Answer
The solution set of the equation $6\left| 1-2x \right|-7=11$ is $\left\{ -1,2 \right\}$.
Work Step by Step
Let us consider the following equation involving the absolute value,
$6\left| 1-2x \right|-7=11$
Solve the equation as given below:
Add 7 to both sides:
$\begin{align}
& 6\left| 1-2x \right|-7+7=11+7 \\
& 6\left| 1-2x \right|=18
\end{align}$
Divide by 6 on both sides,
$\begin{align}
& \frac{6\left| 1-2x \right|}{6}=\frac{18}{6} \\
& \left| 1-2x \right|=3
\end{align}$
By using the definition of an absolute function, $\left| x \right|=a\Rightarrow x=a\text{ or }x=-a $
$\begin{align}
& \left| 1-2x \right|=3 \\
& 1-2x=3\text{ or 1}-2x=-3
\end{align}$
Solve the equation $1-2x=3$.
Subtract 1 from both sides:
$\begin{align}
& 1-2x-1=3-1 \\
& -2x=2
\end{align}$
And divide by $-2$ on both sides,
$\begin{align}
& \frac{\left( -2x \right)}{\left( -2 \right)}=\frac{2}{\left( -2 \right)} \\
& x=-1
\end{align}$
Then, solve the second part of the equation $1-2x=-3$
Subtract 1 from both sides,
$\begin{align}
& 1-2x-1=-3-1 \\
& -2x=-4
\end{align}$
Divide by $-2$ on both sides,
$\begin{align}
& \frac{\left( -2x \right)}{\left( -2 \right)}=\frac{\left( -4 \right)}{\left( -2 \right)} \\
& x=2
\end{align}$
Therefore, $ x=-1$ is the solution of the equation.
Check $ x=-1$ in the original equation,
$\begin{align}
& 6\left| 1-2\times \left( -1 \right) \right|-7=11 \\
& 6\left| 1+2 \right|-7=11 \\
& 6\left| 3 \right|-7=11 \\
& 6\times 3-7=11
\end{align}$
And simplify further,
$\begin{align}
& 18-7=11 \\
& 11=11\text{ }\left( \text{true} \right)
\end{align}$
Therefore, $ x=-1$ is the solution of the equation.
Thus, the solution set of the equation $6\left| 1-2x \right|-7=11$ is $\left\{ -1,2 \right\}$.
