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If x and y are negative and x^4-y^4<0 which of the following...

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If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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New post 21 Feb 2018, 01:21
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Question Stats:

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If x and y are negative and \(x^4-y^4<0\) which of the following must be true?

I: \(x<y\)

II: \(xy<y^2\)

III: \((x+y)^2<(x-y)^2\)

A.) I only
B.) II only
C.) III only
D.) I & II
E.) II & III
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If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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New post 21 Feb 2018, 01:28
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paddy41 wrote:
If x and y are negative and \(x^4-y^4<0\) which of the following must be true?

I: \(x<y\)

II: \(xy<y^2\)

III: \((x+y)^2<(x-y)^2\)

A.) I only
B.) II only
C.) III only
D.) I & II
E.) II & III


Give: \(x^4-y^4<0\):

Re-arrange: \(x^4<y^4\);

Take the fourth root: \(|x| < |y|\);

Since both are negative, then \(-x < -y\);

Re-arrange: \(y < x\).


Evaluate options:

I: \(x<y\). Not true.

II: \(xy<y^2\): reduce by y and flip the sign because y is negative: \(x > y\). TRUE.

III: \((x+y)^2<(x-y)^2\): expand \(x^2 + 2xy + y^2 < x^2 - 2xy + y^2\). Simplify: \(xy < 0\). Since both x and y are negative, then \(xy = negative*negative = positive\), thus this options is also not true.


Answer: B.

Hope it's clear.
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Re: If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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New post 21 Feb 2018, 01:36
Step 1: Rephrasing \(x^4-y^4<0\) to \((x^2-y^2)(x^2+y^2)<0\). From this we know that \(x^2-y^2\) has to be negative.
Step 2: Rephrasing \(x^2-y^2<0\) to \((x-y)(x+y)<0\). Since x and y are both negative we know that \((x-y)>0\) which means \(x>y\)

Evaluating the answers:

I. not true
II. If we divide both sides by y we end up at \(x>y\) - true
III. If we factor out we end up at \(4xy<0\). Since x and y are both negative this can´t be true

Answer B.
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Re: If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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New post 21 Feb 2018, 01:46
Bunuel wrote:
paddy41 wrote:

Take the fourth root: \(|x| < |y|\);

Since both are negative, then \(-x < -y\);



Hi Bunuel,

can you shortly elaborate on the underlying rule of the above steps?

Thank you!
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Re: If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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New post 21 Feb 2018, 01:50
paddy41 wrote:
Bunuel wrote:
paddy41 wrote:

Take the fourth root: \(|x| < |y|\);

Since both are negative, then \(-x < -y\);



Hi Bunuel,

can you shortly elaborate on the underlying rule of the above steps?

Thank you!


Absolute value properties:

When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\).

10. Absolute Value



For more check Ultimate GMAT Quantitative Megathread



Hope it helps.
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Re: If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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New post 22 Feb 2018, 02:45
paddy41 wrote:
If x and y are negative and \(x^4-y^4<0\) which of the following must be true?

I: \(x<y\)

II: \(xy<y^2\)

III: \((x+y)^2<(x-y)^2\)

A.) I only
B.) II only
C.) III only
D.) I & II
E.) II & III


From question stem, We can organize exmaples

Let x = -1 & y = -2

Let x = \(\frac{-1}{4}\) & y = \(\frac{-1}{2}\)

Let x = \(\frac{-1}{2}\) & y = -1

I: \(x<y\)........From examples above.......... NOT True

II: \(xy<y^2\)....\(xy-y^2<0\).......y<0 & x>y........From examples above........True

III: \((x+y)^2<(x-y)^2\)..........Take first example to DISPROVE........ NOT True

Answer: B
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Re: If x and y are negative and x^4-y^4<0 which of the following...   [#permalink] 22 Feb 2018, 02:45
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