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# Is x greater than x^3? (1) x is negative. (2) x^2 - x^3 > 2

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Re: Is x greater than x^3? (1) x is negative. (2) x^2 - x^3 > 2 [#permalink]
Bunuel wrote:
Is x greater than x^3?

(1) x is negative.
(2) x^2 - x^3 > 2

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.

We can modify the question as follows.
$$x > x^3$$
$$⇔ x^3 - x < 0$$
$$⇔ x(x^2-1) < 0$$
$$⇔ x(x+1)(x-1) < 0$$
$$⇔ x < -1$$ or $$0 < x < 1$$ from the graph of $$y = x(x+1)(x-1)$$

Condition 1)
$$x < 0$$
Since the range of the question does not include that of the condition 1), this is not sufficient.

Condition 2)
$$x^2 - x^3 > 2$$
$$⇔ x^3 - x^2 + 2 < 0$$
$$⇔ (x+1)(x^2 - 2x + 2 ) < 0$$
$$⇔ x + 1 < 0$$ since $$x^2-2x+2 = (x-1)^2 + 1 ≥ 1 > 0$$.
$$⇔ x < -1$$
Since the range of the question includes that of the condition 2), this is sufficient.