Solving Absolute Value Equations
Solving an Equation of the Form  z = w
The equation z = w says that the distance on a number line from z to 0
the same as the distance from w to 0. So, z and w must be equal or they
must be opposites.
This can be summarized in the following principle:
Principle
Absolute Value Equations of the Form  z = w
In the equation z = w,
let z and w represent any algebraic expressions.
Then, either z and w are equal: z = w or z and w are opposites: z = w.
Note:
If z and w are equal, then they are at the
same location on a number line.
If z and w are opposites, then they are the
same distance from zero on a number line,
but on opposite sides of 0.
We will use this principle to solve this equation:
Replace x + 6 with z and 2x with w.
Rewrite the absolute value equation as two
equations. Replace z with x + 6 and w with 2x.
Subtract x from both sides.
In the equation on the right, divide
both sides by 3. 
x + 6 = 2x z = w
z = w or z = w x + 6 = 2x or x + 6 = 2x 6 = x or 6 = 3x 6 = x
or 2 = x 
Thus, the solutions to x + 6 = 2x are x = 6 and x = 2.
