Chapter 1, Equations and Inequalities - 1.7 Solve Absolute Value Equations and Inequalities - 1.7 Exercises - Skill Practice - Page 55: 32

$$z=-9\quad \mathrm{or}\quad \:z=27$$

Since the value inside the absolute value can be positive or negative and get the same number, we must consider both options. Doing this, we find: $$\frac{2}{3}z-6=-12\quad \mathrm{or}\quad \frac{2}{3}z-6=12 \\ z=-9\quad \mathrm{or}\quad \:z=27$$