BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Inequalities

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
User avatar
Senior | Next Rank: 100 Posts
Posts: 75
Joined: Fri Jun 26, 2015 7:43 am

Inequalities

by gmat_for_life » Sat Mar 12, 2016 12:31 pm
If x and y are nonzero integers, is (x^-1 + y^-1)^-1 > [(x^-1)(y^-1)]^-1 ?

(1) x = 2y

(2) x + y > 0

The official answer is A. However, if the above algebraic expression is simplified to xy(1-x-y)>0? option A seems insufficient. If we assume x and y as -2 and -1, the simplified equation is greater than 0. However if we assume x and y as 2,1, the same equation is less than 0. Could you guys please let me know where I'm going wrong?

Regards,
Amit
Top
Source: — Data Sufficiency |
User avatar
Junior | Next Rank: 30 Posts
Posts: 26
Joined: Tue Feb 02, 2016 10:25 am
Thanked: 1 times

by John fran kennedi » Sat Mar 12, 2016 9:47 pm
Could you please write down how you did simplify it ?
Top
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Sun Mar 13, 2016 4:29 am
gmat_for_life wrote:If x and y are nonzero integers, is (x^-1 + y^-1)^-1 > [(x^-1)(y^-1)]^-1 ?

(1) x = 2y

(2) x + y > 0
(xˉ¹ + yˉ¹)ˉ¹ > (xˉ¹yˉ¹)ˉ¹ ?

(1/x + 1/y)ˉ¹ > (1/x * 1/y)ˉ¹ ?

[ (x+y)/xy ]ˉ¹ > xy ?

xy/(x+y) > xy ?

Statement 1:
Substituting x=2y into xy/(x+y) > xy, we get:
(2y)(y) / (2y+y) > (2y)(y) ?

2y² / 3y > 2y² ?

y² / 3y > y² ?

Since y is NONZERO, y² > 0, allowing us to divide both sides by y²:
1/3y > 1 ?

1/y > 3 ?

Since y is an INTEGER, it is not possible that 1/y > 3.
Thus, the answer to the question stem is NO.
SUFFICIENT.

Statement 2:
Substituting x=1 and y=1 into xy/(x+y) > xy, we get:
(1*1)/(1+1) > (1*1) ?
1/2 > 1 ?
In this case, the answer to the question stem is NO.

Substituting x=-1 and y=3 into xy/(x+y) > xy, we get:
(-1*3)/(-1+3) > (-1*3) ?
-3/2 > -3 ?
In this case, the answer to the question stem is YES.

Since the answer is NO in the first case but YES in the second case, INSUFFICIENT.

The correct answer is A.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top
GMAT Instructor
Posts: 2630
Joined: Wed Sep 12, 2012 3:32 pm
Location: East Bay all the way
Thanked: 625 times
Followed by:119 members
GMAT Score:780

by Matt@VeritasPrep » Thu Mar 17, 2016 11:04 pm
We could also use

Is (xˉ¹ + yˉ¹)ˉ¹ > (xˉ¹yˉ¹)ˉ¹ ?

Is (xˉ¹ + yˉ¹)ˉ¹ - (xˉ¹yˉ¹)ˉ¹ > 0 ?

Is 1/(1/x + 1/y) - 1/(1/xy) > 0?

Is xy/(x+y) - xy > 0 ?

Is xy * (1/(x+y) - 1) > 0?

Given S1, we have

Is 2y² * (1/3y - 1) > 0 ?

Is 2y² * (1 - 3y)/3y > 0 ?

Since 2y² > 0, we can divide both sides to reach

Is (1 - 3y)/3y > 0 ?

This will be positive if (1 - 3y) and 3y share a sign. But if y > 1/3, then the first is negative and the second is positive, and if 0 > y the first is positive and the second is negative. So for any nonzero integer, we'll have conflicting signs, and (1 - 3y)/3y CANNOT be positive; SUFFICENT.

Given S2, we have

Is xy * (1/(x+y) - 1) > 0? (from our simplified stem)

Is xy * (1/positive - 1) > 0?

If (x + y) = 1, then 1/(x+y) - 1 is 0, so we can't have a result > 0. But if (x + y) = 2, then 1/(x+y) - 1 is negative, so the question becomes "Is xy also negative?" We don't have enough information to say, so this statement is INSUFFICIENT.
Top
Post new topic Post Reply
• Page 1 of 1