College Algebra, Chapter 1, 1.7, Section 1.7, Problem 36

Solve the inequality $\displaystyle \left| \frac{x+1}{2} \right| \geq 4$. Express the answer using interval notation.

We have,


$
\begin{equation}
\begin{aligned}
\frac{x+1}{2} &\geq 4 && \text{and}& - \left( \frac{x+1}{2} \right) &\geq 4 && \text{Divide both sides by -1}\\
\\
\frac{x+1}{2} &\geq 4 && \text{and}& \frac{x+1}{2} &\leq -4 && \text{Multiply by 2}\\
\\
x+1 &\geq 8 &&\text{and}& x+1 &\leq -8 && \text{Subtract 1}\\
\\
x &\geq 7 && \text{and}& x &\leq -7
\end{aligned}
\end{equation}
$


The solution set is $(-\infty, -7] \bigcup [7,\infty)$