Answer
$x\le-8 \text{ or } x\ge0$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inqeuality, $
9-|x+4|\le5
,$ use the properties of inequality to isolate the absolute value expression. Then use the definition of a greater than absolute value inequality and solve each resulting inequality. Finally, graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Using the properties of inequality to isolate the absolute value expression, the given is equivalent to
\begin{array}{l}\require{cancel}
9-|x+4|\le5
\\\\
-|x+4|\le5-9
\\\\
-|x+4|\le-4
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol) results to
\begin{array}{l}\require{cancel}
-|x+4|\le-4
\\\\
\dfrac{-|x+4|}{-1}\le\dfrac{-4}{-1}
\\\\
|x+4|\ge4
.\end{array}
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}
x+4\ge4
\\\\\text{OR}\\\\
x+4\le-4
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
x+4\ge4
\\\\
x\ge4-4
\\\\
x\ge0
\\\\\text{OR}\\\\
x+4\le-4
\\\\
x\le-4-4
\\\\
x\le-8
.\end{array}
Hence, the solution set is $
x\le-8 \text{ or } x\ge0
.$
