Hi chipbmk,chipbmk wrote:Is X negative?
1)x^3(1-x^2)<0
2)x^2-1<0
OA: C
Please let me know what the best way is to tackle this one.
Thanks!
By reasoning first, and alegebra second!
Statement 1:
x^3(1-x^2)<0
So, the left hand side is negative. Instead of x^3(1-x^2)<0, try to look at it as: (number)(another number) < 0.
How can the product of two numbers be negative: only if one is positive and the other negative. Accordingly, as soon as we see one number can be either pos or neg we know the statement is not sufficient.
We know x^3 will have the same sign as x. So, like any unknown x, x^3 can be pos or neg.
Statement 1 is not sufficient.
Statement 2:
x^2-1<0
So, again, the left hand side is negative (and these two statements look suspiciusly similar).
We know squares are always positive, so, if x were 2 or 3 it will remain positive even if we subtracted 1 from it. Therefore, x must be a fraction (when we square a fraction we get an even smaller fraction). But it can be either a positive or negative fraction.
Insufficient.
combo:
The fact that these expressions are similar is telling us something.
We have a "1-x^2" in statement 1 and "x^2 -1". Notice that the 1 and x^2 have exactly opposite signs. This means:
(1-x^2) = (-1)*(x^2 -1) Or: (x^2-1) = (-1)*(1-x^2)
So, if from statement 2, x^2-1 is (-), then the (1-x^2) in statement 1 is positive. Then, we have:
x^3*(pos) < 0
This means that x^3 is negative. (Product of two numbers is negative only when one is positive and the other negative).
And that means that x is negative. (x^3 and x will have same sign).
So, x is negative, the answer to the question is "yes", and the statements, although insufficient in isolation, are sufficient in combination.
Choose C
