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Is (x-1)^3< (x-1)?

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Is (x-1)^3< (x-1)?

by Max@Math Revolution » Mon Mar 12, 2018 11:51 pm
[GMAT math practice question]

$$Is\ (x-1)^3<(x-1)?$$

$$1)\ x>-1$$
$$2)\ x<0$$
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Source: — Data Sufficiency |
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by GMATGuruNY » Tue Mar 13, 2018 4:37 am
Max@Math Revolution wrote:[GMAT math practice question]

$$Is\ (x-1)^3<(x-1)?$$

$$1)\ x>-1$$
$$2)\ x<0$$
a³ < a if a < -1 or 0 < a < 1.
By extension, (x-1)³ < (x-1) in the following cases:
Case 1: x-1 < -1, with the result that x < 0
Case 2: 0 < x-1 < 1, with the result that 1 < x < 2
Question stem, rephrased:
Is x < 0 or 1 < x < 2?

Statement 1: x > -1
If x=10, then the answer to the rephrased question stem is NO.
If x=-1/2, then the answer to the rephrased question stem is YES.
INSUFFICIENT.

Statement 2: x < 0
Thus, the answer to the rephrased question stem is YES.
SUFFICIENT.

The correct answer is B.
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by Max@Math Revolution » Thu Mar 15, 2018 12:59 am
=>

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. We then recheck the question.

The question can be modified as follows:
(x-1)^3< (x-1)
=> (x-1)^3- (x-1) < 0
=> (x-1)((x-1)^2- 1) < 0
=> (x-1)((x-1)+1)((x-1)-1) < 0
=> x(x-1)(x-2) < 0
=> x < 0 or 1 < x < 2
So, the question asks whether x < 0 or 1 < x < 2.

Condition 1)
Since the set of the question doesn't include that of condition 1), it is not sufficient.

Condition 2)
Since the set of the question includes that of condition 2), it is sufficient.

Therefore, the answer is B.

Answer: B
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