Problem Solving

I have attached the screen shot of a question. I am not sure how option III is correct. GMATPrep solution is also given here. It says, Since X<-3 implies X<-1. But putting X=-2 in Mod will not hold given condition (mod X > 3) true.

|a| = the distance between a and 0.
|a-b| = the distance between a and b.

IF |X|>3, which of the following must be true?
A) X>3
B) X^2>9
c) |X-1|>2
I only
II only
I and II only
II and III only
I, II, and III

Constraint: |x| > 3
This means that the distance between x and 0 is greater than 3.
Any value in the two red ranges below satisfies this constraint:
<----(-3).......(3)---->

I: x>3
The red range on the left illustrates that x does not have to be greater than 3.
Eliminate A, C, and E.

III: |x-1| > 2.
This statement implies that the distance between x and 1 must be greater than 2.
Every value in the red ranges above is more than 2 places away from 1.
Thus, statement III must be true.
Eliminate B.

The correct answer is D.

|x-1| > 2 ==> -x+1 > 2 ==> x < -1 ...so if x = -2 but this is not valid if x < -3 then the condition does not hold...

c) |x-1| > 2 ==> 2 cases: x>3 or x<-1

Given |x|>3 we have x>3 or x<-3
as you can see in c) x>3 is satisfied and x<-1 is satisfied so the answer is correct.
x<-3 implies x<-1
I hope its clear now mgm. If you have further queries kindly advise and I wil be happy to help.

In my view OA should be B. As it is the only condition satisfying lxl<3.