#### Answer

$x\lt-2 \text{ or } x\gt8$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
Solve the given inequality, $
|x-3|+2\gt7
,$ by isolating first the absolute value expression. Then use the definition of a greater than absolute value 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, the given expression is equivalent to
\begin{array}{l}\require{cancel}
|x-3|+2\gt7
\\\\
|x-3|\gt7-2
\\\\
|x-3|\gt5
.\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 given inequality is equivalent to
\begin{array}{l}\require{cancel}
x-3\gt5
\\\\\text{OR}\\\\
x-3\lt-5
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
x-3\gt5
\\\\
x\gt5+3
\\\\
x\gt8
\\\\\text{OR}\\\\
x-3\lt-5
\\\\
x\lt-5+3
\\\\
x\lt-2
.\end{array}
Hence, the solution set is $
x\lt-2 \text{ or } x\gt8
.$