a solution to the inequality y>x−3:

ⓐ (0,0)
ⓑ (2,1)
ⓒ (−1,−5)
ⓓ (−6,−3)
ⓔ (1,0)

506.Determine whether each ordered pair is a solution to the inequality y<x+2:

ⓐ (0,3)
ⓑ (−3,−2)
ⓒ (−2,0)
ⓓ (0,0)
ⓔ (−1,4)

507.Determine whether each ordered pair is a solution to the inequality y<x+5:

ⓐ (−3,0)
ⓑ (1,6)
ⓒ (−6,−2)
ⓓ (0,1)
ⓔ (5,−4)

508.Determine whether each ordered pair is a solution to the inequality x+y>4:

ⓐ (5,1)
ⓑ (−2,6)
ⓒ (3,2)
ⓓ (10,−5)
ⓔ (0,0)

509.

Determine whether each ordered pair is a solution to the inequality x+y>2:

ⓐ (1,1)
ⓑ (4,−3)
ⓒ (0,0)
ⓓ (−8,12)
ⓔ (3,0)

Recognize the Relation Between the Solutions of an Inequality and its Graph

In the following exercises, write the inequality shown by the shaded region.

510.Write the inequality shown by the graph with the boundary line y=3x−4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3x minus 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the right of the line is shaded.

511.Write the inequality shown by the graph with the boundary line y=2x−4.

511.Write the inequality shown by the graph with the boundary line y=2x−4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2x minus 4 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

512.Write the inequality shown by the graph with the boundary line y=12x+1.

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 one-half x plus 1 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

513.Write the inequality shown by the graph with the boundary line y=−13x−2.

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 one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

514.Write the inequality shown by the shaded region in the graph with the boundary line x+y=5.

515.Write the inequality shown by the shaded region in the graph with the boundary line x+y=3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 3 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.

516.Write the inequality shown by the shaded region in the graph with the boundary line 2x+y=−4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus y equals negative 4 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

517.Write the inequality shown by the shaded region in the graph with the boundary line x+2y=−2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 2 y equals negative 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

518.Write the inequality shown by the shaded region in the graph with the boundary line 3xy=6.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.

519.Write the inequality shown by the shaded region in the graph with the boundary line 2xy=4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.

520.Write the inequality shown by the shaded region in the graph with the boundary line 2x−5y=10.

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

521.Write the inequality shown by the shaded region in the graph with the boundary line 4x−3y=12.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x minus 3 y equals 12 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

Graph Linear Inequalities

In the following exercises, graph each linear inequality.

Everyday Math

Writing Exercises

554.Lester thinks that the solution of any inequality with a > sign is the region above the line and the solution of any inequality with a < sign is the region below the line. Is Lester correct? Explain why or why not.

555.Explain why in some graphs of linear inequalities the boundary line is solid but in other graphs it is dashed.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “verify solutions to an inequality in two variables,”, “recognize the relation between the solutions of an inequality and its graph,” and “graph linear inequalities.” The rest of the cells are blank.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?