Question

solve the equation.
$-5 = -|z + 8| - 3$
the solution set is .

Explanation:

Step1: Isolate the absolute value expression

Start with the equation \(-5 = -|z + 8| - 3\). Add 3 to both sides to isolate the term with the absolute value.
\[
-5 + 3 = -|z + 8| - 3 + 3
\]
\[
-2 = -|z + 8|
\]

Step2: Solve for the absolute value

Multiply both sides by -1 to solve for \(|z + 8|\).
\[
(-1)\times(-2) = (-1)\times(-|z + 8|)
\]
\[
2 = |z + 8|
\]

Step3: Solve the absolute value equation

The absolute value equation \(|z + 8| = 2\) means that \(z + 8 = 2\) or \(z + 8 = -2\) (because if \(|a| = b\) where \(b\geq0\), then \(a = b\) or \(a = -b\)).

• For \(z + 8 = 2\): Subtract 8 from both sides.

\[
z + 8 - 8 = 2 - 8
\]
\[
z = -6
\]

• For \(z + 8 = -2\): Subtract 8 from both sides.

\[
z + 8 - 8 = -2 - 8
\]
\[
z = -10
\]

Answer:

\(\{-10, -6\}\)