Is x > 0? (1) (x^2 - 1)(x^3) > 0 (2) x^2 < 1

Hi..
I wrote the opposite signs by mistake..
So (x^2-1)(x^3)>0...
So two case
1) both are positive
x^3>0, x^2>1..X>1
2) both negative
x^3<0, x^2<1...-1<X<0
Insufficient

But statement II says x^2<1..
Individually it is insufficient
But when you add it to the info of statement I..
x^2<1 means x^3 is also <0 or X<0
Sufficient

chetan2u

Just to be sure i fully understand it:

S(1) + S(2): results in the following range for x -->

-1<x<0

Which implies that x is not x>0, am I correct???

Yes combined it tells us -1<X<0

OA:C

\((1) \quad (x^2 - 1)(x^3) > 0\)

It leads to two cases
Case 1
\((x^2-1)>0\) AND \(x^3>0\)
This leads to \(x>1\)

OR

Case 2
\((x^2-1)<0\) AND \(x^3<0\)
This leads to \(-1<x<0\)

So \(x\) can be either \(x>1\) OR \(-1<x<0\)
Statement \(1\) alone is insufficient

\((2) \quad x^2 < 1\)
This leads to \(-1<x<1\)
Statement \(2\) alone is insufficient

Combining \((1)\) and \((2)\), we get \(-1<x<0\).
After combining \((1)\) and \((2)\), there is a definite answer to the question: Is x > 0? No

This is a yes/no Q.

Start with statement 2 as it is easier to evaluate. According to statement 2 x could be a positive fraction such that x lies between 0 & 1 or a negative fraction which lies between -1 and 0. Clearly we have two answers (a Yes & a No) to the same questions hence statement 2 alone is not sufficient to answer whether x>0? Strike out option B & D.

Statement 1 alone would give the same result as in statement 2; if x is a negative fraction which lies between -1 and 0 then we have a no but if x >1 then we have a yes. Hence insufficient.

Combining 1 and 2 we get only one answer that is x is a negative fraction which lies between -1 and 0. And we have a No to the Q. Ans C is correct.

Target question: Is x > 0?

Statement 1: (x² - 1)(x³) > 0
Let's TEST some values.
There are several values of x that satisfy statement 1. Here are two:
Case a: x = 2. In this case, the answer to the target question is YES, x is greater than 0
Case b: x= -2. In this case, the answer to the target question is NO, x is not greater than 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x² < 1
Let's TEST some values.
There are several values of x that satisfy statement 2. Here are two:
Case a: x = 0.5. In this case, the answer to the target question is YES, x is greater than 0
Case b: x= -0.5. In this case, the answer to the target question is NO, x is not greater than 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 2 tells us that x² < 1
Subtract 1 from both sides to get: x² - 1 < 0
In other words, x² - 1 is negative
Statement 1 tells us that (x² - 1)(x³) > 0
Divide both sides of the inequality by (x² - 1) to get: x³ < 0 [ since we divided both sides by a NEGATIVE value, we REVERSED the direction of the inequality symbol]
If x³ < 0, then we can be certain that x is negative
The answer to the target question is NO, x is not greater than 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

RELATED VIDEO

1) Either (x^2 - 1) and x^3 are both positive or both negative. For x>0, its always true. Now, when x < 0, x^3 will be negative, and to make x^2 -1 <0, we have to take x as -1 < x < 0 (x cannot be greater than 0 as in that case x^3 will be positive and the inequality will be less than 0). x can be either positive or negative.
2) -1 < x < 1. Not sufficient.

Together, x lies in the range -1< x< 0. Sufficient.
C is the answer