x^3 < x^2 Describe the possible values of x

Hi Sri,

Rather than go about this algebraically, you can TEST VALUES to figure out what's "possible" and what's "not"

X^3 < X^2

Let's start with the obvious: POSITIVE INTEGERS. Will ANY positive integers "fit" this inequality?

X = 1 gives us 1 < 1; this does NOT fit
X = 2 gives us 8 < 4; this does NOT fit

As X becomes a larger positive integer, we run into the same problem: X^3 is NOT < X^2. We've eliminated a set of values. X CANNOT be greater than or equal to 1.

Next, let's try NEGATIVE INTEGERS. There's a great Number Property here…

(Negative)^3 = Negative
(Negative)^2 = Positive

From this, we can deduce that ANY NEGATIVE (integer or non-integer) will "FIT" this inequality.

We're not done though. We still have to deal with 0 and POSITIVE FRACTIONS.

If X = 0, then we have 0 < 0; this does NOT fit.

If X = 1/2, then we have 1/8 < 1/4; this DOES fit. We can also make a deduction from this…

Cubing a positive fraction that is less than 1 IS less than squaring that same positive fraction.

From these TESTS, we've deduced the following:
X CANNOT be a positive integer nor can it be greater than 1
X CANNOT be 0

X CAN be ANY negative
X CAN be ANY positive fraction less than 1

GMAT assassins aren't born, they're made,
Rich