I don't understand why statement (1) is insufficient!!

Since module X is always >0, we can change the equation into x<x*module x

Form here we consider three possible variants:
a) x<0
then x>x*module x
(for instance, -2>-2*2)
but this contradicts our initial condition, so we drop it

b) 0<x<1
then x>x*module x
(for example, 1/2>1/2*1/2)
we see contradiction again, hence, this variant is not workable

c) x>1
then x<x*module x
(e.g. 2<2*2)
this is the only variant that satisfies our problem, we conlcude that module x>1, hence statement (1) should be sufficient.

Could anyone tell me what is the flaw in my reasoning? Will greatly appreciate your input.

I got it!

Instead of variant a), we should consider

a.1) -1<x<0
then x<x*module x
(see, -1/2<-1/2*1/2)
satisfies our statement

a.2) x<-1
x>x*module x
(-2>-2*2)
violates statement

So, we have two possible options when x<x*module x, in particular, option a.1) and c). And no guidance as which one to chose. Hence, statement (1) is insufficient.

In statement 1 I tried a negative fraction and a negative integer for x:

(-1/4)/(1/4)=-1 x<1 |x|<1

(-2)/2=-1 x<1 |x|>1

Therefore you cannot tell.
Hence statement 1 is insufficient.

For statement 1
case i. if x >0, |x| > 1 therefore x > 1.
case ii. if x < 0, |x| < 1 therefore -1< x <0.
so we have x between -1 and 0 or x > 1.
stmt 1 is insufficient.

from stmt 2 we get x <0. insufficient.

combining stmt 1 and 2 we get -1< x<0 so |x| <1.