how can you find an absolute value inequality from a number line?

Question

for example, someone gives you a number line with 0 in the middle, and on the left side of the zero, there is a -5 and an arrow going downwards (so x<-5), and on the right side of the zero, you have a 1/2 and an arrow going up (so x>1/2). how can you write an absolute value equation from the information given?

Answer 1

The midpoint of the line segment from -5 to 1/2 is (-5 + 1/2)/2 = -9/4.  The distance from -5 to -9/4 and the distance from -9/4 to 1/2 is 11/4.  

 

So, if x is either less than -5 or greater than 1/2, then x is more than 11/4 units from -9/4

 

This can be written as l x - (-9/4) l > 11/4

 

Simplify:    l x + 9/4 l > 11/4

 

                  l 4x + 9 l > 11