Inequalities ques of CAT exam #4..plz explain.

[aksheymehta31]

If x is not equal to 0, is |x| less than 1?

(1) x/|x|< x

(2) |x| > x

Here is how i solved it.

In the question they have asked is |x| less than 1 that is
-1< x <1.

By solving first option we get that for positive values
x>1 and for negative values x<-1,that means
from 1st option we get that x does not lie between -1 and 1,hence we get the answer as NO.

From the second option we get that its true only for all the negative values hence its not sufficient.

So the ans according to me should be A but the official
answer is C.

Please explain the official answer.

[parthatayi]

Lets consider the first condition:
x/|x|< x
Multiply by |x| on both sides as |x| is always positive:
x<|x|.x
Case (i) if x>o
=> x<x^2
=>x^2 - x>0
=> x>1 or x<0

Case(ii)
if x<0
=>(x^2 +1)<0
=> -1<x<0

From option 1 we know that
x is either -1<x<0 or x>1

from option 2 we can conclude that x<0

Therefore combining both option 1 and option we can confirm that -1<x<0

[RonPurewal]

statement (2) means that x is negative.
this is not enough information to tell whether |x| is less than 1.
insufficient.

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to interpret statement (1), note that the fraction x/|x| is equal to 1 for any positive value of x, and equal to -1 for any negative value of x.
therefore, to solve this equation, and just consider the positive and negative cases separately.
if x is a positive number, then this inequality can be rewritten as 1 < x.
if x is a negative number, then this inequality can be rewritten as -1 < x. since this only applies to negative values, we can amend this to give -1 < x < 0.

therefore, statement (1) means that EITHER x > 1 OR -1 < x < 0.
for the first possibility, |x| is greater than 1; for the second, |x| is less than 1. insufficient.

--

together:
the only interval that satisfies both statements is -1 < x < 0, in which all numbers satisfy |x| < 1.
sufficient.