Chapter 2 - Linear and Quadratic Functions - Section 2.8 Equations and Inequalities Involving the Absolute Value Function - 2.8 Assess Your Understanding - Page 182: 22

$z=-1$ or $z=2$.

Using the definition of the absolute value, we have $$ |1-2z|+6=9\Longrightarrow |1-2z|=9-6=3\\ \Longrightarrow 1-2z=3~or~ 1-2z=-3 .$$ Thus, we solve the two possible equations: $2z=-2~or~2z=4$ and hence $z=-1$ or $z=4/2=2$.