Chapter 1.4, Problem 40E

To determine

To solve: the given equation and check the solution.

Answer to Problem 40E

The solution is $w=-2\text{ or }w=-\frac{1}{5}$ .

Explanation of Solution

Given:

  $2\left|4w-1\right|=3\left|4w+2\right|$

Calculation:

  $\begin{array}{l}2\left|4w-1\right|=3\left|4w+2\right|\text{ Given}\\ 2\left(4w-1\right)=3\left(4w+2\right)\text{ or }2\left(4w-1\right)=-3\left(4w+2\right)\text{ Write two linear equations}\end{array}$

Solve equation 1:

  $\begin{array}{l}2\left(4w-1\right)=3\left(4w+2\right)\\ 8w-2=12w+6\text{ Use distributive Property}\\ 8w-12w-2=12w-12w+6\text{ Subtract both sides by 12}w\\ -4w-2+2=6+2\text{ Add both sides by 2}\\ \frac{-4w}{-4}=\frac{8}{-4}\text{ Divide both sides by -4}\\ w=-2\end{array}$

Solve equation 2:

  $\begin{array}{l}2\left(4w-1\right)=-3\left(4w+2\right)\\ 8w-2=-12w-6\text{ Use distributive Property}\\ 8w+12w-2=-12w+12w-6\text{ Add both sides by 12}w\\ 20w-2+2=-6+2\text{ Add both sides by 2}\\ \frac{20w}{20}=\frac{-4}{20}\text{ Divide both sides by 20}\\ w=-\frac{1}{5}\end{array}$

Check $\left(w=-2\right)$ :

  $\begin{array}{l}2\left(4w-1\right)=3\left(4w+2\right)\\ 2\left(4\left(-2\right)-1\right)=3\left(4\left(-2\right)+2\right)\text{ Substitute}\\ 2\left|-8-1\right|=3\left|-8+2\right|\text{ Simplify}\\ 2\left|-9\right|=3\left|-6\right|\text{ Simplify}\\ 2\left(-9\right)=3\left(-6\right)\text{ Get absolute values}\\ -18=-18\text{ Simplify}\end{array}$

Check $\left(w=-\frac{1}{5}\right)$ :

  $\begin{array}{l}2\left(4\left(w\right)-1\right)=3\left(4w+2\right)\\ 2\left(4\left(-\frac{1}{5}\right)-1\right)=3\left(4\left(-\frac{1}{5}\right)+2\right)\text{ Substitute}\\ 2\left|-\frac{4}{5}-1\right|=3\left|-\frac{4}{5}+2\right|\text{ Simplify}\\ 2\left|-\frac{4}{5}-\frac{5}{5}\right|=3\left|-\frac{4}{5}+\frac{10}{2}\right|\text{ Simplify}\\ \text{2}\left|-\frac{9}{5}\right|=3\left|-\frac{6}{5}\right|\text{ Simplify }\\ 2\left(\frac{9}{5}\right)=3\left(\frac{6}{5}\right)\text{ Get absolute values}\\ \frac{18}{5}=\frac{18}{5}\text{ Simplify}\end{array}$

Since $p=2$ does not satisfy the original equation, we must reject it since it is an extraneous root.

Conclusion:

Therefore, the solution is $w=-2\text{ or }w=-\frac{1}{5}$ .