Question

Solve each equation. Check your solution.

$\mathbf{2}\left|\mathbf{2}\mathbf{y}\mathbf{-}\mathbf{6}\right|\mathbf{+}\mathbf{4}\mathbf{=}\mathbf{8}$

Short answer

The solution set of given equation is $\left\{\mathbf{2}\mathbf{,}\mathbf{4}\right\}$.

Step-by-step solution

– Description of step.

Subtract 4 from both the sides of the given equation.

$\begin{array}{c}2\left|2y-6\right|+4-4=8-4\\ 2\left|2y-6\right|=4\end{array}$

– Description of step.

Divide each side of the given equation by 2 and simplify.

$\begin{array}{c}\frac{2\left|2y-6\right|}{2}=\frac{4}{2}\\ \left|2y-6\right|=2\end{array}$

– Description of step.

For any real numbers p and q, where $q\ge 0$,

Case 1) if $\left|p\right|=q$ then $p=q$ or

Case 2) if $\left|p\right|=q$ then $p=-q$

– Description of step.

Apply case 1 mentioned in step 3 to $\left|2y-6\right|=2$ and simplify.

$\begin{array}{c}2y-6=2\\ 2y-6+6=2+6\\ \frac{2y}{2}=\frac{8}{2}\\ y=4\end{array}$

– Description of step.

Apply case 2 mentioned in step 3 to $\left|2y-6\right|=2$ and simplify.

$\begin{array}{c}2y-6=-2\\ 2y-6+6=-2+6\\ \frac{2y}{2}=\frac{4}{2}\\ y=2\end{array}$

– Description of step.

To check whether the obtained value of x is the solution of given equation, substitute $4$ for $y$ into $2\left|2y-6\right|+4=8$ and simplify.

$\begin{array}{d}2\left|2\left(4\right)-6\right|+4=8\\ 2\left|8-6\right|+4\stackrel{?}{=}8\\ 2\cdot 2+4\stackrel{?}{=}8\\ 4+4\stackrel{?}{=}8\\ 8=8\end{array}$

– Description of step.

To check whether the obtained value of x is the solution of given equation, substitute $2$for $y$into $2\left|2y-6\right|+4=8$and simplify.

$\begin{array}{d}2\left|2\left(2\right)-6\right|+4=8\\ 2\left|4-6\right|+4\stackrel{?}{=}8\\ 2\cdot 2+4\stackrel{?}{=}8\\ 4+4\stackrel{?}{=}8\\ 8=8\end{array}$

Therefore, the solution set of given equation is $\left\{2,4\right\}$.