question is is X<0??

s1) x^3(1-x^2)<0--->either x^3<0 or (1-x^2)<0 ---->either x<0 or x>0 hence insuff
s2) x^2-1<0--->x^2<1--->x can be >0 or <0 insuff

combining x^2 is always>0 so if x^2<1 then x is a fractional number therefore (1-x^2) is always +ve which means x^3<0---->x<0
sufficient

abhi758 wrote:Is X negative?

1. x^3(1-x^2)<0
2. x^2-1<0

Please explain your answer..

Ron, how do I go about applying your strategy of picking numbers combining both statements to check for option C or E right in the begining? Am looking for a 2 min solution here. Look forward to your reply.

C

There are two parts of the equation that are important to watch out for.
(x^3) always takes on the sign of X
(1-x^2) depends on how large X is. If x^2 is >1, it is positive, if x^2 is smaller than 1 it becomes negative.
note that X cannot be 0
pick:
-------- (x^3)*(1-x^2)-------
x=2: (Pos)*(neg) = neg
x=0.5: (Pos)*(pos) = pos
x=-0.5: (neg)*(pos) = neg
x=-2: (neg)*(neg) = pos

So, with both x=2 and x=-2 the equation holds, but you can't say whether X is negative, it could be both. => insufficient

Statement 2: restricts the value of X: x^2-1<0 => x^2<1, so this holds 1>X>-1 (because for any X larger than 1 or smaller than -1, the statement wouldn't hold). So X doesn't have to be negative. => insufficient

Together:
so: since 1>X>-1, it follows that (1-x^2) will always be positive!

Therefore X^3 must be negative for statement 1 to hold, which must X is negative!
Therefore, together the statements are sufficient

Agree with C.

Is x<0 ?

St1] x^3(1-x^2)<0
either x <0, while 1-x^2 >0 OR x>0, while 1-x^2 <0

Case1: x<0,while 1-x^2 >0
let's analyse whether 1-x^2 can be positive while x is negative.
1-x^2 > 0
Can 1>x^2 when x<0? Yes this is possible e.g. x= -0.5
Hence the answer to the main question is Yes.

Case2: x>0, while 1-x^2 <0
Can 1-x^2 < 0 while x>0?
Can 1<x^2 while x>0? Yes this is possible. e.g. x = 5
Hence our answer to the main question is No.

Insuff.

St2] x^2-1<0
This means x^2<1 means that x can be a positive or negative fraction eg. + or - 0.5.
Insuff.

Together the result matches our case1 presented above, and x has to be a negative fraction.

I discussed this question here: https://www.beatthegmat.com/og-11-154-be ... tml#202306