Chapter 1.4, Problem 24E

To determine

To solve the equation:

  $9\left|4p+2\right|+8=35$

And also graph the solution.

Answer to Problem 24E

  $p=0.25,\text{ and }p=-1.25$

  

Explanation of Solution

Given:

  $9\left|4p+2\right|+8=35$

Concept Used:

To solve $\left|ax+b\right|=c,c\ge 0$ , solve the related equations:

  $ax+b=c,\text{ or }ax+b=-c$

When $c<0$ the absolute value equation $\left|ax+b\right|=c$ has no solution, since absolute value always indicates a value that is not negative.

Calculation:

In order to solve the equation:

  $9\left|4p+2\right|+8=35$

First, simplify the equation so that the only absolute value term is on one side and rest is on other side of the equation, as

  $\begin{array}{c}9\left|4p+2\right|+8=35\\ \Rightarrow 9\left|4p+2\right|+8-8=35-8\\ \Rightarrow 9\left|4p+2\right|=27\\ \Rightarrow \frac{9\left|4p+2\right|}{9}=\frac{27}{9}\\ \Rightarrow \left|4p+2\right|=3\end{array}$

Now, form the related equations of above:

  $4p+2=3,\text{ and }4p+2=-3$

Now, solve above equations first subtract 2 from each sides, as

  $\begin{array}{c}4p+2-2=3-2,\text{ and }4p+2-2=-3-2\\ \Rightarrow 4p=1,\text{ and }4p=-5\end{array}$

Now, isolate p as

  $\begin{array}{c}4p=1,\text{ and }4p=-5\\ \Rightarrow p=\frac{1}{4},\text{ and }p=\frac{-5}{4}\\ \Rightarrow \\ \Rightarrow p=0.25,\text{ and }p=-1.25\end{array}$

Thus, solutions for given equation are $p=0.25,\text{ and }p=-1.25$ .

Now, represent the solution of the given equation by representing points $-1.25$ , and 0.25 on the number line using a graphing tool.