#### Answer

$m\le2
\text{ OR }
m\ge4$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|m-3|\ge1
,$ use the definition of absolute value inequality. Then use the properties of inequality to isolate the variable. Graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
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}
m-3\ge1
\\\\\text{OR}\\\\
m-3\le-1
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
m-3\ge1
\\\\
m\ge1+3
\\\\
m\ge4
\\\\\text{OR}\\\\
m-3\le-1
\\\\
m\le-1+3
\\\\
m\le2
.\end{array}
Hence, the solution set is $
m\le2
\text{ OR }
m\ge4
.$
The graph above confirms the solution set of the inequality.