Algebra Tutorials!
Wednesday 29th of June
Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# 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: 3|5x| ≥ 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. 3|5x| ≥ 60  |5x| ≥ 20 |w| ≥ 20 w ≤ -20 or w ≥ 20 5x ≤ -20 or 5x ≥ 20   x ≤ -4 or x ≥ 4

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 3|5x| ≥ 3|5(-6)| ≥ 3|-30| ≥ 3 Â· 30 ≥ 90 ≥ 6060 ? 60 ? 60 ? 60 ? Yes Is Is Is Is 3|5x| ≥ 3|5(7)| ≥ 3|35| ≥ 3 Â· 35 ≥ 105 ≥ 6060 ? 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.