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

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Intern
Joined: 11 Jan 2015
Posts: 33
If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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21 Feb 2018, 01:21
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Difficulty:

75% (hard)

Question Stats:

53% (02:23) correct 47% (01:50) wrong based on 77 sessions

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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
Math Expert
Joined: 02 Sep 2009
Posts: 58434
If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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21 Feb 2018, 01:28
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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

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.

Hope it's clear.
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Intern
Joined: 11 Jan 2015
Posts: 33
Re: If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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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 $$x>y$$

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

Intern
Joined: 11 Jan 2015
Posts: 33
Re: If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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21 Feb 2018, 01:46
Bunuel 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!
Math Expert
Joined: 02 Sep 2009
Posts: 58434
Re: If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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21 Feb 2018, 01:50
Bunuel 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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Joined: 26 Mar 2013
Posts: 2345
Re: If x and y are negative and x^4-y^4<0 which of the following...  [#permalink]

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22 Feb 2018, 02:45
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

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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