Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 34

Solve the inequality $|-2x - 4| \geq 5$, and graph the solution set.

The absolute value inequality is rewritten as

$|-2x - 4| \geq 5$ or $-2x - 4 \leq -5$,

because $-2x - 4$ must represent a number that is more than $5$ units from on either side of the number line. We can solve the compound inequality.


$\begin{equation} \begin{aligned} |-2x - 4| \geq & 5 && \qquad \text{or} &&& -2x - 4 \leq & -5 && \\ -2x \geq & 9 && \qquad \text{or} &&& -2x \leq & -1 && \text{Add } 4 \\ x \leq & \frac{-9}{2} && \qquad \text{or} &&& x \geq & \frac{1}{2} && \text{Divide $-2$. Reverse the inequality symbols.} \end{aligned} \end{equation}$


The solution set is $\displaystyle \left( - \infty, \frac{-9}{2} \right) \bigcup \left[ \frac{1}{2}, \infty \right)$.