Graphs

# 38 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

Before you get started, take this readiness quiz.

### 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.

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

where are not both zero.

Do you remember that an inequality with one variable had many solutions? The solution to the inequality 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).

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

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

Determine whether each ordered pair is a solution to the inequality :

ⓐ ⓑ ⓒ ⓓ ⓔ

- ⓐ

Simplify.

So, is not a solution to . - ⓑ

Simplify.

So, is a solution to . - ⓒ

Simplify.

So, is not a solution to . - ⓓ

Simplify.

So, is not a solution to . - ⓔ

Simplify.

So, is a solution to .

Determine whether each ordered pair is a solution to the inequality :

ⓐ ⓑ ⓒ ⓓ ⓔ

ⓐ yes ⓑ yes ⓒ yes ⓓ yes ⓔ no

Determine whether each ordered pair is a solution to the inequality :

ⓐ ⓑ ⓒ ⓓ ⓔ

ⓐ yes ⓑ yes ⓒ no ⓓ no ⓔ yes

### 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) again. The point 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).

The solution to is the shaded part of the number line to the right of .

Similarly, the line separates the plane into two regions. On one side of the line are points with . On the other side of the line are the points with . We call the line a boundary line.

The line with equation is the boundary line that separates the region where from the region where .

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

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 (Figure)

Boundary line is not included in solution. | Boundary line is included in solution. |

Boundary line is dashed. |
Boundary line is solid. |

Now, let’s take a look at what we found in (Figure). We’ll start by graphing the line , and then we’ll plot the five points we tested. See (Figure).

In (Figure) we found that some of the points were solutions to the inequality and some were not.

Which of the points we plotted are solutions to the inequality ? The points and are solutions to the inequality . Notice that they are both on the same side of the boundary line .

The two points and are on the other side of the boundary line , and they are not solutions to the inequality . For those two points, .

What about the point ? Because , the point is a solution to the equation . So the point 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 . The point 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 . All points on the left are solutions.

Similarly, all points on the right side of the boundary line, the side with and , are not solutions to . See (Figure).

The graph of the inequality is shown in (Figure) below. The line divides the plane into two regions. The shaded side shows the solutions to the inequality .

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

The boundary line shown is . Write the inequality shown by the graph.

The line is the boundary line. On one side of the line are the points with and on the other side of the line are the points with .

Let’s test the point and see which inequality describes its side of the boundary line.

At , which inequality is true:

Since, is true, the side of the line with , is the solution. The shaded region shows the solution of the inequality .

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality .

We could use any point as a test point, provided it is not on the line. Why did we choose ? Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that .

Write the inequality shown by the graph with the boundary line .

Write the inequality shown by the graph with the boundary line .

The boundary line shown is . Write the inequality shown by the graph.

The line is the boundary line. On one side of the line are the points with and on the other side of the line are the points with .

Let’s test the point and see which inequality describes its side of the boundary line.

At , which inequality is true:

So the side with is the side where .

(You may want to pick a point on the other side of the boundary line and check that .)

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The graph shows the solution to the inequality .

Write the inequality shown by the shaded region in the graph with the boundary line .

Write the inequality shown by the shaded region in the graph with the boundary line .

### Graph Linear Inequalities

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

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

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

- Identify and graph the boundary line.
- If the inequality is , 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.

Graph the linear inequality .

First we graph the boundary line . The inequality is so we draw a dashed line.

Then we test a point. We’ll use again because it is easy to evaluate and it is not on the boundary line.

Is a solution of ?

The point is a solution of , so we shade in that side of the boundary line.

Graph the linear inequality .

Graph the linear inequality .

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

Graph the linear inequality .

First we graph the boundary line . It is in slope–intercept form, with . The inequality is so we draw a solid line.

Now, we need a test point. We can see that the point is not on the boundary line.

Is a solution of ?

The point is not a solution to , so we shade in the opposite side of the boundary line. See (Figure).

Graph the linear inequality .

Graph the linear inequality .

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?

Graph the linear inequality .

First we graph the boundary line . It is a horizontal line. The inequality is > so we draw a dashed line.

We test the point .

is not a solution to .

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

Graph the linear inequality .

Graph the linear inequality .

### Key Concepts

**To Graph a Linear Inequality**- Identify and graph the boundary line.

If the inequality is , 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.

- Identify and graph the boundary line.

### Section 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.

Determine whether each ordered pair is a solution to the inequality :

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Determine whether each ordered pair is a solution to the inequality :

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

ⓐ yes ⓑ no ⓒ no ⓓ yes ⓔ no

Determine whether each ordered pair is a solution to the inequality :

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Determine whether each ordered pair is a solution to the inequality :

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

ⓐ yes ⓑ no ⓒ no ⓓ yes ⓔ yes

Determine whether each ordered pair is a solution to the inequality :

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Determine whether each ordered pair is a solution to the inequality :

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

ⓐ no ⓑ no ⓒ no ⓓ yes ⓔ yes

**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

Write the inequality shown by the graph with the boundary line

Write the inequality shown by the graph with the boundary line

Write the inequality shown by the graph with the boundary line

Write the inequality shown by the shaded region in the graph with the boundary line

Write the inequality shown by the shaded region in the graph with the boundary line

Write the inequality shown by the shaded region in the graph with the boundary line

Write the inequality shown by the shaded region in the graph with the boundary line

Write the inequality shown by the shaded region in the graph with the boundary line

Write the inequality shown by the shaded region in the graph with the boundary line

Write the inequality shown by the shaded region in the graph with the boundary line

Write the inequality shown by the shaded region in the graph with the boundary line

**Graph Linear Inequalities**

In the following exercises, graph each linear inequality.

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

#### Everyday Math

**Money.** Gerry wants to have a maximum of ?100 cash at the ticket booth when his church carnival opens. He will have ?1 bills and ?5 bills. If *x* is the number of ?1 bills and *y* is the number of ?5 bills, the inequality models the situation.

- ⓐ Graph the inequality.
- ⓑ List three solutions to the inequality where both
*x*and*y*are integers.

**Shopping.** Tula has ?20 to spend at the used book sale. Hardcover books cost ?2 each and paperback books cost ?0.50 each. If *x* is the number of hardcover books Tula can buy and *y* is the number of paperback books she can buy, the inequality models the situation.

- ⓐ Graph the inequality.
- ⓑ List three solutions to the inequality where both
*x*and*y*are whole numbers.

- ⓐ

- ⓑ Answers will vary.

#### Writing Exercises

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.

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.

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

### Chapter 4 Review Exercises

#### Rectangular Coordinate System

**Plot Points in a Rectangular Coordinate System**

In the following exercises, plot each point in a rectangular coordinate system.

ⓐ

ⓑ

ⓒ

ⓓ

ⓐ

ⓑ

ⓒ

ⓓ

ⓐ

ⓑ

ⓒ

ⓓ

ⓐ

ⓑ

ⓒ

ⓓ

**Identify Points on a Graph**

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

ⓐ ⓑ ⓒ ⓓ

**Verify Solutions to an Equation in Two Variables**

In the following exercises, which ordered pairs are solutions to the given equations?

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

a, b

**Complete a Table of Solutions to a Linear Equation in Two Variables**

In the following exercises, complete the table to find solutions to each linear equation.

0 | ||

1 | ||

0 | ||

4 | ||

0 | 3 | |

4 | 1 | (4, 1) |

4 |

0 | ||

1 | ||

0 | ||

0 | ||

0 | ||

2 | 0 | |

**Find Solutions to a Linear Equation in Two Variables**

In the following exercises, find three solutions to each linear equation.

Answers will vary.

Answers will vary.

#### Graphing Linear Equations

**Recognize the Relation Between the Solutions of an Equation and its Graph**

In the following exercises, for each ordered pair, decide:

- ⓐ Is the ordered pair a solution to the equation?
- ⓑ Is the point on the line?

(3, 1)

(6, 4)

ⓐ yes; yes ⓑ yes; no

**Graph a Linear Equation by Plotting Points**

In the following exercises, graph by plotting points.

**Graph Vertical and Horizontal lines**

In the following exercises, graph each equation.

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

and

and

#### Graphing with Intercepts

**Identify the x– and y-Intercepts on a Graph**

In the following exercises, find the *x*– and *y*-intercepts.

**Find the x– and y-Intercepts from an Equation of a Line**

In the following exercises, find the intercepts of each equation.

**Graph a Line Using the Intercepts**

In the following exercises, graph using the intercepts.

#### Slope of a Line

**Use Geoboards to Model Slope**

In the following exercises, find the slope modeled on each geoboard.

In the following exercises, model each slope. Draw a picture to show your results.

**Use to find the Slope of a Line from its Graph**

In the following exercises, find the slope of each line shown.

1

**Find the Slope of Horizontal and Vertical Lines**

In the following exercises, find the slope of each line.

undefined

0

**Use the Slope Formula to find the Slope of a Line between Two Points**

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

**Graph a Line Given a Point and the Slope**

*In the following exercises, graph each line with the given point and slope.*

;

;

*x*-intercept ;

*y*-intercept 1;

**Solve Slope Applications**

In the following exercises, solve these slope applications.

The roof pictured below has a rise of 10 feet and a run of 15 feet. What is its slope?

A mountain road rises 50 feet for a 500-foot run. What is its slope?

#### Intercept Form of an Equation of a Line

**Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line**

In the following exercises, use the graph to find the slope and *y*-intercept of each line. Compare the values to the equation .

slope and *y*-intercept

**Identify the Slope and y-Intercept from an Equation of a Line**

In the following exercises, identify the slope and *y*-intercept of each line.

**Graph a Line Using Its Slope and Intercept**

In the following exercises, graph the line of each equation using its slope and *y*-intercept.

In the following exercises, determine the most convenient method to graph each line.

horizontal line

intercepts

plotting points

**Graph and Interpret Applications of Slope–Intercept**

Katherine is a private chef. The equation models the relation between her weekly cost, *C*, in dollars and the number of meals, *m*, that she serves.

- ⓐ Find Katherine’s cost for a week when she serves no meals.
- ⓑ Find the cost for a week when she serves 14 meals.
- ⓒ Interpret the slope and
*C*-intercept of the equation. - ⓓ Graph the equation.

Marjorie teaches piano. The equation models the relation between her weekly profit, *P*, in dollars and the number of student lessons, *s*, that she teaches.

- ⓐ Find Marjorie’s profit for a week when she teaches no student lessons.
- ⓑ Find the profit for a week when she teaches 20 student lessons.
- ⓒ Interpret the slope and
*P*–intercept of the equation. - ⓓ Graph the equation.

ⓐ −?250 ⓑ ?450 ⓒ The slope, 35, means that Marjorie’s weekly profit, *P*, increases by ?35 for each additional student lesson she teaches. The *P*–intercept means that when the number of lessons is 0, Marjorie loses ?250. ⓓ

**Use Slopes to Identify Parallel Lines**

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.

not parallel

**Use Slopes to Identify Perpendicular Lines**

In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

perpendicular

#### Find the Equation of a Line

**Find an Equation of the Line Given the Slope and y-Intercept**

In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.

slope and

slope and

slope and

slope and

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

**Find an Equation of the Line Given the Slope and a Point**

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

, point

, point

Horizontal line containing

, point

**Find an Equation of the Line Given Two Points**

In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.

and

and

and .

and

**Find an Equation of a Line Parallel to a Given Line**

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

line , point

line , point

line , point

line , point

**Find an Equation of a Line Perpendicular to a Given Line**

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

line , point

line , point

line , point

line point

#### Graph Linear Inequalities

**Verify Solutions to an Inequality in Two Variables**

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

Determine whether each ordered pair is a solution to the inequality :

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Determine whether each ordered pair is a solution to the inequality :

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

ⓐ yes ⓑ no ⓒ yes ⓓ yes ⓔ no

**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 .

Write the inequality shown by the graph with the boundary line .

Write the inequality shown by the shaded region in the graph with the boundary line .

Write the inequality shown by the shaded region in the graph with the boundary line

**Graph Linear Inequalities**

In the following exercises, graph each linear inequality.

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

Graph the linear inequality .

### Practice Test

Plot each point in a rectangular coordinate system.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Which of the given ordered pairs are solutions to the equation ?

ⓐ

ⓑ

ⓒ

ⓐ yes ⓑ yes ⓒ no

Find three solutions to the linear equation .

Find the *x*– and *y*-intercepts of the equation .

Find the slope of each line shown.

undefined

Find the slope of the line between the points and .

Graph the line with slope containing the point .

Graph the line for each of the following equations.

Find the equation of each line. Write the equation in slope–intercept form.

slope and *y*-intercept

, point

containing and

parallel to the line , containing the point

perpendicular to the line , containing the point

Write the inequality shown by the graph with the boundary line .

Graph each linear inequality.

### Glossary

- boundary line
- The line with equation that separates the region where from the region where .

- linear inequality
- An inequality that can be written in one of the following forms:

where are not both zero.

- solution of a linear inequality
- An ordered pair is a solution to a linear inequality the inequality is true when we substitute the values of
*x*and*y*.