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 $(-\infty, -7] \bigcup [7,\infty)$