If x^3 < x^2, what is the value of the integer x?

[BTGmoderatorDC]

If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

OA D

Source: Veritas Prep

[fskilnik@GMATH]

If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x
Source: Veritas Prep

$$x\,\,{\mathop{\rm int}} \,\,\left( * \right)$$
$${x^3} < {x^2}\,\,\,\,\, \Leftrightarrow \,\,\,\,\,{x^2}\left( {x - 1} \right) < 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x \ne 0\,\,\,{\rm{and}}\,\,x < 1\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,x \le - 1\,\,\,\,\left( {**} \right)\,\,$$
$$? = x$$

$$\left( 1 \right)\,\,\left| x \right| < 2\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\left| x \right| = 0\,\,\,{\rm{or}}\,\,\,\,\left| x \right| = 1\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,x = - 1\,\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.\,\,\,\,$$

$$\left( 2 \right)\,\,{x^3} = x\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x\left( {x + 1} \right)\left( {x - 1} \right) = x\left( {{x^2} - 1} \right)\,\, = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,x \in \left\{ {0,-1,1} \right\}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,x = - 1\,\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.\,\,$$

The correct answer is (D).


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[[email protected]]

Hi All,

We're told that X is an INTEGER and X^3 is LESS than X^2. We're asked for the value of X. This question can be solved with a mix of Number Properties and TESTing VALUES.

To start, there are only certain types of values that will fit the given information that X^3 is less than X^2:
-ANY negative value
-Positive fractions (0 < X < 1)

1) |X| < 2

With Fact 1, we have just 3 possible values for X: -1, 0 and +1. However, only one of those values (-1) fits the given information that X^3 < X^2. Thus, -1 is the value of X.
Fact 1 is SUFFICIENT

2) X^3 = X

With Fact 2, we again have just 3 possible values for X: -1, 0 and +1. From our work in Fact 1, we know that just one of those values 'fits' everything that we're told: - 1.
Fact 2 is SUFFICIENT

Final Answer: D

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