Understanding Compound Inequalities
A compound inequality includes two inequalities in one statement. A statement such as means and . There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.Example 7: Solving a Compound Inequality
Solve the compound inequality: .Solution
The first method is to write two separate inequalities: and . We solve them independently.
Then, we can rewrite the solution as a compound inequality, the same way the problem began.
In interval notation, the solution is written as .
The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.
We get the same solution: .
Try It 7
Solve the compound inequality . SolutionExample 8: Solving a Compound Inequality with the Variable in All Three Parts
Solve the compound inequality with variables in all three parts: .Solution
Lets try the first method. Write two inequalities:
The solution set is or in interval notation . Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line.
![A number line with the points -4 and 5/6 labeled. Dots appear at these points and a line connects these two dots.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/09/25200410/CNX_CAT_Figure_02_07_003.jpg)