Solving Absolute Value Inequalities
Solving an Absolute Value Inequality of the Form
 x < a
Example 1
Solve: 64x > 72
Solution
Step 1 Isolate the absolute value.
Divide both sides by 6 and reverse
the direction of the inequality symbol.
Simplify.
Step 2 Make the substitution w = 4x.
Step 3 Use the Absolute Value
Principle to solve for w.
Step 4 Replace w with 4x.
Replace w with 4x.
Step 5 Solve for x.
Divide all three parts by 4. 
64x > 72
4x < 12
w < 12
12 < w < 12
12 < 4x < 12
3 < x < 3 
Notes:
1) When multiplying or dividing both sides of
an inequality by a negative number,
reverse the direction of the inequality.
2) The compound inequality 3 < x < 3 is
read â€œ3 is less than xâ€ and â€œx is less
than 3â€.
Step 6 Check the answer.
We leave the check to you.
So, the solution is 3 < x
< 3.
Example 2
Solve: 4 + 52x
 4 < 36
Solution
Step 1 Isolate the absolute value.
Add 4 to both sides.
Divide both sides by 5.
Step 2 Make the substitution w = 2x
 4.
Step 3 Use the Absolute Value
Principle to solve for w.
Step 4 Replace w with 2x  4.
Step 5 Solve for x.
Add 4 to all three parts.
Divide each part by 2. 
4 + 52x
 4 < 36
52x  4 < 40
2x  4 < 8
w < 8
8 < w <
8
8 < 2x 
4 < 8
4 < 2x < 12
2 < x <
6 
Step 6 Check the answer.
We leave the check to you.
So, the solution is 2 < x
< 6.
Note:
8 < 2x  4
< 8 is a compound inequality since it contains
two inequality symbols.
A compound inequality is solved when the
variable has been isolated in the middle
part.
For example, 2 < x
< 6 is solved.
Remember, absolute value always represents a nonnegative number.
Therefore, some absolute value inequalities have no solution, as the next
example shows.
Example 3
Solve: x + 6 < 4
Solution
Step 1 Isolate the absolute value.
Subtract 6 from both sides. 
x + 6 < 4
x < 2 
For any value of x, we know that x is a nonnegative number. Thus,
x cannot be less than 2.
Therefore, x + 6 < 4 has no solution.
