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# If x and y are positive, is y < 2?

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Intern
Joined: 02 Jul 2010
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If x and y are positive, is y < 2?  [#permalink]

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04 Nov 2010, 15:56
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45% (medium)

Question Stats:

71% (01:55) correct 29% (02:11) wrong based on 195 sessions

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If x and y are positive. is y< 2 ?

(1) x > 2y
(2) x < y + 2
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Joined: 24 Jun 2008
Posts: 1418
Re: If x and y are positive, is y < 2?  [#permalink]

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04 Nov 2010, 16:18
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tatane90 wrote:
If x and y are positive. is y< 2 ?

(1) x > 2y
(2) x < y + 2

And what is the level fo this question ?

Neither statement is sufficient alone; from statement 1, we might have y=1 and x=10, or y=3 and x=10, and for statement 2 we might have y=1 and x=1, or y=5 and x=1.

Combining the statements, there are a few ways one could look at this question. You have the following two inequalities, rewriting the second one so both inequalities face in the same direction:

x > 2y
y + 2 > x

Recall that we can add two inequalities which face the same way, just as we add equations (be careful though - you cannot subtract inequalities in this way). Doing that here we have:

x + y + 2 > 2y + x
y + 2 > 2y
2 > y

Or, perhaps more simply, you could 'chain' the inequalities together. Here we know that x > 2y, and x < y+2, so we must have that

2y < x < y + 2

So certainly 2y < y + 2, or y < 2.

Most test takers find abstract inequalities questions difficult, so any similar question would be at least a medium-high level question. One takeaway here: if you see an inequalities problem that looks like a 2 equations/2 unknowns problem, and if you don't see anything else to do, then try lining up your inequalities and adding them to see what happens. Often that will give you the answer you're looking for, as it did here. Of course there are often more direct approaches, like the one I used in the second solution above, but if you don't see that kind of solution quickly, adding your inequalities might get you there.
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Joined: 17 Apr 2010
Posts: 56
Re: If x and y are positive, is y < 2?  [#permalink]

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05 Nov 2010, 07:52
its C

x > 2y
y+2 > x

---> x+y+2 > x+y+y
deleting X+Y both sides
2>x
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Re: If x and y are positive, is y < 2?  [#permalink]

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11 Dec 2010, 08:19
1
If x and y are positive, is y < 2 ?

(1) x > 2y
(2) x < y + 2
Math Expert
Joined: 02 Sep 2009
Posts: 54367
Re: If x and y are positive, is y < 2?  [#permalink]

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11 Dec 2010, 08:42
3
2
If x and y are positive, is y< 2 ?

(1) x > 2y
(2) x < y + 2

Clearly each statement alone is not sufficient. When combined, you can line up both equation like $$2y<x<y+2$$ (as 2y<x and x<y+2, so 2y<x<y+2), get rid of $$x$$ you'll get $$2y<y+2$$ --> $$y<2$$, so the answer to the question is y<2 is YES. Sufficient.

Or you as the signs of the inequalities are in the opposite directions then we can subtract (2) from (1) (remember: you can only add inequalities when their signs are in the same direction and you can only apply subtraction when their signs are in the opposite directions) --> $$x-x>2y-(y+2)$$ --> $$y<2$$.

Hope it's clear.
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Re: If x and y are positive, is y < 2?  [#permalink]

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11 Dec 2010, 09:07
Thanks for the explanation.
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Re: If x and y are positive, is y < 2?  [#permalink]

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06 Oct 2013, 10:51
Bunuel wrote:
If x and y are positive, is y< 2 ?

(1) x > 2y
(2) x < y + 2

Clearly each statement alone is not sufficient. When combined, you can line up both equation like $$2y<x<y+2$$ (as 2y<x and x<y+2, so 2y<x<y+2), get rid of $$x$$ you'll get $$2y<y+2$$ --> $$y<2$$, so the answer to the question is y<2 is YES. Sufficient.

Or you as the signs of the inequalities are in the opposite directions then we can subtract (2) from (1) (remember: you can only add inequalities when their signs are in the same direction and you can only apply subtraction when their signs are in the opposite directions) --> $$x-x>2y-(y+2)$$ --> $$y<2$$.

Hope it's clear.

That's great didn't know the subtraction technique. So when you subtract then the inequality sign stays the same as the minuend's in all cases right?
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Re: If x and y are positive, is y < 2?  [#permalink]

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06 Oct 2013, 10:54
jlgdr wrote:
Bunuel wrote:
If x and y are positive, is y< 2 ?

(1) x > 2y
(2) x < y + 2

Clearly each statement alone is not sufficient. When combined, you can line up both equation like $$2y<x<y+2$$ (as 2y<x and x<y+2, so 2y<x<y+2), get rid of $$x$$ you'll get $$2y<y+2$$ --> $$y<2$$, so the answer to the question is y<2 is YES. Sufficient.

Or you as the signs of the inequalities are in the opposite directions then we can subtract (2) from (1) (remember: you can only add inequalities when their signs are in the same direction and you can only apply subtraction when their signs are in the opposite directions) --> $$x-x>2y-(y+2)$$ --> $$y<2$$.

Hope it's clear.

That's great didn't know the subtraction technique. So when you subtract then the inequality sign stays the same as the minuend's in all cases right?

Right.

Hope this helps.
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Re: If x and y are positive, is y < 2?  [#permalink]

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06 Oct 2015, 01:42
Could you please explain how you subtracted the inequality? I understood till 2y < y+2, didn't follow after that. Would really help if you could.
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Re: If x and y are positive, is y < 2?  [#permalink]

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06 Oct 2015, 04:33
Could you please explain how you subtracted the inequality? I understood till 2y < y+2, didn't follow after that. Would really help if you could.

Subtract y from both sides of 2y < y+2 to get y < 2.
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Re: If x and y are positive, is y < 2?  [#permalink]

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31 Jan 2016, 21:56
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If x and y are positive. is y< 2 ?

(1) x > 2y
(2) x < y + 2

When it comes to inequality, >0 can be an equation. Thus, in the original condition, there are 2 variables(x,y) and 1 equation(x,y>0), which should match with the number of equation. So you need 1 more equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
For 1), you can figure out x>2y>y, but not y<2?, which is not sufficient.
For 2), in x<y+2, you cannot figure out y<2?, which is not sufficient.
When 1) & 2), 2y<x<y+2, 2y<y+2, y<2 -> yes, which is sufficient.

 For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: If x and y are positive, is y < 2?  [#permalink]

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01 Aug 2017, 07:53
tatane90 wrote:
If x and y are positive. is y< 2 ?

(1) x > 2y
(2) x < y + 2

Given : x > 0 and y > 0
DS : y < 2

Statement 1 :
x> 2y

for y = 1 , x>2.
for y = 2 , x >4

NOT SUFFICIENT.

Statement 2 : x < y+2
for y = 1, x<3
for y = 2, x<4
NOT SUFFICIENT

Combined : 2y < x < y +2
For y = 1 2 <x < 3.. So there are values of x
For y =2 4<x<4 No real values of x
For y =3 6 <x < 5 No real values of x
So for y>2, there are no real values of x....
Hence Y<2

SUFFICIENT

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Re: If x and y are positive, is y < 2?  [#permalink]

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01 Aug 2017, 08:44
tatane90 wrote:
If x and y are positive. is y< 2 ?

(1) x > 2y
(2) x < y + 2

None of the statements alone is sufficient y has dependency on x, about which we do not know anything.

Multiple statement 2 by (-1) to flip the sign that makes it compatible for addition with statement 1.
x > 2y
-x>-y-2
----------
0>y-2

=> y<2

Choice C is the right option!
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Re: If x and y are positive, is y < 2?  [#permalink]

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16 Apr 2019, 12:39
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Re: If x and y are positive, is y < 2?   [#permalink] 16 Apr 2019, 12:39
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