Solving Absolute Value Inequalities
Solving an Absolute Value Inequality of the
Form  x > a
Principle
Absolute Value Inequalities of the Form  x > a and  x ≥ a
Let a represent a positive real number.
â€¢ If x > a, then x < a or x > a. 

â€¢ If x ≥ a, then x
≤ a or x ≥ a. 

â€¢ If x > 0, then the solution is all real numbers, except 0.
â€¢ If x ≥ 0, then the solution is all real numbers.
â€¢ If x > a, then the solution is all real numbers.
â€¢ If x ≥ a, then the solution is all real numbers.
Note:
The absolute value of a number or expression is always
greater than a negative number.
Next, letâ€™s solve some absolute value inequalities of the form x > a and
x ≥ a.
Example 1
Solve: 35x ≥ 60.
Solution
Step 1 Isolate the absolute value.
Divide both sides by 3.
Step 2 Make the substitution w = 5x.
Step 3 Use the Absolute Value
Principle to solve for w.
Step 4 Replace w with 5x.
Step 5 Solve for x.
Divide each side by 5. 
35x
≥ 60
5x ≥ 20
w ≥ 20
w ≤ 20 or w ≥ 20
5x ≤ 20 or 5x ≥ 20
x ≤ 4 or x ≥ 4 


Step 6 Check the answer.
There are infinitely many solutions. It is a good idea to check one
number from each part of the solution. Letâ€™s check x = 6 and
x = 7.
Check x = 6 
Check x = 7 
Is
Is
Is
Is 
35x ≥
35(6) ≥
330 ≥
3 Â· 30 ≥
90 ≥ 
60 60 ?
60 ?
60 ?
60 ? Yes 
Is
Is
Is
Is 
35x ≥
35(7) ≥
335 ≥
3 Â· 35 ≥
105 ≥ 
60 60 ?
60 ?
60 ?
60 ? Yes 
So, the solution is x ≤ 4 or x
≥ 4.
Note:
You may be tempted to write x ≤ 4 or x
≥ 4
as
4
≥ x
≥ 4.
However, this implies that 4 is greater
than 4, which is false.
We cannot write x ≤ 4 or x
≥ 4 as a
single compound inequality.
