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# If x/y > 2, is 3x + 2y < 18 ? (1) x - y is less than

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If x/y > 2, is 3x + 2y < 18 ? (1) x - y is less than [#permalink]

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25 Aug 2008, 00:40
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If x/y > 2, is 3x + 2y < 18 ?

(1) x - y is less than 2.
(2) y - x is less than 2.

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Senior Manager
Joined: 28 Aug 2006
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25 Aug 2008, 01:03
Given x/y>2.....implies both x and y can be positive or both of them are negative
If both of them are negative, 3x+2Y<0.
If both of them are positive we have x>2y..............(1)

Statement1: x-y<2
so x<2+y......(2)
From (1) and (2)
2y<x<2+y
So 2y<2+y
Hence y<2.......(3)
From (2) and (3), x<4 --------(4)
So from (3) and (4), it is clear that 3x+2y<16.
Hence statement (1) alone is sufficient

Coming to statement 2, we can think along the same lines and easily find two contradicting examples.

Let x=100 and y=1
Clearly x/y >2, and y-x<2 and the value of 3x+2y=302 (>18)

Let x=3 and y=1
Clearly x/y>2 and y-x<2 and the value of 3x+2y=11(<18)
So statement(2) is not sufficient

Hence A.

Hi, Kevin........I am happy to come back again with your question
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Last edited by cicerone on 27 Aug 2008, 00:36, edited 1 time in total.
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25 Aug 2008, 06:15
cicerone wrote:
Given x>2y..............(1) ( in the question)

You cannot cross multiply inequalities like that, y can be negative.

The answer should be A, but I tried with number picking, no idea how to solve otherwise.
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25 Aug 2008, 07:44
Nerdboy wrote:
cicerone wrote:
Given x>2y..............(1) ( in the question)

You cannot cross multiply inequalities like that, y can be negative.

The answer should be A, but I tried with number picking, no idea how to solve otherwise.

Agree with you. Its possible that Y CAN BE NEGATIVE

x/y > 2 --> implies that x and y must have same singn -- both +Ve or both v-e

When positive.. "cicerone" solution lead to "A"
When both negative . 3x + 2y obviously <0
3x + 2y <18
Sufficient.. A
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Senior Manager
Joined: 28 Aug 2006
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27 Aug 2008, 00:35
x2suresh wrote:
Nerdboy wrote:
cicerone wrote:
Given x>2y..............(1) ( in the question)

You cannot cross multiply inequalities like that, y can be negative.

The answer should be A, but I tried with number picking, no idea how to solve otherwise.

Agree with you. Its possible that Y CAN BE NEGATIVE

x/y > 2 --> implies that x and y must have same singn -- both +Ve or both v-e

When positive.. "cicerone" solution lead to "A"
When both negative . 3x + 2y obviously <0
3x + 2y <18
Sufficient.. A

Thank you friend.....My intention is to discuss the negative case.......but I forgot to mention that.....thanks once again....i will edit my post
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Joined: 26 Dec 2008
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Schools: Booth (Admit R1), Sloan (Ding R1), Tuck (R1)

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27 Dec 2008, 19:50
1
KUDOS
I tried solving this problem graphically and got A as well..

Solution: Following the steps below

1. Draw 3x + 2y = 18 (x intercept 6, y intercept 9)..The question asks if region BELOW 3x + 2y = 18 is satisfied by given data
2. Draw y = 1/2 x and shade the region BELOW the line. Find intersection with last line (intersects at (4.5, 2.25)).
3. Draw x - y = 2
4. condition A: x - y < 2 --> shade the region above x - y = 2 and find intersection with shaded region belonging to y = 1/2 x --> shaded intersection is BELOW 3x + 2y = 18 --> sufficient
5. condition B: x - y > 2 --> shade region below x - y = 2 and find intersection with shaded region belonging to y = 1/2 x --> shaded intersection is below as well as above 3x + 2y = 18 --> insufficient

Although this method appears complicated when written out, it actually is quite fast if you're comfortable with plotting lines / curves
Director
Joined: 29 Aug 2005
Posts: 836

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29 Dec 2008, 05:24
xyz21 wrote:
I tried solving this problem graphically and got A as well..

Solution: Following the steps below

1. Draw 3x + 2y = 18 (x intercept 6, y intercept 9)..The question asks if region BELOW 3x + 2y = 18 is satisfied by given data
2. Draw y = 1/2 x and shade the region BELOW the line. Find intersection with last line (intersects at (4.5, 2.25)).
3. Draw x - y = 2
4. condition A: x - y < 2 --> shade the region above x - y = 2 and find intersection with shaded region belonging to y = 1/2 x --> shaded intersection is BELOW 3x + 2y = 18 --> sufficient
5. condition B: x - y > 2 --> shade region below x - y = 2 and find intersection with shaded region belonging to y = 1/2 x --> shaded intersection is below as well as above 3x + 2y = 18 --> insufficient

Although this method appears complicated when written out, it actually is quite fast if you're comfortable with plotting lines / curves

Solving graphically can be very easy - thanks for sharing your method xyz. Would you mind to post plotted graph that illustrates this?
Below I have tried to solve the problem algebraically.

Question stem:
x/y>2 => x>0, y>0; or x<0, y<0. When x<0, y<0 =>the answer to "3x+2y<18?" will be "Yes", so we are really looking at the case when x>0, y>0. Let me know if someone disagrees with my thinking here.

Let's denote "Greater Than" as "GT" and "Less Than" as "LT"
Given x>0; y>0, x/y>2 =>(x-2y)/y>0 => x>2y and y>0. We can depict x>2y as x=LT2y. ----- (1)
Substituting (1) for x in 3x+2y<18 => 3(LT2y)+2y<18 => LT6y+2y<18 => LT8y<18 => LTy<9/4.

Given the above, the question is really asking is 0<y<9/4 true?

Stmt1: x-y<2; or given (1), LT2y-y<2 => LTy<2. Given y>0, stmt 1 => 0<y<2 Suff

Stmt2: y-x<2; or x>y-2. Given (1), LT2y>y-2 => LTy>-2 Insuff

Director
Joined: 29 Aug 2005
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29 Dec 2008, 05:25
-deleted-
sorry for repeated posts.

--== Message from GMAT Club Team ==--

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Last edited by seofah on 29 Dec 2008, 05:39, edited 1 time in total.
Re: DS: Inequalities   [#permalink] 29 Dec 2008, 05:25
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# If x/y > 2, is 3x + 2y < 18 ? (1) x - y is less than

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