Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 80

Evaluate the equation $|7x + 12| = |x - 8|$.

This equation is satisfied either if $7x + 12$ and $x - 8$ are equal to each other or if $7x + 12$ and $x - 8$ are negatives of each other


$
\begin{equation}
\begin{aligned}

7x + 12 =& x - 8 && \text{or} &&& 7x + 12 =& -(x - 8)
\\
7x - x =& -8-12 && \text{or} &&& 7x + x =& 8-12
\\
6x =& -20 && \text{or} &&& 8x =& -4
\\
x =& - \frac{20}{6} && \text{or} &&& x =& - \frac{4}{8}
\\
\\
x =& - \frac{10}{3} && \text{or} &&& x =& - \frac{1}{2}


\end{aligned}
\end{equation}
$


The solution set is $\displaystyle \left \{ - \frac{10}{3}, - \frac{1}{2} \right \}$.