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:
If |x + 6 | = 7, then (see rule 5 above)
a) x + 6 = 7
or
b) x + 6 = -7
Solve equation a)
x + 6 = 7
x = 1
Solve equation b)
x + 6 = -7
x = -13
Check solutions:
solution x = 1
Left Side of Equation for x = 1.
|1 + 6 |
= | 7 |
= 7
Right Side of Equation for x = 1.
7
x = -13
Left Side of Equation for x = 1.
|-13 + 6 |
= | -7 |
= 7
Right Side of Equation for x = 1.
7
The solutions to the given equation are x = 1 and x = -13
Matched Exercise 1: Solve the equation
Example 2 Solve the equation and check the answers found.
-2 |x / 2 + 3 | - 4 = -10
Solution to Example 2:
Given
-2 |x / 2 + 3 | - 4 = -10
We first write the equation in the form | A | = B. Add 4 to both sides and group like terms
-2|x / 2 + 3 | = -6
Divide both sides by -2
|x / 2 + 3 | = 3
We now proceed as in example 1 above, the equation
|x / 2 + 3 | = 3 gives two equations.
a) x / 2 + 3 = 3
or
b) x / 2 + 3 = -3
Solve equation a)
x / 2 + 3 = 3
to obtain
x = 0
Solve equation b)
x / 2 + 3 = -3
to obtain
x = -12
Check solutions:
x = 0
Left Side of Equation for x = 0.
-2 |x / 2 + 3 | - 4
= -2| 3 | - 4
= -10
Right Side of Equation for x = 1.
-10
x = -12
Left Side of Equation for x = -12.
-2 |x / 2 + 3 | - 4
= -2 |-12 / 2 + 3 | - 4
= -2 |-6 + 3 | - 4
= -2(3) - 4
= -10
Right Side of Equation for x = -12.
-10
The solutions to the given equation are x = 0 and x = -12
Matched Exercise 2: Solve the equation
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