Graphs and Functions
Graph Linear Inequalities in Two Variables
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 in two variables
- Solve applications using linear inequalities in two variables
Before you get started, take this readiness quiz.
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.
A linear inequality is an inequality that can be written in one of the following forms:
Where A and B are not both zero.
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 (Figure).
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.
An ordered pair is a solution to 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 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 (Figure).
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 a is included in the solution:
Similarly, for an inequality in two variables,