Solving Absolute Value Inequalities
Solving an Absolute Value Inequality of the
Form  x > a
Example 1
Solve: 10 + 42x  7 > 50
Solution
Step 1 Isolate the absolute value.
Add 10 to both sides.
Divide both sides by 4.
Step 2 Make the substitution w = 2x  7.
Step 3 Use the Absolute Value
Principle to solve for w. Step 4 Replace w with 2x  7.
Step 5 Solve for x.
Add 7 to both sides.
Divide both sides by 2.
Step 6 Check the answer.
We leave the check to you.
So, the solution is x < 4 or x
> 11. 
10 + 42x  7 > 50
42x  7 > 60
2x  7 > 15
w > 15
w < 15 or w > 15
2x  7 < 15 or 2x  7 > 15
2x < 8 or 2x > 22
x < 4 or x > 11 
Example 2
Solve: 5x  12 < 28
Solution Step 1 Isolate the absolute value.
Add 12 to both sides.
Divide both sides by 5 and reverse the
direction of the inequality symbol. 
5x  12 < 28
5x < 40
x > 8 
Since x represents a nonnegative number, it is greater than
8 for all
values of x.
Therefore, the solution is all real numbers.
