Chapter 1.4, Problem 39E

To determine

To solve: the given equation and check the solution.

Answer to Problem 39E

The solution is $p=10\text{ or }p=\frac{2}{3}$ .

Explanation of Solution

Given:

  $4\left|p-3\right|=\left|2p+8\right|$

Calculation:

  $\begin{array}{l}4\left|p-3\right|=\left|2p+8\right|\text{ Given}\\ 4\left(p-3\right)=2p+8\text{ or }4\left(p-3\right)=-\left(2p+8\right)\text{ Write two linear equations}\end{array}$

Solve equation 1:

  $\begin{array}{l}4\left(p-3\right)=2p+8\\ 4p-12=2p+8\text{ Use distributive Property}\\ 4p-2p-12=2p-2p+8\text{ Subtract both sides by 2p}\\ 2p-12+12=8+12\text{ Add both sides by 12}\\ \frac{2p}{2}=\frac{20}{2}\text{ Divide both sides by 2}\\ p=10\end{array}$

Solve equation 2:

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

Check $\left(p=10\right)$ :

  $\begin{array}{l}4\left|p-3\right|=\left|2p+8\right|\\ 4\left|\left(10\right)-3\right|=\left|2\left(10\right)+8\right|\text{ Substitute}\\ 4\left|7\right|=\left|20+8\right|\text{ Simplify}\\ 4\left|7\right|=\left|28\right|\text{ Simplify}\\ 4\left|7\right|=28\text{ Get absolute values}\\ 28=28\text{ Simplify}\end{array}$

Check $\left(p=\frac{2}{3}\right)$ :

  $\begin{array}{l}4\left|p-3\right|=\left|2p+8\right|\\ 4\left|\left(\frac{2}{3}\right)-3\right|=\left|2\left(\frac{2}{3}\right)+8\right|\text{ Substitute}\\ 4\left|-\frac{7}{3}\right|=\left|\frac{4}{3}+8\right|\text{ Simplify}\\ 4\left|-\frac{7}{3}\right|=\left|\frac{28}{3}\right|\text{ Simplify}\\ 4\left(\frac{7}{3}\right)=\frac{28}{3}\text{ Get absolute values}\\ \frac{28}{3}=\frac{28}{3}\text{ Simplify}\end{array}$

Conclusion:

Therefore, the solution is $p=10\text{ or }p=\frac{2}{3}$ .