Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 58

Solve $|-2x - 6 | \leq 5$. Graph the solution set.


The expression $-2x - 6$ must represent a number that is less than or equal $5$ units from on either side of the number line. That is, $-2x - 6$ between $-5$ and $5$(inclusive). So we have


$
\begin{equation}
\begin{aligned}

-5 \leq & -2x - 6 \leq 5
&&
\\
1 \leq & -2x \leq 11
&& \text{Add each side by } 6
\\
- \frac{1}{2} \geq & x \geq - \frac{11}{2}
&& \text{Divide each side by $-2$. Change signs $$}

\end{aligned}
\end{equation}
$




The solution set is $\displaystyle \left[ - \frac{11}{2}, - \frac{1}{2} \right]$.