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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Is x^3-x^2+x-1 > 0? tagged by: Max@Math Revolution ##### This topic has 2 expert replies and 0 member replies ### GMAT/MBA Expert ## Is x^3-x^2+x-1 > 0? ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult [GMAT math practice question] Is x^3-x^2+x-1 > 0? 1) x^5 > x^2 2) x^3 + x > x^2 + 1 _________________ Math Revolution Finish GMAT Quant Section with 10 minutes to spare. The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. Only$149 for 3 month Online Course
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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 x⁵>x² implies that x is NONZERO, we can safely divide each side by x², which must be positive:
x⁵/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.

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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.

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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