Absolute Value Equations
Follow these steps to solve an absolute value equality which contains one absolute value:
1. Isolate the absolute value on one side of the equation.
2. Is the number on the other side of the equation negative? If you answered yes, then the
equation has no solution. If you answered no, then go on to step 3.
3. Write two equations without absolute values. The first equation will set the quantity
inside the bars equal to the number on the other side of the equal sign; the second
equation will set the quantity inside the bars equal to the opposite of the number on the
other side.
4. Solve the two equations.
Follow these steps to solve an absolute value equality which contains two absolute values (one
on each side of the equation):
1. Write two equations without absolute values. The first equation will set the quantity
inside the bars on the left side equal to the quantity inside the bars on the right side. The
second equation will set the quantity inside the bars on the left side equal to the opposite
of the quantity inside the bars on the right side.
2. Solve the two equations.
Let's look at some examples.
Example 1: Solve |2x - 1| + 3 = 6
Step 1: Isolate the absolute value
|2x - 1| + 3 = 6
|2x - 1| = 3
Step 2: Is the number on the other side of
the equation negative?
No, it’s a positive number, 3, so continue
on to step 3
Step 3: Write two equations without
absolute value bars
2x - 1 = 3
2x - 1 = -3
Step 4: Solve both equations
2x - 1 = 3
2x - 1 = -3
2x = 4
2x = -2
x=2
x = -1
Example 2: Solve |3x - 6| - 9 = -3
Step 1: Isolate the absolute value
|3x - 6| - 9 = -3
|3x - 6| = 6
Step 2: Is the number on the other side of
the equation negative?
No, it’s a positive number, 6, so continue
on to step 3
Step 3: Write two equations without
absolute value bars
3x - 6 = 6
3x - 6 = -6
Step 4: Solve both equations
3x - 6 = 6
3x - 6 = -6
3x = 12
3x = 0
x=4
x=0
Example 3: Solve |5x + 4| + 10 = 2
Step 1: Isolate the absolute value
|5x + 4| + 10 = 2
|5x + 4| = -8
Step 2: Is the number on the other side of
the equation negative?
Yes, it’s a negative number, -8. There is no
solution to this problem.
Example 4: Solve |x - 7| = |2x - 2|
Step 1: Write two equations without
absolute value bars
x - 7 = 2x - 2
x - 7 = -(2x - 2)
Step 4: Solve both equations
x - 7 = 2x - 2
x - 7 = -2x + 2
-x - 7 = -2
3x - 7= 2
-x = 5
3x = 9
x = -5
x=3
Example 5: Solve |x - 3| = |x + 2|
Step 1: Write two equations without
absolute value bars
x-3=x+2
x - 3 = -(x + 2)
Step 4: Solve both equations
x-3=x+2
x - 3 = -x - 2
- 3 = -2
2x - 3= -2
false statement
2x = 1
No solution from
this equation
x = 1/2
Step 1: Write two equations without
absolute value bars
x-3=3-x
x - 3 = -(3 - x)
Step 4: Solve both equations
x-3=3-x
x - 3 = -(3 - x)
2x - 3 = 3
x - 3= -3 + x
2x = 6
-3 = -3
x=3
All real numbers are
solutions to this
equation
So the only solution to this problem is x = 1/2
Example 6: Solve |x - 3| = |3 - x|
Since 3 is included in the set of real numbers, we will just say that the solution to this equation is
All Real Numbers