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Manager  Joined: 29 Nov 2011
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If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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11 00:00

Difficulty:   65% (hard)

Question Stats: 61% (01:57) correct 39% (01:56) wrong based on 400 sessions

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If xy < 4, is x < 2 ?

(1) y > 1
(2) y > x

Originally posted by Smita04 on 18 Mar 2012, 04:40.
Last edited by Bunuel on 18 Mar 2012, 06:12, edited 1 time in total.
Math Expert V
Joined: 02 Sep 2009
Posts: 59730
Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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6
4
If xy < 4, is x < 2 ?

Notice that in order xy<4 to hold true at least one of the multiples must be less than 2 (if both x and y are more than or equal to 2 then xy>2).

(1) y > 1. If for example $$y=1.5>1$$ then $$x$$ can be 1, so less than 2 or 2 so not less than 2. Not sufficient.

(2) y > x. According to above: $$y>x\geq{2}$$ is not possible since in this case $$xy>4$$, so $$x$$ must be less than 2. Sufficient.

Hope it's clear.
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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2
Smita04 wrote:
If xy < 4, is x < 2 ?

(1) y > 1
(2) y > x

From F.S 1, when y = 4, x = 0.5, xy<4 and x<2.Hence a YES for the question stem. However, for y = 3/2 and x = 2, xy<4 and x=2, hence a NO for the question stem.Insufficient.

From F. S 2, we know that y>x.

Had y been equal to x, we would have x*x<4 --> |x|<2 --> -2<x<2

However, as y>x, to maintain the above given inequality, the value of x will have to reduce even further. Thus, x <2. Sufficient.

B.
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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Bunuel wrote:
If xy < 4, is x < 2 ?

Notice that in order xy<4 to hold true at least one of the multiples must be less than 2 (if both x and y are more than or equal to 2 then xy>2).

(1) y > 1. If for example $$y=1.5>1$$ then $$x$$ can be 1, so less than 2 or 2 so not less than 2. Not sufficient.

(2) y > x. According to above: $$y>x\geq{2}$$ is not possible since in this case $$xy>2$$, so $$x$$ must be less than 2. Sufficient.

Hope it's clear.

Hi Bunuel,

I guess the xy>2 should be replaced by xy>4 in above solution
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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1
Have a somewhat different solution for this one.

Given that XY <4 ...... Is X<2???

(1). Y > 1 => Y = GT 1 (Where GT = Greater than)

XY < 4 => X < (4/Y) (We can divide by a number >1 both sides)

=> X < (4/GT1) = > X < LT4 = > X < 4 Not sufficient as X can be 3 , 2 ,1 and so on so forth.

(2). Y>X

=> X GTX < 4

Substituting values X=2 , 2*3 < 4 Not true
X=1 , 1*2 < 4 true

Hence (B) !!
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If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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1
getbetter wrote:
Sure let me explain step by step:

1) xy <4 ———(1)

y > 1 -> 1<y -> -y<-1
multiply both sides by 4 : -4y<-4 ————(2)

(1)+(2)
xy - 4y < 0
y(x-4) <0
y<0 and x < 4
not suff.

2) xy<4

multiply both sides by ‘y’: xy^2 < 4y ————(1)
y > x
Multiply both sides by 4: 4x < 4y —————(2)

(1)-(2):

x(y^2 -4) <0

x <0 suff.

Ok, this looks better now.

Text in green: correct!
Text in red: incorrect

1) xy <4 ———(1)

y > 1 -> 1<y -> -y<-1
multiply both sides by 4 : -4y<-4 ————(2)

(1)+(2)
xy - 4y < 0
y(x-4) <0
------ (3)
y<0 and x < 4
not suff.

From 3, you know that y (x-4)<0 ---> this means 2 cases:

either y<0 and x>4 (not possible as given y>1)
or y>0 and x<4 . Even if we have x<4, x can be 3 or 1.5 , thus x <2 may or may not be true. Thus this statement is not sufficient.

2) xy<4

multiply both sides by ‘y’: xy^2 < 4y ————(1)
y > x
Multiply both sides by 4: 4x < 4y —————(2)

(1)-(2):

x(y^2 -4) <0

x <0 suff.

How do you know whether y < or > 0 when you multiply xy<4 by y? if y is >0 then yes what you did is correct, but if y < 0 , then xy *y > 4y. The thing to remember here is that DO NOT multiply an inequality by a variable when you do not know the sign of that variable.

Alternately, what you can do is :

xy < 4 and y >x , ---> as x<y ----> $$x^2 < 4$$ (as x<y and xy < 4 ----> $$x^2< xy <y^2$$) ----> -2<x<2 and thus this statement is sufficient to say x<2. Thus B is the correct answer.

Manager  Joined: 29 Nov 2011
Posts: 71
Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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Bunuel wrote:
If xy < 4, is x < 2 ?

Notice that in order xy<4 to hold true at least one of the multiples must be less than 2 (if both x and y are more than or equal to 2 then xy>=2).

(1) y > 1. If for example $$y=1.5>1$$ then $$x$$ can be 1, so less than 2 or 2 so not less than 2. Not sufficient.

(2) y > x. According to above: $$y>x\geq{2}$$ is not possible since in this case $$xy>2$$, so $$x$$ must be less than 2. Sufficient.

Hope it's clear.

Yes Bunuel. It's clear. Thanks! Math Expert V
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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up4gmat wrote:
Bunuel wrote:
If xy < 4, is x < 2 ?

Notice that in order xy<4 to hold true at least one of the multiples must be less than 2 (if both x and y are more than or equal to 2 then xy>2).

(1) y > 1. If for example $$y=1.5>1$$ then $$x$$ can be 1, so less than 2 or 2 so not less than 2. Not sufficient.

(2) y > x. According to above: $$y>x\geq{2}$$ is not possible since in this case $$xy>2$$, so $$x$$ must be less than 2. Sufficient.

Hope it's clear.

Hi Bunuel,

I guess the xy>2 should be replaced by xy>4 in above solution

Typo edited. Thank you. +1.
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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Smita04 wrote:
If xy < 4, is x < 2 ?

(1) y > 1
(2) y > x

Alternate solution (rather alternate interpretation) for this problem -
xy < 4 or x < 4/y

(1) y > 1: Interpret 4/y as 4/(1 and increasing). This implies x can be <2 or >2 - So not sufficient

(2) y > x: Interpret xy < 4 as x$$2$$<4 and since y is greater than x, it implies that x < 2 - So sufficient

Ans.
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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- A is insufficient.
- Option B. what if x or y is negative integer. I don't see you mentioned it in your answer.
Intern  Joined: 21 Oct 2009
Posts: 23
Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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nguyenduong wrote:
- A is insufficient.
- Option B. what if x or y is negative integer. I don't see you mentioned it in your answer.

nguyenduong,

Interpretation should be based on given (true) statements. Given here xy<4 and for pt. (2) y>x.

Take any signs, y has to be greater than x or greater part of the multiplication with x resulting in a product smaller than 4.
Consider if xy was 4 (xy=4) and y=x then both x & y would have been + /- 2. Now when the product reduces (product < 4), x has to be always less than the equal contribution i.e. 2.

Hope I could explain.
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Posts: 59730
Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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nguyenduong wrote:
- A is insufficient.
- Option B. what if x or y is negative integer. I don't see you mentioned it in your answer.

The question asks whether x<2. Now, for (2) if x is negative, then x<0<2 and if y is negative, then x<y<0<2. In either case we have an YES answer to the question.
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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Hi Folks, does the method below looks ok?

1) xy <4
-4y<-4
y(x-4) <0
y<0 and x < 4
not suff

2) xy<4
xy^2 <4y

4x < 4y
x(y^2 -4) <0

x <0 suff.
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If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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getbetter wrote:
Hi Folks, does the method below looks ok?

1) xy <4
-4y<-4
y(x-4) <0
y<0 and x < 4
not suff

2) xy<4
xy^2 <4y
4x < 4y
x(y^2 -4) <0

x <0 suff.

1) Where are you getting the above text in red from? How do you get y (x-4) < 0 ? Even if I assume that what you have written in correct, where are you getting y<0 from?

2) Text in blue is only possible if y>0. You need to reverse the inequality if y <0. Statement 2 just mentions that y>x and it does not tell us that x>0 to make y>0 at the same time. Also, how are you getting x(y^2-4)<0 from 4x<4y?

Based on your solution, you were lucky to get to the correct answer and it does not look correct. Maybe, if you could write out all the steps clearly, we can take a look again.
Intern  Joined: 04 Mar 2015
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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Sure let me explain step by step:

1) xy <4 ———(1)

y > 1 -> 1<y -> -y<-1
multiply both sides by 4 : -4y<-4 ————(2)

(1)+(2)
xy - 4y < 0
y(x-4) <0
y<0 and x < 4
not suff.

2) xy<4

multiply both sides by ‘y’: xy^2 < 4y ————(1)
y > x
Multiply both sides by 4: 4x < 4y —————(2)

(1)-(2):

x(y^2 -4) <0

x <0 suff.
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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given: xy<4 ;is x<2?

from(1) y>1=> y can be any +ve number(not specifically Integer);say y= 1.5 than x>2 but if y=3 than x<2 Hence Not sufficient
from(2) y>x; Take the worst case when y=x than both will have value 4; but as y grows more than 2 value of x has to go below 2 in order for Y to be greater than x. Since already xy<4 hence x has to be less than 2 Hence Sufficient
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Re: If xy < 4, is x < 2 ? (1) y > 1 (2) y > x  [#permalink]

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