#### Answer

$-1\le h\le7$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|h-3|\le4
,$ use the definition of absolute value inequality. Then use the properties of inequality to isolate the variable. Graph the solution.
In the graph a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-4\le h-3\le4
.\end{array}
Using the properties of inequality to isolate the variable results to
\begin{array}{l}\require{cancel}
-4+3\le h-3+3\le4+3
\\\\
-1\le h\le7
.\end{array}
The graph above confirms the solution set of the inequality.