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# If which x < y < 0, of the following inequalities must be true?

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Math Expert
Joined: 02 Sep 2009
Posts: 49431
If which x < y < 0, of the following inequalities must be true?  [#permalink]

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15 Jul 2018, 09:00
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55% (hard)

Question Stats:

56% (01:26) correct 44% (01:19) wrong based on 79 sessions

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If which x < y < 0, of the following inequalities must be true?

A. y + 1 < x
B. y - 1 < x
C. xy^2 < x
D. xy < y^2
E. xy < x^2

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Re: If which x < y < 0, of the following inequalities must be true?  [#permalink]

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15 Jul 2018, 10:10
Bunuel wrote:
If which x < y < 0, of the following inequalities must be true?

A. y + 1 < x
B. y - 1 < x
C. xy^2 < x
D. xy < y^2
E. xy < x^2

Plugging in numbers is the best one for this question.

Note: both x and y are negative. Between x and y, x is lower than y

Let assume the value of x and y .

x = -2 x= -0.5
y = -1 y = -0.25

Option E is always true .

E) xy <$$x^2$$

= (-1)(-2) < (-2)^2

=2 < 4

Plugging In Fraction :

(-0.5)(-0.25) < $$(0.5)^2$$

0.125 < 0.25......true.

Thus E is the best answer.
Math Expert
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Posts: 6808
If which x < y < 0, of the following inequalities must be true?  [#permalink]

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15 Jul 2018, 10:31
1
If which x < y < 0, of the following inequalities must be true?

A. y + 1 < x
Let x=-2 and y=-1........-1+1<-2...0<-2....NO

B. y - 1 < x
Let x=-2 and y=-1......-1-1<-2......-2<-2.....NO

C. xy^2 < x
$$x-xy^2>0......x(1-y^2)>0.......x<0, so:(1-y^2)<0......y^2>1$$...Not necessary, y can be -1/2

D. xy < y^2
$$y^2-xy>0.......y(y-x)>0......y<0, so:(y-x)<0......y<x$$....NO

E. xy < x^2
$$x^2-xy>0.....x(x-y)>0......X<0,so:(x-y)<0.....X<y$$...yes always

E
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Re: If which x < y < 0, of the following inequalities must be true?  [#permalink]

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16 Jul 2018, 03:35
Bunuel wrote:
If which x < y < 0, of the following inequalities must be true?

A. y + 1 < x
B. y - 1 < x
C. xy^2 < x
D. xy < y^2
E. xy < x^2

Bunuel Sir,

Kindly provide OE.

I am getting the answer as E but the OE provided is D.

Thanking you.
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Math Expert
Joined: 02 Sep 2009
Posts: 49431
Re: If which x < y < 0, of the following inequalities must be true?  [#permalink]

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16 Jul 2018, 04:15
PKN wrote:
Bunuel wrote:
If which x < y < 0, of the following inequalities must be true?

A. y + 1 < x
B. y - 1 < x
C. xy^2 < x
D. xy < y^2
E. xy < x^2

Bunuel Sir,

Kindly provide OE.

I am getting the answer as E but the OE provided is D.

Thanking you.

The OA is E. Edited. Thank you.
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Re: If which x < y < 0, of the following inequalities must be true?  [#permalink]

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16 Jul 2018, 08:32
chetan2u wrote:
If which x < y < 0, of the following inequalities must be true?

A. y + 1 < x
Let x=-2 and y=-1........-1+1<-2...0<-2....NO

B. y - 1 < x
Let x=-2 and y=-1......-1-1<-2......-2<-2.....NO

C. xy^2 < x
$$x-xy^2>0......x(1-y^2)>0.......x<0, so:(1-y^2)<0......y^2>1$$...Not necessary, y can be -1/2

D. xy < y^2
$$y^2-xy>0.......y(y-x)>0......y<0, so:(y-x)<0......y<x$$....NO

E. xy < x^2
$$x^2-xy>0.....x(x-y)>0......X<0,so:(x-y)<0.....X<y$$...yes always

E

Hi thanks for the explanation but I'm still trying to understand how you simplified option E... Isn't the answer supposed to be x>0 (x-y) >0. Why the sign change?

Posted from my mobile device
Math Expert
Joined: 02 Aug 2009
Posts: 6808
Re: If which x < y < 0, of the following inequalities must be true?  [#permalink]

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16 Jul 2018, 08:53
1
Kem12 wrote:
chetan2u wrote:
If which x < y < 0, of the following inequalities must be true?

A. y + 1 < x
Let x=-2 and y=-1........-1+1<-2...0<-2....NO

B. y - 1 < x
Let x=-2 and y=-1......-1-1<-2......-2<-2.....NO

C. xy^2 < x
$$x-xy^2>0......x(1-y^2)>0.......x<0, so:(1-y^2)<0......y^2>1$$...Not necessary, y can be -1/2

D. xy < y^2
$$y^2-xy>0.......y(y-x)>0......y<0, so:(y-x)<0......y<x$$....NO

E. xy < x^2
$$x^2-xy>0.....x(x-y)>0......X<0,so:(x-y)<0.....X<y$$...yes always

E

Hi thanks for the explanation but I'm still trying to understand how you simplified option E... Isn't the answer supposed to be x>0 (x-y) >0. Why the sign change?

Posted from my mobile device

Hi..
We have X(x-y)>0....
There are two cases..
1) both X and x-y are positive since +*+=+>0
2) both X and x-y are negative since -*-=+>0
And it is given X<0, so case 1 is out..
In case II X<0, and therefore x-y<0...X<y

Hope it clarifies
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If which x < y < 0, of the following inequalities must be true?  [#permalink]

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19 Jul 2018, 12:38
Bunuel wrote:
If which x < y < 0, of the following inequalities must be true?

A. y + 1 < x
B. y - 1 < x
C. xy^2 < x
D. xy < y^2
E. xy < x^2

We can see that both x and y are negative, and y is greater than x. So A is not true. B is not true, either, since we don’t know how much y is greater than x. If we divide both sides by x in the inequality in C (and switch the inequality sign since x is negative), we have:

y^2 > 1

However, since we don’t know the value of y, we can’t determine whether its square is in fact greater than 1. So C might not be true. If we divide both sides by y in the inequality in D (and switch the inequality sign since y is negative), we have:

x > y

This is not true since we know y > x. So the correct answer must be E. However, let’s verify it by dividing both sides of the inequality by x:

y > x

This is true since we know y is indeed greater than x.

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Re: If which x < y < 0, of the following inequalities must be true? &nbs [#permalink] 19 Jul 2018, 12:38
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