#### Answer

$x=\left\{ 2,4 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|6-2x|+1=3
,$ use the properties of equality to isolate the absolute value expression. Then use the properties of absolute value equality.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
|6-2x|=3-1
\\\\
|6-2x|=2
.\end{array}
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
6-2x=2
\\\\\text{OR}\\\\
6-2x=-2
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
6-2x=2
\\\\
-2x=2-6
\\\\
-2x=-4
\\\\
x=\dfrac{-4}{-2}
\\\\
x=2
\\\\\text{OR}\\\\
6-2x=-2
\\\\
-2x=-2-6
\\\\
-2x=-8
\\\\
x=\dfrac{-8}{-2}
\\\\
x=4
.\end{array}
Hence, $
x=\left\{ 2,4 \right\}
.$