Chapter 1, Problem 17CT

To determine

To solve: the equation $-2\left|5w-7\right|+9=-7$ .

Answer to Problem 17CT

$w=3$ Or $w=-\frac{1}{5}$

Explanation of Solution

Consider the given equation:

  $-2\left|5w-7\right|+9=-7$

Subtract 9 from both sides,

  $\begin{array}{c}-2\left|5w-7\right|+9-9=-7-9\\ -2\left|5w-7\right|=-16\end{array}$

Divide both sides by $-2$ ,

  $\begin{array}{c}\frac{-2\left|5w-7\right|}{-2}=\frac{-16}{-2}\\ \left|5w-7\right|=8\end{array}$

Now write the related linear equations,

  $\begin{array}{c}5w-7=8\\ 5w-7+7=8+7\\ 5w=15\end{array}$

  $\begin{array}{c}\frac{5w}{5}=\frac{15}{5}\\ w=3\end{array}$

Or

  $\begin{array}{c}5w-7=-8\\ 5w-7+7=-8+7\\ 5w=-1\\ w=-\frac{1}{5}\end{array}$

Check:

First substitute $w=3$ in the equation $-2\left|5w-7\right|+9=-7$ ,

  $\begin{array}{c}-2\left|5\left(3\right)-7\right|+9=-7\\ -2\left|15-7\right|+9=-7\\ -2\left|8\right|+9=-7\end{array}$

  $\begin{array}{c}-2\left(8\right)+9=-7\\ -16+9=-7\\ -7=-7\end{array}$

This statement is correct, so $w=3$ is a solution of given equation.

Now substitute $w=-\frac{1}{5}$ in the equation $-2\left|5w-7\right|+9=-7$ ,

  $\begin{array}{c}-2\left|5\left(-\frac{1}{5}\right)-7\right|+9=-7\\ -2\left|-1-7\right|+9=-7\\ -2\left|-8\right|+9=-7\end{array}$

  $\begin{array}{c}-2\left(8\right)+9=-7\\ -16+9=-7\\ -7=-7\end{array}$

This statement is correct, so $w=-\frac{1}{5}$ is also a solution of given equation.