Answer
FALSE
Work Step by Step
If $a=1,$ $b=2,$ and $c=3,$ then the expression, $a|b-c|,$ evaluates to
\begin{array}{l}\require{cancel}
a|b-c|=1|2-3|
\\\\
a|b-c|=1|-1|
\\\\
a|b-c|=1(1)
\\\\
a|b-c|=1
,\end{array}
while the expression $|ab|-|ac|$ evaluates to
\begin{array}{l}\require{cancel}
|ab|-|ac|=|1(2)|-|1(3)|
\\\\
|ab|-|ac|=|2|-|3|
\\\\
|ab|-|ac|=2-3
\\\\
|ab|-|ac|=-1
.\end{array}
Hence, the given statement, $a|b-c|=|ab|-|ac|,$ is $\text{
FALSE
.}$
