Chapter 1, Problem 16CR

To determine

To calculate:

The solution of the following equation and verify the solution:

  $-2\left|5w-7\right|+9=-7$

Answer to Problem 16CR

The solution of $-2\left|5w-7\right|+9=-7$ is $-\frac{1}{5}$ and $3$ .

Explanation of Solution

Given information:

  $-2\left|5w-7\right|+9=-7$

Formula used:

The absolute rule is given by the following expression:

If $\left|u\right|=a$ , $a>0$ then $u=a$ or $u=-a$

Calculation:

The given equation is:

  $-2\left|5w-7\right|+9=-7$

… (1)

Subtract $9$ from both sides of equation:

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

Divide by $-2$ :

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

On applying absolute rule: $\left(a=8\right)>0$ then;

Case 1:

  $5w-7=8$

Add $3$ on both sides of equation:

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

Divide by $5$ :

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

OR

Case 2:

  $5w-7=-8$

Add $3$ on both sides of equation:

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

Divide by $5$ :

  $\begin{array}{l}\frac{5}{5}w=\frac{-1}{5}\\ w=\frac{-1}{5}\end{array}$

Therefore, the solution of $-2\left|5w-7\right|+9=-7$ is $w=-\frac{1}{5},3$ .

Now, check for solutions:

Put $w=-\frac{1}{5}$ :

  $\begin{array}{l}-2\left|5w-7\right|+9=-7\\ -2\left|5\left(-\frac{1}{5}\right)-7\right|+9=-7\\ -2\left|-1-7\right|+9=-7\\ -2\left|-8\right|+9=-7\\ -2\times 8+9=-7\left[\because \left|-a\right|=a\right]\\ -16+9=-7\\ -7=-7\end{array}$

Put $w=3$ :

  $\begin{array}{l}-2\left|5w-7\right|+9=-7\\ -2\left|5\left(3\right)-7\right|+9=-7\\ -2\left|15-7\right|+9=-7\\ -2\left|8\right|+9=-7\\ -2\times 8+9=-7\left[\because \left|a\right|=a,a\ge 0\right]\\ -16+9=-7\\ -7=-7\end{array}$

Hence, solutions are verified.