3.4 Graph Linear Inequalities in Two Variables

Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at linear inequalities in two variables which are very similar to linear equations in two variables.

Linear 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 made a profit.

Recall that an inequality with one variable had many solutions. For example, 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 3.10.

Similarly, linear inequalities in two variables have many solutions. Any ordered pair $\left(x,y\right)$ that makes an inequality true when we substitute in the values is a solution to a linear inequality.

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 shown previously 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 3.11.

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:

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

$$\begin{array}{cccccc}Ax+By<C\hfill & & & & & Ax+By\le C\hfill \\ Ax+By>C\hfill & & & & & Ax+By\ge C\hfill \\ \text{Boundary line is}Ax+By=C\hfill & & & & & \text{Boundary line is}Ax+By=C\hfill \\ \text{Boundary line is not included in solution.}\hfill & & & & & \text{Boundary line is included in solution.}\hfill \\ \mathbf{\text{Boundary line is dashed.}}\hfill & & & & & \mathbf{\text{Boundary line is solid.}}\hfill \end{array}$$

Now, let’s take a look at what we found in Example 1. We’ll start by graphing the line $y=x+4,$ and then we’ll plot the five points we tested, as shown in the graph. See Figure 3.12.

In Example 1 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 $\left(1,6\right)$ and $\left(\mathrm{-8},12\right)$ 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 $\left(0,0\right)$ and $\left(\mathrm{-5},\mathrm{-15}\right)$ 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 $\left(2,6\right)?$ Because $6=2+4,$ the point is a solution to the equation $y=x+4,$ but not a solution to the inequality $y>x+4.$ So the point $\left(2,6\right)$ is on the boundary line.

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

$$\begin{array}{ccc}\hfill y& >\hfill & x+4\hfill \\ \hfill 10& \stackrel{?}{>}\hfill & 0+4\hfill \\ \hfill 10& >\hfill & 4\hfill \end{array}$$

So, $\left(0,10\right)$ is a solution to $y>x+4.$

Any point you choose above the boundary line is a solution to the inequality $y>x+4.$ All points above the boundary line are solutions.

Similarly, all points below the boundary line, the side with $\left(0,0\right)$ and $\left(\mathrm{-5},\mathrm{-15}\right),$ are not solutions to $y>x+4,$ as shown in Figure 3.13.

The graph of the inequality $y>x+4$ is shown in 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.

Graph Linear Inequalities in Two Variables

Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.

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

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.

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.

Recall that:

$$\begin{array}{cccccc}x=a\hfill & & & & & \text{vertical line}\hfill \\ y=b\hfill & & & & & \text{horizontal line}\hfill \end{array}$$

Solve Applications using Linear Inequalities in Two Variables

Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

Key Concepts

• How to graph a linear inequality in two variables.
  1. Identify and graph the boundary line.
     If the inequality is $\le \text{or}\ge ,$ the boundary line is solid.
     If the inequality is $<\text{or}>,$ the boundary line is dashed.
  2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  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.

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.

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

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

Write the inequality shown by the graph with the boundary line $y=3x-4.$

Write the inequality shown by the graph with the boundary line $y=2x-4.$

Write the inequality shown by the graph with the boundary line $y=\frac{1}{2}x+1.$

Write the inequality shown by the graph with the boundary line $y=-\frac{1}{3}x-2.$

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

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

Write the inequality shown by the shaded region in the graph with the boundary line $3x-y=6.$

Write the inequality shown by the shaded region in the graph with the boundary line $2x-y=4.$

Graph Linear Inequalities in Two Variables

In the following exercises, graph each linear inequality.

Solve Applications using Linear Inequalities in Two Variables

ⓐ $11x+16.5y\ge 330$
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ⓒ Answers will vary.

ⓐ $15x+10y\ge 500$
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ⓒ Answers will vary.

Writing Exercises

Self Check

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

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?