Is x negative?

[BTGmoderatorDC]

Is x negative?

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

OA C

Source: Official Guide

[Jay@ManhattanReview]

Is x negative?

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

OA C

Source: Official Guide

Let's take each statement one by one.

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

Since x^3(1 - x^2) is less than 0, x cannot be 0.

From x^3(1 - x^2) < 0, we have x^3 - x^5 < 0. Divinding the inequality throught by x^2, we get x < x^2.

From x < x^2, we get

Case 1: Say x = 2, then x^2 = 4; thus, we do not have x as negative. The answer is no.
Case 2: Say x = -1/2, then x^2 = 1/4; thus, we have x as negative. The answer is yes.

No unique answer. Insufficient.

(2) x^2 - 1 < 0

x^2 < 1

Case 1: Say x = -1/2, then x^2 = 1/4; thus, we have x as negative. The answer is yes.
Case 2: Say x = 1/2, then x^2 = 1/4; thus, we do not have x as negative. The answer is no.

No unique answer. Insufficient.

(1) and (2)

From both, we have x < x^2 < 1; thus, Case 1 of Statement 1 and Case 2 of Statement 2 are not applicable. Thus, x is negative. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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