Chapter 2, Problem 10CT

To determine

To solve: the absolute inequality $\left|2q+8\right|>4$ and graph the solution.

Answer to Problem 10CT

  $q<-6\text{ or }q>-2$

Explanation of Solution

Concept Used:

Given the absolute inequality $\left|ax+b\right|>c$ where $c>0$ , it will either be:

$ax+b<-c$ Or $ax+b>c$

Calculation:

Consider the absolute inequality,

  $\left|2q+8\right|>4$

Write its corresponding compound inequality,

$2q+8<-4$ Or $2q+8>4$

Solve the two inequalities individually,

First consider $2q+8<-4$ ,

Subtract 8 from both sides of the inequality,

  $\begin{array}{c}2q+8-8<-4-8\\ 2q<-12\end{array}$

Divide both sides of the inequality by 2,

  $\begin{array}{c}\frac{2q}{2}<\frac{-12}{2}\\ q<-6\end{array}$

Now consider $2q+8>4$ ,

Subtract 8 from both sides of the inequality,

  $\begin{array}{c}2q+8-8>4-8\\ 2q>-4\end{array}$

Divide both sides of the inequality by 2,

  $\begin{array}{c}\frac{2q}{2}>\frac{-4}{2}\\ q>-2\end{array}$

So, the solution of given inequality is $q<-6\text{ or }q>-2$ .

Now, graph $q<-6\text{ or }q>-2$

Now because of $>$ and $<$ symbols, both -6 and -2 are represented on number line by an open circle.

And the line to the left of -6 is shaded to represent $q<-6$ and the line to the right of -2 is shaded to represent $q>-2$

The graph of $q<-6\text{ or }q>-2$ is given below: