Chapter 1.6, Problem 40PPS

To determine

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

Answer to Problem 40PPS

The solution is given by $w\ge \frac{7}{2}\text{or}w\le -\frac{23}{2}$ .

Explanation of Solution

Given information:

Given inequality

  $\frac{\left|2w+8\right|}{5}\ge 3$

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

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

Calculation:

Consider the given inequality,

  $\frac{\left|2w+8\right|}{5}\ge 3$

Now, solve the inequality as shown,

  $\begin{array}{c}\frac{\left|2w+8\right|}{5}\ge 3\\ \left|2w+8\right|\ge 15\\ 2w+8\ge 15\text{or}2w+8\le -15\\ 2w+8-8\ge 15-8\text{or}2w+8-8\le -15-8\\ 2w\ge 7\text{or}2w\le -23\\ w\ge \frac{7}{2}\text{or}w\le -\frac{23}{2}\end{array}$

Hence, the solution of the inequality is $w\ge \frac{7}{2}\text{or}w\le -\frac{23}{2}$ .

Now, graph the inequality as shown:

  

The closed ball suggests that the solution region is excluded the points $\frac{7}{2}$ and $-\frac{23}{2}$ .

The sign indicates the region is left side to the point $-\frac{23}{2}$ and right side to the point $\frac{7}{2}$ on number line.