# Graphs

## Graphs of 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.
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 .
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.

### 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  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 .
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.
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 
Now, let’s take a look at what we found in . We’ll start by graphing the line , and then we’ll plot the five points we tested. See .
In  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 .
The graph of the inequality  is shown in  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.

### 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.
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.
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?

### Key Concepts

1. To Graph a Linear Inequality



### 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.
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Graph Linear Inequalities
In the following exercises, graph each linear inequality.

### Everyday Math


### Writing Exercises


### 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?
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.
Find Solutions to a Linear Equation in Two Variables
In the following exercises, find three solutions to each linear equation.

### Graphing Linear Equations

Recognize the Relation Between the Solutions of an Equation and its Graph
In the following exercises, for each ordered pair, decide:
1. ⓐ Is the ordered pair a solution to the equation?
2. ⓑ Is the point on the line?
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.

### Graphing with Intercepts

Identify the 
In the following exercises, find the x- and y-intercepts.
Find the 
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 
In the following exercises, find the slope of each line shown.
Find the Slope of Horizontal and Vertical Lines
In the following exercises, find the slope of each line.
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.
Solve Slope Applications
In the following exercises, solve these slope applications.

### 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 .
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.
Graph and Interpret Applications of Slope–Intercept
Use Slopes to Identify Parallel Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.
Use Slopes to Identify Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

### Find the Equation of a Line

Find an Equation of the Line Given the Slope and 
In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.
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.
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.
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.
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.

### 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.
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Graph Linear Inequalities
In the following exercises, graph each linear inequality.

### Practice Test

Find the slope of each line shown.
Graph the line for each of the following equations.
Find the equation of each line. Write the equation in slope–intercept form.
Graph each linear inequality.