Answer
$\left( -\infty, -\dfrac{9}{2} \right]
\cup
\left[ \dfrac{1}{2},\infty \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|-2x-4| \ge 5
,$ use the definition of absolute value inequalities. Then use the properties of inequality to isolate the variable.
For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$
For graphing inequalities, use a hollowed dot for the symbols $\lt$ or $\gt.$ Use a solid dot for the symbols $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-2x-4 \ge 5
\\\\\text{OR}\\\\
-2x-4 \le -5
.\end{array}
Using the properties of inequality to isolate the variable results to
\begin{array}{l}\require{cancel}
-2x-4 \ge 5
\\\\
-2x \ge 5+4
\\\\
-2x \ge 9
\\\\\text{OR}\\\\
-2x-4 \le -5
\\\\
-2x \le -5+4
\\\\
-2x \le -1
.\end{array}
Dividing by a negative number (and consequently reversing the sign), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-2x \ge 9
\\\\
x \le \dfrac{9}{-2}
\\\\
x \le -\dfrac{9}{2}
\\\\\text{OR}\\\\
-2x \le -1
\\\\
x \ge \dfrac{-1}{-2}
\\\\
x \ge \dfrac{1}{2}
.\end{array}
In interval notation, the solution set is $
\left( -\infty, -\dfrac{9}{2} \right]
\cup
\left[ \dfrac{1}{2},\infty \right)
.$
The colored graph is the graph of the solution set.