Here are the steps to follow when solving absolute value inequalities:
(quantity inside absolute value) < (number on other
side)
OR
(quantity inside absolute value) > (number on other side)
The same setup is used for a ³
sign.
If your absolute value is less than a number, then set up a threepart compound inequality that looks like this:
(number on other side) < (quantity inside absolute value) < (number on other side)
The same setup is used for a £
sign
This process can be a little confusing at first, so be
patient while learning how to do these problems. Let’s look at some examples.
Example 1: x + 4  6 < 9
Step 1: Isolate the absolute value 
x + 4 < 15 
Step 2: Is the number on the other side negative?  No, it’s a positive number, 15. We’ll move on to step 3. 
Step 3: Set up a compound inequality 
15 < x + 4 < 15 
Step 4: Solve the compound inequality 

Example 2: 2x – 1  7 ³
3
Step 1: Isolate the absolute value 
2x – 1 ³ 4 
Step 2: Is the number on the other side a negative number?  No, it’s a positive number, 4. We’ll move on to step 3. 
Step 3: Set up a compound inequality 
2x – 1 £ 4 or 2x – 1 ³ 4 
Step 4: Solve the inequalities 
2x £ 3 or 2x ³ 5 x £ 3/2 or x ³ 5/2 
Example 3: 5x + 6 + 4 < 1
Step 1: Isolate the absolute value 
5x + 6 < 3 
Step 2: Is the number on the other side a negative number?  Yes, it’s a negative number, 3.
We’ll look at the signs of each side of the inequality to determine the solution to the problem: 5x + 6 < 3 positive < negative This statement is never true, so there is no solution to this problem. 
Example 4: 3x – 4 + 9 > 5
Step 1: Isolate the absolute value 
3x – 4 > 4 
Step 2: Is the number on the other side a negative number?  Yes, it’s a negative number, 4.
We’ll look at the signs of each side of the inequality to determine the solution to the problem: 3x – 4 > 4 positive > negative This statement is always true, so the solution to the problem is All Real Numbers 