Chapter 1.6, Problem 38PPS

To determine

To solve: the given inequality and graph the solution on number line.

Answer to Problem 38PPS

The solution is given by $-\frac{9}{4}<p<\frac{5}{4}$ .

Explanation of Solution

Given information:

Given inequality

  $6\left|4p+2\right|-8<34$

Concept used:

Absolute value of a function a is given by

  $\left|a\right|=\{\begin{array}{l}aa>0\\ -aa<0\\ 0a=0\end{array}$ \

And

  $\begin{array}{l}\left|-a\right|=a\\ \left|a\right|=a\end{array}$

More over

  $\begin{array}{l}\left|x+a\right|<b\\ \Rightarrow -b<x+a<b\end{array}$

Calculation:

Consider the given inequality,

  $6\left|4p+2\right|-8<34$

Now, solve the inequality as shown,

  $\begin{array}{c}6\left|4p+2\right|-8<34\\ 6\left|4p+2\right|-8+8<34+8\\ 6\left|4p+2\right|<42\\ \left|4p+2\right|<7\\ -7<4p+2<7\\ -7-2<4p+2-2<7-2\\ -9<4p<5\\ -\frac{9}{4}<p<\frac{5}{4}\end{array}$

Hence, the solution of the inequality is $-\frac{9}{4}<p<\frac{5}{4}$ .

Now, graph the inequality as shown:

  

The open ball suggests that the solution region is not included the points $-\frac{9}{4}$ and $\frac{5}{4}$ .

The sign indicates the region is left side to the point $\frac{5}{4}$ and right side to the point $-\frac{9}{4}$ on number line.