Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 76

Solve the equation $\displaystyle \left| \frac{2}{3}x + \frac{1}{6} \right| + \frac{1}{2} = \frac{5}{2}$.


$
\begin{equation}
\begin{aligned}

\left| \frac{2}{3}x + \frac{1}{6} \right| + \frac{1}{2} =& \frac{5}{2}
\\
\\
\left| \frac{2}{3}x + \frac{1}{6} \right| =& \frac{5}{2} - \frac{1}{2}
\\
\\
\left| \frac{2}{3}x + \frac{1}{6} \right| =& 2

\end{aligned}
\end{equation}
$


We can solve this compound equation as follows


$
\begin{equation}
\begin{aligned}

\frac{2}{3}x + \frac{1}{6} =& 2 && \text{or} &&& \frac{2}{3}x + \frac{1}{6} =& -2
&&
\\
\\
4x + 1 =& 12 && \text{or} &&& 4x + 1 =& -12
&& \text{Multiply each side by } 6
\\
\\
4x =& 11 && \text{or} &&& 4x =& -13
&& \text{Subtract each side by } 1
\\
\\
x =& \frac{11}{4} && \text{or} &&& x =& - \frac{13}{4}
&& \text{Divide each side by } 4

\end{aligned}
\end{equation}
$



The solution set is $\displaystyle \left \{ - \frac{13}{4}, \frac{11}{4} \right \}$.