Question

Solve each equation. Check your solutions.

$\mathbf{5}\left|\mathbf{x}\mathbf{-}\mathbf{3}\right|\mathbf{=}\mathbf{65}$

Short answer

The solution set of given equation is $\left\{\mathbf{16}\mathbf{,}\mathbf{-}\mathbf{10}\right\}$.

Step-by-step solution

– Description of step.

Divide each side of the given equation by 5 and simplify.

$\begin{array}{l}\frac{5\left|x-3\right|}{5}=\frac{65}{5}\\ \left|x-3\right|=13\end{array}$

– 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 2 to $\left|x-3\right|=13$ and simplify.

$\begin{array}{l}x-3=13\\ x-3+3=13+3\\ x=16\end{array}$

– Description of step.

Apply case 2 mentioned in step 2 to $\left|x-3\right|=13$and simplify.

$\begin{array}{l}x-3=-13\\ x-3+3=-13+3\\ x=-10\end{array}$

– Description of step.

To check whether the obtained value of $x$is the solution of given equation, substitute 16 for $x$into $\left|x-3\right|=13$and simplify.

$\begin{array}{l}\left|x-3\right|=13\\ \left|16-3\right|\stackrel{?}{=}13\\ \left|13\right|\stackrel{?}{=}13\\ 13=13\end{array}$

– Description of step.

To check whether the obtained value of $x$is the solution of given equation, substitute $-10$ for $x$ into $\left|x-3\right|=13$ and simplify.

$\begin{array}{l}\left|x-3\right|=13\\ \left|-10-3\right|\stackrel{?}{=}13\\ \left|-13\right|\stackrel{?}{=}13\\ 13=13\end{array}$

Therefore, the solution set of given equation is$\left\{16,-10\right\}$.