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