Answer
The solution set of the equation $-4\left| 2x+1 \right|+12=0$ is $\left\{ -2,1 \right\}$.
Work Step by Step
Consider the provided equation,
$-4\left| 2x+1 \right|+12=0$
Subtract, $12$ from both sides,
$\begin{align}
& -4\left| 2x+1 \right|+12-12=-12+0 \\
& -4\left| 2x+1 \right|=-12
\end{align}$
Multiply, both sides by $-\frac{1}{4}$ ,
$\begin{align}
& -\frac{1}{4}\left( -4\left| 2x+1 \right| \right)=-\frac{1}{4}\cdot \left( -12 \right) \\
& \left| 2x+1 \right|=3
\end{align}$
Consider, the following absolute rule,
If $\left| u \right|=a$ , $a>0$ then, $u=a$ or $u=-a$
Then, according to the expression $\left| 2x+1 \right|=3$ ,
$2x+1=3\text{ or }2x+1=-3$
Take, $2x+1=3$ and subtract 1 from both sides:
$\begin{align}
& 2x+1-1=3-1 \\
& 2x=2
\end{align}$
Multiply, both sides by $\frac{1}{2}$:
$\begin{align}
& \frac{1}{2}\left( 2x \right)=\frac{1}{2}\left( 2 \right) \\
& x=1
\end{align}$
Now, take $2x+1=-3$ and subtract 1 from both sides:
$\begin{align}
& 2x+1-1=-3-1 \\
& 2x=-4
\end{align}$
Multiply, both sides by $\frac{1}{2}$:
$\begin{align}
& \frac{1}{2}\left( 2x \right)=\frac{1}{2}\left( -4 \right) \\
& x=-2
\end{align}$
Thus, the solution set of the expression $-4\left| 2x+1 \right|+12=0$ is $\left\{ -2,1 \right\}$.
Check:
Now, check whether the solutions satisfy the provided equation $-4\left| 2x+1 \right|+12=0$ or not.
To check the value at $x=-2$ substitute, $x=-2$ in the equation $-4\left| 2x+1 \right|+12=0$.
$\begin{align}
& -4\left| 2\left( -2 \right)+1 \right|+12\overset{?}{\mathop{=}}\,0 \\
& -4\left| -4+1 \right|+12\overset{?}{\mathop{=}}\,0 \\
& -4\left| -3 \right|+12\overset{?}{\mathop{=}}\,0 \\
& -12+12\overset{?}{\mathop{=}}\,0
\end{align}$
On further simplification,
$0=0$
True.
Now, to check the value at $x=1$ substitute, $x=1$ in $-4\left| 2x+1 \right|+12=0$
$\begin{align}
& -4\left| 2\left( 1 \right)+1 \right|+12\overset{?}{\mathop{=}}\,0 \\
& -4\left| 2+1 \right|+12\overset{?}{\mathop{=}}\,0 \\
& -4\left| 3 \right|+12\overset{?}{\mathop{=}}\,0 \\
& -12+12\overset{?}{\mathop{=}}\,0
\end{align}$
On further simplification,
$0=0$
True.
Hence, the solution set of the provided equation $-4\left| 2x+1 \right|+12=0$ is $\left\{ -2,1 \right\}$.
