#### Answer

$x\le2 \text{ or } x\ge8$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inqeuality, $
12-|x-5|\le9
,$ 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}
12-|x-5|\le9
\\\\
-|x-5|\le9-12
\\\\
-|x-5|\le-3
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol) results to
\begin{array}{l}\require{cancel}
-|x-5|\le-3
\\\\
\dfrac{-|x-5|}{-1}\le\dfrac{-3}{-1}
\\\\
|x-5|\ge3
.\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-5\ge3
\\\\\text{OR}\\\\
x-5\le-3
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
x-5\ge3
\\\\
x\ge3+5
\\\\
x\ge8
\\\\\text{OR}\\\\
x-5\le-3
\\\\
x\le-3+5
\\\\
x\le2
.\end{array}
Hence, the solution set is $
x\le2 \text{ or } x\ge8
.$