## Precalculus (6th Edition) Blitzer

The solution set of the equation $-4\left| 2x+1 \right|+12=0$ is $\left\{ -2,1 \right\}$.
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 $\left\{ -2,1 \right\}$.