How to write an absolute inequality for a graph?

Question

Absolute value inequality for : x<-9 or 3<x

Answer 1

Given inequalities are x < -9 and 3 > x

Graph of the inequlities are

Answer 2

The inequalities are x  < -9 or  3 < x

x  < -9 or x  > 3

First look the end points -9 and 3

Negitive 9 and 3 are 12 units a part. Half of the 12 is 6.

Adjust this inequality so that it relates to -6 or 6.

x  < -9 or x  > 3

Since -9 + 3 = 6 and 3 +3 = 6, need to add 3 all around.

x + 3 < -9 + 3 or x  + 3 > 3 + 3

x + 3 < -6 or x + 3 > 6

Since the last line above greater than format the absolute-value inequality will be of the form "absolute value of something is greater than or equal to 6 ". I can convert this nicely to:

| x  + 3| > 6

Absolute inequality is | x  + 3| > 6.

Graph of absolute inequality

.

Comments

Graph of the absolute value inequality :