Author 
Message 
TAGS:

Hide Tags

Retired Moderator
Status: The last round
Joined: 18 Jun 2009
Posts: 1296
Concentration: Strategy, General Management

If (x/y)>2, is 3x+2y<18? (1) xy is less than 2 (2) [#permalink]
Show Tags
15 Jan 2010, 09:32
6
This post received KUDOS
26
This post was BOOKMARKED
Question Stats:
34% (03:29) correct
66% (03:01) wrong based on 272 sessions
HideShow timer Statistics



Senior Manager
Joined: 21 Jul 2009
Posts: 364
Schools: LBS, INSEAD, IMD, ISB  Anything with just 1 yr program.

Re: Tough Inequality Challange [#permalink]
Show Tags
15 Jan 2010, 13:47
Hussain15 wrote: If (x/y)>2, is 3x+2y<18?
(1) xy is less than 2 (2) yx is less than 2
It will be great to see how do you guys approach this lethal one. Given x > 2y. Have to substantiate if 3x + 2y < 18. Stmt1: x < 2 + y. Keep substituting different values for y, we get ranges for x based on the stimulus condition and this statement, substitute these different values and we notice that certain values are applicable while many others aren't applicable to substantiate the posed question. Therefore, NS. Stmt2: can be rephrased as x > y  2. Do the same method as above, same situation, no definitive answer. Therefore, NS. combining both the statements, still substituting all possible values for y and deriving ranges for x, we can't really substantiate the given equation. My answer is E. I wonder if there is a simpler way of solving problems of this kind. I used the brute force approach of substituting valid numbers for y and ended up getting wierder ranges for x and again, choose something which accidentally would substantiate the equation and mostly certain other numbers that do not. Took me more than a 10 mins handling work simultaneously, and if such questions appear on the real deal, I might as well give up on GMAT and pursue a PhD in Pure Math.
_________________
I am AWESOME and it's gonna be LEGENDARY!!!



Math Expert
Joined: 02 Sep 2009
Posts: 39723

Re: Tough Inequality Challange [#permalink]
Show Tags
15 Jan 2010, 14:02
14
This post received KUDOS
Expert's post
6
This post was BOOKMARKED
Hussain15 wrote: If (x/y)>2, is 3x+2y<18?
(1) xy is less than 2 (2) yx is less than 2
It will be great to see how do you guys approach this lethal one. I would solve this question with graphic approach, by drawing the lines. With this approach you will "see" that the answer is A. But we can do it with algebra as well. \(\frac{x}{y}>2\) tells us that \(x\) and \(y\) are either both positive or both negative, which means that all points \((x,y)\) satisfying given inequality are either in I or III quadrant. When they are both negative (in III quadrant) inequality \(3x+2y<18\) is always true, so we should check only for I quadrant, or when both \(x\) and \(y\) are positive. In I quadrant, as \(x\) and \(y\) are both positive, we can rewrite \(\frac{x}{y}>2\) as \(x>2y>0\) (remember \(x>0\) and \(y>0\)). So basically question becomes: If \(x>0\) and \(y>0\) and \(x>2y>0\), is \(3x+2y<18\)? (1) \(xy<2\). Subtract inequalities \(x>2y\) and \(xy<2\) (we can do this as signs are in opposite direction) > \(x(xy)>2y2\) > \(y<2\). Now add inequalities \(xy<2\) and \(y<2\) (we can do this as signs are in the same direction) > \(xy+y<2+2\) > \(x<4\). We got \(y<2\) and \(x<4\). If we take maximum values \(x=4\) and \(y=2\) and substitute in \(3x+2y<18\), we'll get \(12+4=16<18\). Sufficient. (2) \(yx<2\) and \(x>2y\): \(x=3\) and \(y=1\) > \(3x+2y=11<18\) true. \(x=11\) and \(y=5\) > \(3x+2y=43<18\) false. Not sufficient. Answer: A.
_________________
New to the Math Forum? Please read this: 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? Extrahard Quant Tests with Brilliant Analytics



Retired Moderator
Status: The last round
Joined: 18 Jun 2009
Posts: 1296
Concentration: Strategy, General Management

Re: Tough Inequality Challange [#permalink]
Show Tags
15 Jan 2010, 22:35
Bunuel wrote: Hussain15 wrote: If (x/y)>2, is 3x+2y<18?
(1) xy is less than 2 (2) yx is less than 2
It will be great to see how do you guys approach this lethal one. I would solve this question with graphic approach, by drawing the lines. With this approach you will "see" that the answer is A. But we can do it with algebra as well. x/y>2 tells us that x and y are either both positive or both negative, which means that all points (x,y) satisfying given inequality are in I or III quadrants. When they are both negative (in III quadrant) inequality 3x+2y<18 is always true, so we should check only for I quadrant. In I quadrant x and y are both positive and we can rewrite x/y>2 as x>2y>0 (remember x>0 and y>0). (1) xy<2. Subtract inequalities x>2y and xy<2 (we can do this as signs are in opposite direction) > x(xy)>2y2 > y<2. Now add inequalities xy<2 and y<2 (we can do this as signs are in the same direction) > xy+y<2+2 > x<4. We got y<2 and x<4. If we take maximum values x=4 and y=2 and substitute in 3x+2y<18, we'll get 12+4=12<18. Sufficient. (2) yx<2 and x>2y: x=3 and y=1 > 3x+2y=11<18 true. x=11 and y=5 > 3x+2y=43<18 false. Not sufficient. Answer: A. OA is "A". Thanks for detailed answer. You have plugged the numbers in option 2, can it be done algeberically??
_________________
[ From 470 to 680My Story ] [ My Last Month Before Test ] [ GMAT Prep Analysis Tool ] [ US. Business School Dashboard ] [ Int. Business School Dashboard ]
I Can, I Will
GMAT Club Premium Membership  big benefits and savings



Math Expert
Joined: 02 Sep 2009
Posts: 39723

Re: Tough Inequality Challange [#permalink]
Show Tags
17 Jan 2010, 01:55



CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2783
Location: Malaysia
Concentration: Technology, Entrepreneurship
GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35

Re: 3x+2y<18? [#permalink]
Show Tags
08 May 2010, 11:50
neoreaves wrote: If x/y >2, is 3x+2y<18?
1. xy is less than 2 2. yx is less than 2 Nice explanation bunnel... thanks
_________________
Fight for your dreams :For all those who fear from Verbal lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
Support GMAT Club by putting a GMAT Club badge on your blog/Facebook
GMAT Club Premium Membership  big benefits and savings
Gmat test review : http://gmatclub.com/forum/670to710alongjourneywithoutdestinationstillhappy141642.html



Intern
Joined: 11 Oct 2009
Posts: 26

Re: Tough Inequality Challange [#permalink]
Show Tags
21 May 2010, 09:10
1
This post received KUDOS
I spent 5 min for this question with incorrect ans .. There was no way I could have solved this question .. Very nice explanation Brunel ..
But I failed to understand the theory of addition and substraction for equalities with same sign and opposite signs respective .. Can you pls throw some light ..



Math Expert
Joined: 02 Sep 2009
Posts: 39723

Re: Tough Inequality Challange [#permalink]
Show Tags
21 May 2010, 09:28
4
This post received KUDOS
Expert's post
2
This post was BOOKMARKED
sandeepuc wrote: I spent 5 min for this question with incorrect ans .. There was no way I could have solved this question .. Very nice explanation Brunel ..
But I failed to understand the theory of addition and substraction for equalities with same sign and opposite signs respective .. Can you pls throw some light .. You can only add inequalities when their signs are in the same direction:If \(a>b\) and \(c>d\) (signs in same direction: \(>\) and \(>\)) > \(a+c>b+d\). Example: \(3<4\) and \(2<5\) > \(3+2<4+5\). You can only apply subtraction when their signs are in the opposite directions:If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) > \(ac>bd\) (take the sign of the inequality you subtract from). Example: \(3<4\) and \(5>1\) > \(35<41\). Hope it helps.
_________________
New to the Math Forum? Please read this: 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? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 04 Feb 2009
Posts: 237
Location: Ukraine
Schools: Ross 2013
WE 1: Pharmaceutical industry 5 years, C level

Re: Tough Inequality Challange [#permalink]
Show Tags
28 May 2010, 06:07
great explanation Bunuel



Intern
Joined: 01 Jun 2010
Posts: 23
Location: United States
Schools: Harvard Business School (HBS)  Class of 2014
GPA: 3.53

Re: Tough Inequality Challange [#permalink]
Show Tags
01 Jun 2010, 15:26
Bunuel wrote: Hussain15 wrote: If (x/y)>2, is 3x+2y<18?
(1) xy is less than 2 (2) yx is less than 2
It will be great to see how do you guys approach this lethal one. I would solve this question with graphic approach, by drawing the lines. With this approach you will "see" that the answer is A. But we can do it with algebra as well. \(\frac{x}{y}>2\) tells us that \(x\) and \(y\) are either both positive or both negative, which means that all points \((x,y)\) satisfying given inequality are either in I or III quadrant. When they are both negative (in III quadrant) inequality \(3x+2y<18\) is always true, so we should check only for I quadrant, or when both \(x\) and \(y\) are positive. In I quadrant, as \(x\) and \(y\) are both positive, we can rewrite \(\frac{x}{y}>2\) as \(x>2y>0\) (remember \(x>0\) and \(y>0\)). So basically question becomes: If \(x>0\) and \(y>0\) and \(x>2y>0\), is \(3x+2y<18\)? (1) \(xy<2\). Subtract inequalities \(x>2y\) and \(xy<2\) (we can do this as signs are in opposite direction) > \(x(xy)>2y2\) > \(y<2\). Now add inequalities \(xy<2\) and \(y<2\) (we can do this as signs are in the same direction) > \(xy+y<2+2\) > \(x<4\). We got \(y<2\) and \(x<4\). If we take maximum values \(x=4\) and \(y=2\) and substitute in \(3x+2y<18\), we'll get \(12+4=16<18\). Sufficient. (2) \(yx<2\) and \(x>2y\): \(x=3\) and \(y=1\) > \(3x+2y=11<18\) true. \(x=11\) and \(y=5\) > \(3x+2y=43<18\) false. Not sufficient. Answer: A. +1 already for a great explanation. Followup question: Would you mind detailing a graphical approach to this problem? I haven't taken a math course in 7 years so am a little rusty. Knowing how to solve such problems with a graph seems like it would be very useful.
_________________
HBS Class of 2014



Math Expert
Joined: 02 Sep 2009
Posts: 39723

Re: Tough Inequality Challange [#permalink]
Show Tags
01 Jun 2010, 15:47



Manager
Joined: 07 Oct 2006
Posts: 71
Location: India

Re: Tough Inequality Challange [#permalink]
Show Tags
09 Jun 2010, 11:30
Excellent explanations by Brunel (algebra) and Walker (graph).. Kudos to both of you...
_________________
 Please give kudos, if my post is helpful.
For English Grammar tips, consider visiting http://www.grammarquizzes.com/index.html.



Intern
Joined: 09 Dec 2009
Posts: 32

Re: Tough Inequality Challange [#permalink]
Show Tags
26 Jun 2010, 02:16
Hi Bunuel, Thanks for the wonderful explanation. But in the actual exam when you have the clock ticking, how do we decide whether to try plugging in numbers or try solving it using algebra.I am not sure if there is any definite strategy for this but any inputs from your experience will help.



Intern
Joined: 14 Jun 2010
Posts: 16
Location: Singapore
Concentration: Strategy
WE: Information Technology (Consulting)

Re: Tough Inequality Challange [#permalink]
Show Tags
28 Jun 2010, 00:19
Given x/y > 2. i. xy < 2: for this to be possible x and y have to be negative. Now since x and y are both negative, the equation in question will always result in a negative number. Hence, SUFFICIENT.
ii. yx <2: For this to be possible x and y have to be positive. Now since x and y both are positive and xy>2, multiple solutions exist. Hence, NOT SUFFICIENT.
Therefore, answer is A.



Math Expert
Joined: 02 Sep 2009
Posts: 39723

Re: Tough Inequality Challange [#permalink]
Show Tags
28 Jun 2010, 05:54
sunnyarora wrote: Given x/y > 2. i. xy < 2: for this to be possible x and y have to be negative. Now since x and y are both negative, the equation in question will always result in a negative number. Hence, SUFFICIENT.
ii. yx <2: For this to be possible x and y have to be positive. Now since x and y both are positive and xy>2, multiple solutions exist. Hence, NOT SUFFICIENT.
Therefore, answer is A. OA for this question is A, but your reasoning is not correct: For (1) \(x=2>0\) and \(y=0.5>0\) satisfy both \(\frac{x}{y}>2\) and \(xy<2\), so x and y can be positive as well. For (2) \(x=2<0\) and \(y=0.5>0\) satisfy both \(\frac{x}{y}>2\) and \(yx<2\), so x and y can be negative as well. Solution for this problem is in earlier posts.
_________________
New to the Math Forum? Please read this: 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? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 04 Jun 2010
Posts: 3

Re: Tough Inequality Challange [#permalink]
Show Tags
28 Jun 2010, 08:08
we can solve just drawing th elines in geometric terms ,... for condition 1...draw lines of x>2y and xy<2 and check the area whether it is lying below the line 3x+2y<18 same can be done for second condition Bunuel wrote: sunnyarora wrote: Given x/y > 2. i. xy < 2: for this to be possible x and y have to be negative. Now since x and y are both negative, the equation in question will always result in a negative number. Hence, SUFFICIENT.
ii. yx <2: For this to be possible x and y have to be positive. Now since x and y both are positive and xy>2, multiple solutions exist. Hence, NOT SUFFICIENT.
Therefore, answer is A. OA for this question is A, but your reasoning is not correct: For (1) \(x=2>0\) and \(y=0.5>0\) satisfy both \(\frac{x}{y}>2\) and \(xy<2\), so x and y can be positive as well. For (2) \(x=2<0\) and \(y=0.5>0\) satisfy both \(\frac{x}{y}>2\) and \(yx<2\), so x and y can be negative as well. Solution for this problem is in earlier posts.



Manager
Joined: 03 Aug 2011
Posts: 239
Location: United States
Concentration: General Management, Entrepreneurship
GPA: 3.38
WE: Engineering (Computer Software)

Re: If (x/y)>2, is 3x+2y<18? (1) xy is less than 2 (2) [#permalink]
Show Tags
06 Apr 2012, 15:47
i followed the exact same procedure except at the very end. i tend to have problems switching back/forth between doing algebra, then stepping back and using logic, or stepping back and plugging numbers. in fact, I think a lot of the mistakes I make when I take the exam is trying to switch between techniques.
my problem stem 3x+2y<18 ==> is x < 18/3 2y/3 ==> is x < 6  2y/3?
for (1). negative values of x/y are already sufficient, so these are for positive x and positive y
 first i try to isolate x because i took the problem stem and isolated x
x/y > 2 ==> x > 2*y
x y < 2 ==> x < 2 + y
therefore
2* y < x < 2 + y
in order for the bolded problem stem to be true then the below must be true. x < 2+ y but is x < 6  2y/3?
2 + y <= 6  2y/3
y + 2y/3 <= 4 5y/3 <= 4 y <= 12/5 y <= 2 and 2/5 if the above is true then we know we are OK
then take that inequality with x in the middle and relate the y part only
2*y < 2+y therefore y < 2
so y < 2 shows that it is certainly <= 2 and 2/5



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16010

Re: If (x/y)>2, is 3x+2y<18? (1) xy is less than 2 (2) [#permalink]
Show Tags
23 Oct 2013, 22:09
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources



Intern
Joined: 04 Jul 2013
Posts: 1

Re: If (x/y)>2, is 3x+2y<18? (1) xy is less than 2 (2) [#permalink]
Show Tags
19 Jun 2014, 10:19
Hi Bunuel,
Thank you for the great solution. With regards to using graphs to solving the problem, do we get a grid kind of pad to be able to plot accurately and with ease?



Math Expert
Joined: 02 Sep 2009
Posts: 39723

Re: If (x/y)>2, is 3x+2y<18? (1) xy is less than 2 (2) [#permalink]
Show Tags
20 Jun 2014, 05:48




Re: If (x/y)>2, is 3x+2y<18? (1) xy is less than 2 (2)
[#permalink]
20 Jun 2014, 05:48



Go to page
1 2
Next
[ 28 posts ]




