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x is not equal to 0

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Source: — Data Sufficiency |
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by neelgandham » Fri May 18, 2012 12:23 pm
For the benefit of unregistered users, here is the question.

If x is not equal to 0, is |x| less than 1?
(1) x/|x| < x
x < x * |x|
Case 1: x<-1
Let x =-2, then
Is x < x * |x|?
Is -2 < -2 *|-2|?
Is -2<-4?
Answer : NO

Case 2: -1<x<0
Let x =-1/2, then
Is x < x * |x|?
Is -1/2 < -1/2 *|-1/2|?
Is -1/2<-1/4?
Answer : YES

Case 3: 0<x<1
Let x =1/2, then
Is x < x * |x|?
Is 1/2 < 1/2 *|1/2|?
Is 1/2<1/4?
Answer : NO

Case 4: x>1
Let x =2, then
Is x < x * |x|?
Is 2 < 2 *|-2|?
Is 2<4?
Answer : YES

The solution set is -1<x<0 and x>1. So, |x| need not necessarily be less than one. Statement 1 is insufficient to answer the question.
(2) |x| > x
Implies x<0.
So, |x| need not necessarily be less than one. Statement 2 is insufficient to answer the question.
1 + 2
The intersection of '-1<x<0 and x>1' and 'x<0' is -1<x<0. So, |x|<1. Statement 1 + 2 combined is sufficient to answer the question.

IMO C
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by Anurag@Gurome » Fri May 18, 2012 10:19 pm
GmatKiss wrote:If x is not equal to 0, is |x| < 1?

(1) x/|x|< x

(2) |x| > x
|x| < 1 means is -1 < x < 1?

(1) x/|x|< x

Case I: If x < 0, then x/(-x) < x or -1 < x
But x < 0, so -1 < x < 0.

Case II: If x > 0, then x/x < x or 1 < x

So, -1 < x < 0 or x > 1, which is NOT sufficient to find if -1 < x < 1 or not.

(2) |x| > x
If x ≥ 0, then |x| = x, so we know that x is < 0 or negative.
But still we cannot find if -1 < x < 1 or not; NOT sufficient.

Combining (1) and (2), we know that -1 < x < 0
So, -1 < x < 1 holds true; SUFFICIENT.

The correct answer is C.
Anurag Mairal, Ph.D., MBA
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