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# In the xy-plane, region R consists of all the points (x ,y) such that

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Intern
Joined: 06 Aug 2006
Posts: 17
In the xy-plane, region R consists of all the points (x ,y) such that  [#permalink]

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Updated on: 27 Jul 2015, 14:05
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Difficulty:

95% (hard)

Question Stats:

43% (01:17) correct 57% (01:52) wrong based on 207 sessions

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In the xy-plane, region R consists of all the points (x ,y) such that 2x + 3y <= 6. Is the point (r,s) in region R?

(1) 3r + 2s = 6
(2) r <= 3 and s <= 2

Originally posted by loki on 03 Feb 2010, 10:09.
Last edited by Bunuel on 27 Jul 2015, 14:05, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
Manager
Joined: 26 May 2005
Posts: 189
Re: In the xy-plane, region R consists of all the points (x ,y) such that  [#permalink]

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03 Feb 2010, 10:54
need to prove if 2r+3s<=6
(1) 3r + 2s = 6
not sufficient. for example take points (2,0), (0,3) and both yield different result
(2) r <= 3 and s <= 2
not sufficient. for example take points(3,2), (2,0) and both yield different result

combining
3r+2s=6
2r + 3s + (r-s) = 6
2r + 3s = 6 - (r-s) = 6 + (s-r) ==> as long as s<=r, the stmt is true

if r=3, s = -3/2 - true
if r=1, s = 3/2 - false
Not sufficient

E
Senior Manager
Joined: 29 Jun 2017
Posts: 405
Re: In the xy-plane, region R consists of all the points (x ,y) such that  [#permalink]

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10 Jul 2017, 03:56
this is hard questions from og 2016.

one way to solve is to pick up some specific number as is the case with the explanation is og.

this way confuse me because we have to choose many pairs of numbers.

I find out a way to solve. draw two lines . remember, for inequality, all the value of y which is above the line of equality satisfying the inequality.

this is hard at first but you will do it quickly after understanding the concept.
Math Expert
Joined: 02 Sep 2009
Posts: 51101
Re: In the xy-plane, region R consists of all the points (x ,y) such that  [#permalink]

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11 Nov 2018, 23:21
loki wrote:
In the xy-plane, region R consists of all the points (x ,y) such that 2x + 3y <= 6. Is the point (r,s) in region R?

(1) 3r + 2s = 6
(2) r <= 3 and s <= 2

In the xy-plane, region R consists of all the points (x, y) such that $$2x + 3y =< 6$$ . Is the point (r,s) in region R?

Q: is $$2r+3s\leq{6}$$?

(1) $$3r + 2s = 6$$ --> very easy to see that this statement is not sufficient:
If $$r=2$$ and $$s=0$$ then $$2r+3s=4<{6}$$, so the answer is YES;
If $$r=0$$ and $$s=3$$ then $$2r+3s=9>6$$, so the answer is NO.
Not sufficient.

(2) $$r\leq{3}$$ and $$s\leq{2}$$ --> also very easy to see that this statement is not sufficient:
If $$r=0$$ and $$s=0$$ then $$2r+3s=0<{6}$$, so the answer is YES;
If $$r=3$$ and $$s=2$$ then $$2r+3s=12>6$$, so the answer is NO.
Not sufficient.

(1)+(2) We already have an example for YES answer in (1) which valid for combined statements:
If $$r=2<3$$ and $$s=0<2$$ then $$2r+3s=4<{6}$$, so the answer is YES;
To get NO answer try max possible value of $$s$$, which is $$s=2$$, then from (1) $$r=\frac{2}{3}<3$$ --> $$2r+3s=\frac{4}{3}+6>6$$, so the answer is NO.
Not sufficient.

Hope it's clear.

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/in-the-xy-pl ... fl=similar
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Re: In the xy-plane, region R consists of all the points (x ,y) such that &nbs [#permalink] 11 Nov 2018, 23:21
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# In the xy-plane, region R consists of all the points (x ,y) such that

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