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If x ≠ y, is 1/(x - y) < y - x?

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If x ≠ y, is 1/(x - y) < y - x? [#permalink]

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New post 24 Jan 2011, 06:34
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If x ≠ y, is 1/(x - y) < y - x?

(1) x^2 + y^2 < 4xy

(2) y < x
[Reveal] Spoiler: OA

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Re: If x ≠ y, is 1/(x - y) < y - x? [#permalink]

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New post 24 Jan 2011, 07:54
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rxs0005 wrote:
if x <> y

is 1 / x - y < y - x


S1 x^2 + y^2 < 4xy

S2 y < x


If \(x\neq{y}\) is \(\frac{1}{x-y}<y-x\)?

(1) x^2 + y^2 < 4xy --> if \(x=1>y=\frac{1}{2}\) then the answer will be NO as in this case \(LHS=\frac{1}{x-y}>0\) and \(RHS=y-x<0\), so \(LHS>RHS\) but if \(x=\frac{1}{2}<y=1\) then the answer will be YES, as in this case \(LHS=\frac{1}{x-y}<0\) and \(RHS=y-x>0\), so \(LHS<RHS\). Not sufficient.

(2) y < x --> we can rewrite it as \(y-x<0\) or \(x-y>0\) --> in this case \(LHS=\frac{1}{x-y}>0\) and \(RHS=y-x<0\), so \(LHS>RHS\) directly gives the answer NO to he question. Sufficient.

Answer: B.

Or you can simplify the question: is \(\frac{1}{x-y}<y-x\)? --> is \(y-x+\frac{1}{y-x}>0\)? is \(\frac{(y-x)^2+1}{y-x}>0\)? As the nominator (\((y-x)^2+1\)) is always positive then the question basically becomes whether denominator (\(y-x\)) is positive too --> is \(y-x>0\)? or is \(y>x\)?

(1) Not sufficient.
(2) Answers NO to the question. Sufficient.

Answer: B.
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Re: If x ≠ y, is 1/(x - y) < y - x? [#permalink]

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New post 24 Jan 2011, 08:00
The question is \(\frac{1}{(x - y)} < -(x - y)\)
or Is \(1 < -(x - y)^2\)? or Is \((x - y)^2 < -1\)?

(1) \(x^2 + y^2 < 4xy\) implies \(x^2 + y^2 -2xy < 2xy\) or \((x - y)^2 < 2xy\)
Since we don't know any relation between x and y, so statement 1 is NOT SUFFICIENT.

(2) y < x implies (x - y) > 0 or \((x - y)^2 > 0\)
So, answer to the main question is "No"
Statement 2 is SUFFICIENT.

[Reveal] Spoiler:
The correct answer is B.

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If x ≠ y, is 1/(x - y) < y - x? [#permalink]

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New post 24 Jan 2011, 08:35
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Anurag@Gurome wrote:
The question is \(\frac{1}{(x - y)} < -(x - y)\)
or Is \(1 < -(x - y)^2\)? or Is \((x - y)^2 < -1\)?

(1) \(x^2 + y^2 < 4xy\) implies \(x^2 + y^2 -2xy < 2xy\) or \((x - y)^2 < 2xy\)
Since we don't know any relation between x and y, so statement 1 is NOT SUFFICIENT.

(2) y < x implies (x - y) > 0 or \((x - y)^2 > 0\)
So, answer to the main question is "No"
Statement 2 is SUFFICIENT.

[Reveal] Spoiler:
The correct answer is B.


The red part is not correct. You cannot multiply both sides of the inequality by \(x-y\) as you don't know whether this expression is positive or negative. Also answer to the question "is \((x - y)^2 < -1\)?" is always NO as the square of a number can not be negative. Basically if \(x-y>0\) the answer is always NO and if \(x-y<0\) the answer is always YES.

You can simplify the question as follows: is \(\frac{1}{x-y}<y-x\)? --> is \(y-x+\frac{1}{y-x}>0\)? is \(\frac{(y-x)^2+1}{y-x}>0\)? As the nominator (\((y-x)^2+1\)) is always positive then the question basically becomes whether denominator (\(y-x\)) is positive too --> is \(y-x>0\)? or is \(y>x\)?
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DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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Re: If x ≠ y, is 1/(x - y) < y - x? [#permalink]

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New post 24 Jan 2011, 21:03
You are right, thanks for pointing that.
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Re: If x ≠ y, is 1/(x - y) < y - x? [#permalink]

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New post 10 Oct 2017, 00:22
rxs0005 wrote:
If x ≠ y, is 1/(x - y) < y - x?

(1) x^2 + y^2 < 4xy

(2) y < x



Isn't the condition always true?

1/(x - y) < y - x => 1/(-1)(y - x) < y - x => -1 < (y - x)^2

RHS is a square and will always be positive, there equation is true no matter what. Therefore, D should be the answer.

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Re: If x ≠ y, is 1/(x - y) < y - x? [#permalink]

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New post 10 Oct 2017, 00:25
praneetkakani wrote:
rxs0005 wrote:
If x ≠ y, is 1/(x - y) < y - x?

(1) x^2 + y^2 < 4xy

(2) y < x



Isn't the condition always true?

1/(x - y) < y - x => 1/(-1)(y - x) < y - x => -1 < (y - x)^2

RHS is a square and will always be positive, there equation is true no matter what. Therefore, D should be the answer.


Please read the whole thread before posting: https://gmatclub.com/forum/if-x-y-is-1- ... ml#p857343

Hope it helps.
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Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

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Re: If x ≠ y, is 1/(x - y) < y - x?   [#permalink] 10 Oct 2017, 00:25
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