Solving Absolute Value Inequalities
Solving an Absolute Value Inequality of the Form
 x < a
Solving an absolute value inequality is similar to solving an absolute value
equation.
First, letâ€™s solve the equation x = 5.
The solution is all numbers 5 units from 0 on the number line.
So, the solutions are x = 5 and x = 5.
Now, letâ€™s solve the inequality x = 5.
The solution is all numbers that are less than 5 units from 0.
We can show this on a number line by shading between 5 and 5.
We use an open circle, o, at 5 and 5 to indicate that they are not part of
the solution.
The solution of x < 5 can be written as a compound inequality:
5 < x < 5
The inequality x < 5 has infinitely many solutions. Letâ€™s check three of
them.
Check x = 3
Is 3 < 5 ?
Is 3 < 5? Yes 
Check x = 0
Is 0 < 5 ?
Is 0 < 5? Yes 
Check x = 4.5
Is 4.5 < 5 ?
Is 4.5 < 5? Yes 
Here is the general principle for absolute value inequalities.
Principle
Absolute Value Inequalities of the Form  x < a and  x ≤ a
Let a represent a positive real number.
â€¢ If x < a, then a < x < a. 

â€¢ If x ≤ a, then a
≤ x ≤
a. 

â€¢ If x < 0, then there is no solution.
If x ≤ 0, then the solution is x = 0.
â€¢ If x < a, then there is no solution.
If x ≤ a, then there is no solution.
Note:
The absolute value of a number or an expression cannot be
negative.
