Answer
The solution set of equation $\left| \frac{2}{3}x-6 \right|=2$ is $\left\{ \left. 6,12 \right\} \right.$.
Work Step by Step
Consider the provided equation,
$\left| \frac{2}{3}x-6 \right|=2$
Apply modulus property:
$\frac{2}{3}x-6=2$
Add 6 on both sides:
$\begin{align}
& \frac{2}{3}x-6+6=2+6 \\
& \frac{2}{3}x=8
\end{align}$
Multiply by 3 both sides
$2x=24$
Divide by 2 both sides
$x=12$
Or
$\frac{2}{3}x-6=-2$
Add 6 on both sides:
$\begin{align}
& \frac{2}{3}x-6+6=-2+6 \\
& \frac{2}{3}x=4
\end{align}$
Multiply by 3 both sides
$2x=12$
Divide by 2 both sides
$x=6$
Check:
For $x=6$
Put $x=6$ in $\left| \frac{2}{3}x-6 \right|=2$
$\begin{align}
& \left| \frac{2}{3}\left( 6 \right)-6 \right|\overset{?}{\mathop{=}}\,2 \\
& \left| 4-6 \right|\overset{?}{\mathop{=}}\,2 \\
& \left| -2 \right|\overset{?}{\mathop{=}}\,2 \\
& 2=2
\end{align}$
Which is true.
For $x=12$
Put $x=12$ in $\left| \frac{2}{3}x-6 \right|=2$
$\begin{align}
& \left| \frac{2}{3}\left( 12 \right)-6 \right|\overset{?}{\mathop{=}}\,2 \\
& \left| 8-6 \right|\overset{?}{\mathop{=}}\,2 \\
& \left| 2 \right|\overset{?}{\mathop{=}}\,2 \\
& 2=2
\end{align}$
Which is true.
Therefore, the solution set of the equation $\left| \frac{2}{3}x-6 \right|=2$ is $\left\{ \left. 6,12 \right\} \right.$.
