Answer
$z=-7\text{ OR }z=15$
Work Step by Step
Using the properties of equality, the given equation, $
|4-z|-10=1
,$ is equivalent to
\begin{array}{l}\require{cancel}
|4-z|-10+10=1+10
\\
|4-z|=11
.\end{array}
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
4-z=11
\\\\\text{OR}\\\\
4-z=-11
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
4-z=11
\\
4-z-4=11-4
\\
-z=7
\\
-1(-z)=(7)(-1)
\\
z=-7
\\\\\text{OR}\\\\
4-z=-11
\\
4-z-4=-11-4
\\
-z=-15
\\
-1(-z)=(-15)(-1)
\\
z=15
.\end{array}
Hence, $
z=-7\text{ OR }z=15
.$
