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Is x^3-x^2+x-1 > 0?

by Max@Math Revolution » Mon Jun 10, 2019 11:22 pm

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[GMAT math practice question]

Is x^3-x^2+x-1 > 0?

1) x^5 > x^2
2) x^3 + x > x^2 + 1
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by GMATGuruNY » Tue Jun 11, 2019 8:30 am

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Max@Math Revolution wrote:[GMAT math practice question]

Is x^3-x^2+x-1 > 0?

1) x^5 > x^2
2) x^3 + x > x^2 + 1
A critical point occurs when the two sides of an equality are EQUAL.
To determine the ranges where the left side is GREATER than the right side, test one value to the left and right of each critical point.

x³ - x² + x - 1 > 0?
x²(x-1) + (x-1) > 0
(x-1)(x²+1) > 0
The two sides are equal only when x=1.
If we test x=0 and x=2 in x³-x²+x-1>0, only x=2 is viable, implying that the inequality will hold true only for values greater than 1.
Question stem, rephrased:
Is x > 1?

Statement 1:
Since x�>x² implies that x is NONZERO, we can safely divide each side by x², which must be positive:
x�/x² > x²/x²
x³ > 1
The resulting inequality implies that x>1.
Thus, the answer to the rephrased question stem is YES.
SUFFICIENT.

Statement 2:
x³ + x > x² + 1
x³ - x² + x - 1 > 0
Thus, the answer to the original question stem is YES.
SUFFICIENT.

The correct answer is D.
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by Max@Math Revolution » Wed Jun 12, 2019 6:09 pm

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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.

In inequality questions, the law "Question is King" tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient.

x^3-x^2+x-1 > 0
=> x^2(x-1) + (x-1) > 0
=> (x^2+1)(x-1) > 0
=> (x-1) > 0, since x^2+1 > 0
=> x>1
The question asks if x > 1.

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
x^5 > x^2
=> x^5 - x^2 > 0
=> x^2(x^3 - 1) > 0
=> x^2(x-1)(x^2+x+1) > 0
=> x - 1 > 0 since x^2+x+1 >0, and x^2 > 0 if x ≠ 0
x > 1
This condition is equivalent to the question. Therefore, condition 1) is sufficient.

Condition 2)
x^3 + x > x^2 + 1
=> x^3 - x^2 + x - 1 > 0
=> x^2(x-1)+ (x-1) > 0
=> (x^2+1)(x-1) > 0
=> (x-1) > 0, since x^2+1 > 0
=> x > 1
This condition is equivalent to the question. Therefore, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).

If the original condition includes "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations" etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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