Activity
Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.
Here are four inequalities. Study each inequality assigned to your group and work together to:
- Use the graphing tool or technology outside the course to graph each inequality. (The Desmos tool is provided).
- Find some coordinate pairs that represent solutions to the inequality.
- Find some coordinate pairs that do not represent solutions.
- Plot both sets of points. Either use two different colors or two different symbols like X and O.
- Plot enough points until you start to see the region that contains solutions and the region that contains non-solutions. Look for a pattern describing the region where solutions are plotted.
Compare your answer:
Your answer may vary, but here are some samples.
The blue points are solutions and yellow points are not solutions.
Compare your answer:
Your answer may vary, but here are some samples.
The blue points are solutions and yellow points are not solutions.
Compare your answer:
Your answer may vary, but here are some samples.
The blue points are solutions and yellow points are not solutions.
Compare your answer:
Your answer may vary, but here are some samples.
The blue points are solutions and yellow points are not solutions.
Video: Finding Solutions to Inequalities on the Coordinate Plane
Watch the following video to learn more about solutions to inequalities on the coordinate plane.
Self Check
Additional Resources
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.
Linear Inequality
A linear inequality is an inequality that can be written in one of the following forms:
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 the figure below.
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.
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 and .
Example
Determine whether each ordered pair is a solution to the inequality :
Solution
1. No; Substitute 0 for and 0 for .
2. Yes; Substitute 1 for and 6 for .
3. No; Substitute 2 for and 6 for .
4. No; Substitute -5 for and -15 for .
5. Yes; Substitute -8 for and 12 for .
Try it
Try It: Verify Solutions to an Inequality in Two Variables
Determine whether each ordered pair is a solution to the inequality .
Yes
Substitute 0 for and 0 for .
Yes
Substitute 4 for and 9 for .
Yes
Substitute -2 for and 1 for .
Yes
Substitute -5 for and -3 for .
No , so this ordered pair is not a solution.
Substitute 5 for and 1 for . , so this ordered pair is not a solution. , so this ordered pair is not a solution.