Solve Equations with Absolute Value

Solve equations with absolute value; including examples and questions with detailed solutions and explanations.

Review of Absolute Value

The rules you need to know in order to be able to solve the question in this tutorial.
1) | x | = 0 if x = 0
2) | x | = x if x > 0
3) | x | = - x if x < 0
4) The equation | x | = k with k < 0 has no real solutions.
5) The equation | x | = k , k ≥ 0 is equivalent to x = k or x = - k

Examples with Solutions

Example 1
Solve the equation and check the answers found.
|x + 6 | = 7
Solution to Example 1:

Check solutions:
The solutions to the given equation are x = 1 and x = -13
Matched Exercise 1: Solve the equation
|-x - 8 | = 10

Solution to Matched Exercise


Example 2
Solve the equation and check the answers found.
-2 |x / 2 + 3 | - 4 = -10
Solution to Example 2: Check solutions: The solutions to the given equation are x = 0 and x = -12
Matched Exercise 2: Solve the equation
4 |x + 2| - 30 = -10

Solution to Matched Exercise


Example 3
Solve the equation and check the answers found.
|2 x - 2 | = x + 1
Solution to Example 3: Check solutions The solutions to the given equation are x = 3 and x = 1 / 3
Matched Exercise 3:Solve the equation
- 4|x + 2 | = x - 8

Solution to Matched Exercise


Example 4
Solve the equation and check the answers found.
|x2 - 4| = x + 2
Solution to Example 3: Check solutions: Conclusion
The solutions to the given equation are x = -2, x = 1 and x = 3.
Matched Exercise 4: Solve the equation
|x2 - 16 | = x - 4

Solution to Matched Exercise


Solutions to the Above Matched Exercises

Matched Exercise 1

Solve the equation
|-x - 8 | = 10
Answer to Matched Exercise 1:
The above equation has two solutions
x = 2
x = -18

Matched Exercise 2

Solve the equation
4 |x + 2 | - 30 = -10
Answer to Matched Exercise 2:
The above equation has two solutions
x = 3
x = -7

Matched Exercise 3

Solve the equation
- 4 | x + 2 | = x - 8
Answer to Matched Exercise 3:
The above equation has two solutions
x = 0
x = -16/3

Matched Exercise 4

Solve the equation
|x2 - 16 | = x - 4
Answer to Matched Exercise 4:
The above equation has one solution
x = 4

More Exercises with Answers


Solve the following absolute value equations
a) | x - 4 | = 9
b) | x 2 + 4 | = 5
c) | x 2 - 9 | = x + 3
d) | x + 1 | = x - 3
e) | -x | = 2
Answers to Above Exercises
a) -5 , 13
b) -1 , 1
c) -3 , 2 , 4
d) no real solutions
e) -2 , 2

More References and Links

Absolute Value Equations And Inequalities
Solve Equations, Systems of Equations and Inequalities
Step by Step Solver for Equation With Absolute Value.