Chapter 1.6, Problem 39PPS

To determine

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

Answer to Problem 39PPS

The solution is given by $f>\frac{26}{5}\text{or}f<-\frac{22}{5}$ .

Explanation of Solution

Given information:

Given inequality

  $\frac{\left|5f-2\right|}{6}>4$

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

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

Calculation:

Consider the given inequality,

  $\frac{\left|5f-2\right|}{6}>4$

Now, solve the inequality as shown,

  $\begin{array}{c}\frac{\left|5f-2\right|}{6}>4\\ \left|5f-2\right|>24\\ 5f-2>24\text{or}5f-2<-24\\ 5f-2+2>24+2\text{or}5f-2+2<-24+2\\ 5f>26\text{or}5f<-22\\ f>\frac{26}{5}\text{or}f<-\frac{22}{5}\end{array}$

Hence, the solution of the inequality is $f>\frac{26}{5}\text{or}f<-\frac{22}{5}$ .

Now, graph the inequality as shown:

  

The open ball suggests that the solution region is not included the points $\frac{26}{5}$ and $-\frac{22}{5}$ .

The sign indicates the region is left side to the point $-\frac{22}{5}$ and right side to the point $\frac{26}{5}$ on number line.