You might already be familiar with solving inequalities, but what happens when you add an absolute value expression into the mix? Let’s go through the steps it takes to solve absolute value inequalities.
The absolute value symbol looks like this: |a|, where a is any real number. The absolute value sign means that the expression is no longer just a, but the distance a is from 0 on a number line.
A quick example: |2| = 2, because positive 2 is a distance of 2 from 0 on a number line.
Similarly, |-2| = 2, because negative 2 is also a distance of 2 from 0 on a number line.
What Are Absolute Value Inequalities?
So what are absolute value equations and absolute value inequalities? We know that equations have equal signs in them and that inequalities always contain one of the four inequality signs:
- Greater than: >
- Less than: <
- Greater than or equal than: ≥
- Less than or equal than: ≤
So, an absolute value equation is an algebraic expression that contains both an equal sign as well as an absolute value expression, such as 3 = |x| + 1.
An absolute value inequality is an algebraic expression that contains both an inequality symbol as well as an absolute value expression, such as |x| - 2 > 3
Working With Absolute Value Inequalities
Solving absolute value inequalities is a lot like solving absolute value equations.
Let's take |x| - 2 > 3 as an example.
First, we move the absolute value expression (meaning, the absolute value sign and the variable inside it) to one side of the inequality, the same process for all linear equations that we wish to solve for x. Once we add 2 + 3, we end up with |x| > 5.
Then, you remove the absolute value bars by remembering that the same absolute value of a number can come from either a positive value or a negative value.
This generates two inequalities, or a compound inequality, because the solution set (set of numbers that satisfy the inequality) must account for the fact that anything in between the absolute value bars could be either a positive number or a negative number.
Our absolute value inequality is |x| > 5. This means that our compound inequality is x > 5 OR (-x) > 5. Both of those will satisfy the original absolute value inequality.
Now, you are left with two statements with one variable each. You no longer have to deal with any absolute value expressions. You can solve and graph normally.
Reviewing Absolute Value Inequalities
Remember that an absolute value inequality is just an inequality with an absolute value expression inside of it. The same absolute value of a number can come from either a positive or a negative value for the number.
To solve an absolute value inequality, remember these steps:
- Isolate the absolute value expression on one side of the inequality
- Create two inequality statements.
- Lastly, join the solutions for the two inequality statements together.
That’s it! Your next step is to graph the inequality