# Graphs and Functions

## 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  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, linear inequalities in two variables have many solutions. Any ordered pair  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  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 .
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 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.
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, as shown in the graph. 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  but not a solution to the inequality  So the point  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  The point  clearly looks to above the boundary line, doesn’t it? Is it a solution to the inequality?
So,  is a solution to 
Any point you choose above the boundary line is a solution to the inequality  All points above the boundary line are solutions.
Similarly, all points below the boundary line, the side with  and  are not solutions to  as shown in .
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 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  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:

### 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

1. How to graph a linear inequality in two variables.



### 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 Two Variables
In the following exercises, graph each linear inequality.
Solve Applications using Linear Inequalities in Two Variables

### 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?