Hi, I'm happy to help with this.

Fundamentally, absolute value is about

distance. That is something many folks fail to grasp.

|x| is the distance of x from the origin on the number line.

|x - 5| is the distance of x from the point 5 on the number line.

|x + 3| is the distance of x from the point -3 on the number line. (That's because x + 3 = x - (-3))

So, we want to write 1<x<9 as a distance statement.

Step #1: find the middle of that region. The midpoint is at 5. So, this is about distance from 5 ---> |x - 5|

Step #2: how far can we get from 5? Well, up to almost 9, or down to almost 1 -- that is, almost 4. Thus:

distance from 5 < 4

|x - 5| < 4

Once you understand the distance-interpretation of absolute value, you need not spend much time at all on questions like this.

Does that make sense? Please let me know if I can clarify anything here or if you have any more questions.

Here's a challenging absolute value question for practice.

http://gmat.magoosh.com/questions/964Once you submit your answer, that question should be followed by a video explanation of the answer.

Mike

_________________

Mike McGarry

Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)