Question

State an absolute value inequality for each of the following. Then graph each inequality 1. All numbers between -3 and 3 2. All numbers greater than 11 or less than -11

Answer 1

Recall the following properties of the absolute value:

`\begin{gathered} |a|\le b\Leftrightarrow-b\le a\le b, \\ |a|>b\Leftrightarrow a>b\text{ or a<-b.} \end{gathered}`

Therefore, all numbers between -3 and 3 can be represented by the following absolute value inequality:

`|x|\le3.`

All numbers greater than 11 or less than -11 can be represented by the following absolute value inequality:

`|x|>11.`

Answer:

1) All numbers between -3 and 3:

`|x|\le3.`

2) All numbers greater than 11 or less than -11:

`|x|>11.`