Anytime you run into a problem that involves possible fractions and/or possible negative numbers, you want to consider four key ranges:
1. x < -1
2. -1 < x < 0
3. 0 < x < 1
4. x > 1
Let's see how that works for this question:
Is x Between 0 and 1 ?
(1) x^2 is less then X
x^2 is always either zero or positive. It can't be zero, because then x^2 wouldn't be less than X. So we know x^2 is positive.
If 0 < x < 1 (i.e. if x is a proper fraction), then x^2 is less than x. For example (1/2)^2 = 1/4 < 1/2.
If x > 1, then x^2 is greater than x. For example, 2^2 = 4 > 2.
So x MUST be between 0 and 1. SUFFICIENT
(2) X^3 is positive
If a number is cubed, and the result is positive, the number itself must be positive. So if x^3 is positive, then x must be positive.
You can show this by picking numbers in either of our positive ranges:
For 0 < x < 1, pick x = 1/2. (1/2)^3 = 1/8, which is positive
For x > 1, pick x = 2. 2^3 = 8, which is positive.
Because both of these ranges work, you can't say for sure whether x is between 0 and 1. INSUFFICIENT
Rich Zwelling
GMAT Instructor, Veritas Prep
