#### Answer

$p\le2
\text{ OR }
p\ge6$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|p-4|\ge2
,$ 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}
p-4\ge2
\\\\\text{OR}\\\\
p-4\le-2
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
p-4\ge2
\\\\
p\ge2+4
\\\\
p\ge6
\\\\\text{OR}\\\\
p-4\le-2
\\\\
p\le-2+4
\\\\
p\le2
.\end{array}
Hence, the solution set is $
p\le2
\text{ OR }
p\ge6
.$
The graph above confirms the solution set of the inequality.