• 7 CATs FREE!
If you earn 100 Forum Points

Engage in the Beat The GMAT forums to earn
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? ##### This topic has expert replies Elite Legendary Member Posts: 3030 Joined: 24 Jul 2015 Location: Las Vegas, USA Thanked: 19 times Followed by:36 members ### Is x^3-x^2+x-1 > 0? by Max@Math Revolution » Mon Jun 10, 2019 11:22 pm ## 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
Free Resources-30 day online access & Diagnostic Test
Email to : info@mathrevolution.com

GMAT Instructor
Posts: 15495
Joined: 25 May 2010
Location: New York, NY
Thanked: 13060 times
Followed by:1877 members
GMAT Score:790
by GMATGuruNY » Tue Jun 11, 2019 8:30 am

00:00

A

B

C

D

E

## Global Stats

Difficult

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.

xÂ³ - xÂ² + x - 1 > 0?
xÂ²(x-1) + (x-1) > 0
(x-1)(xÂ²+1) > 0
The two sides are equal only when x=1.
If we test x=0 and x=2 in xÂ³-xÂ²+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 xâ�µ>xÂ² implies that x is NONZERO, we can safely divide each side by xÂ², which must be positive:
xâ�µ/xÂ² > xÂ²/xÂ²
xÂ³ > 1
The resulting inequality implies that x>1.
Thus, the answer to the rephrased question stem is YES.
SUFFICIENT.

Statement 2:
xÂ³ + x > xÂ² + 1
xÂ³ - xÂ² + x - 1 > 0
Thus, the answer to the original question stem is YES.
SUFFICIENT.

Mitch Hunt
Private Tutor for the GMAT and GRE
GMATGuruNY@gmail.com

If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.

Available for tutoring in NYC and long-distance.
Student Review #1
Student Review #2
Student Review #3

Elite Legendary Member
Posts: 3030
Joined: 24 Jul 2015
Location: Las Vegas, USA
Thanked: 19 times
Followed by:36 members
by Max@Math Revolution » Wed Jun 12, 2019 6:09 pm

00:00

A

B

C

D

E

## Global Stats

Difficult

=>

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 â‰  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.

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
Free Resources-30 day online access & Diagnostic Test