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:
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.
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.
Thus, the solutions to |x + 6| = |2x| are x = 6 and x = -2.
|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
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