Solving Absolute Value Equations

, if 0, if 0

x xx

x x

• The absolute value of x is defined as

• Example 1

8 8 8

, if 0, if 0

x xx

x x

• Example 2

7 7

• Example 3

4x Solve:

Since 4 4 and 4 4

the solution is

or4,4

4xx

The process used above leads to our method for solving absolute value equations.

• To solve the absolute value equation

1) Write

2) Solve the resulting equations.

expression b

expression b expression b

• Example 4

Solve: 2 7 3 2x

First, add 3 to both sides to get the equation in the form of

expression b

2 7 1x

Write the two equations and solve each.

2 7 1x

2 7 1x 2 7 1x

2 6x

3x

2 8x

4x

3,4x

• Example 5

Now solve the same problem again, only this time use the graphing method.

2 7 3 2x

Move all terms to the left hand side.

2 7 3 2 0x

Let y1 equal the left hand side.

1y

2 7 3 2 0x

Enter the expression for y1 into the calculator

1y

Press Zoom-6 to graph

Use Trace to find the x-intercepts

Press Trace and then enter 3.

Now repeat the process with x = 4, the intercept on the right.

Since the y-value is 0, (3,0) is an x-intercept, and 3 is a solution to the equation.

Again, the y-value is 0.

The solutions to the equation are x = 3,4