Example

Solve for $y$: $\quad \displaystyle \left|5-\frac{2}{3} y \right| -9 = 8$

Solution

In order to use our procedure for solving absolute value equations, we must first get the value value expression on one side by itself. To do so, we add $9$:

Now this absolute value equation is equivalent to the compound equation

We solve each equation separately, starting with the equation on the left.

The first equation has solution $y=-18$.

Now we solve the equation on the right.

The second equation has solution $y=33$.

The solution of the original absolute value equation is the union of the two solutions: