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# M31-46

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Math Expert
Joined: 02 Sep 2009
Posts: 50711

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20 Jun 2015, 10:46
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Difficulty:

55% (hard)

Question Stats:

53% (01:39) correct 47% (01:41) wrong based on 30 sessions

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Is $$x > y$$?

(1) $$a*x^4 + a*|y| < 0$$

(2) $$a*x^3 > a*y^3$$

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Joined: 02 Sep 2009
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20 Jun 2015, 10:46
Official Solution:

Is $$x > y$$?

(1) $$a*x^4 + a*|y| < 0$$.

Factor out $$a$$: $$a(x^4 + |y|) < 0$$. Since $$x^4 + |y| = nonnegative + nonnegative = nonnegative$$, then for $$a*nonnegative$$ to be negative, a must be negative. We know nothing about $$x$$ and $$y$$. Not sufficient.

(2) $$a*x^3 > a*y^3$$.

If $$a$$ is positive, then when reducing by it, we'll get $$x^3 > y^3$$, or which is the same $$x > y$$ BUT if $$a$$ is negative, then when reducing by negative value and flipping the sign, we'd get $$x^3 < y^3$$, or which is the same $$x < y$$. Not sufficient.

(1)+(2) Since from (1) $$a < 0$$, then from (2) $$x < y$$. Sufficient.

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24 Aug 2016, 04:15
I think this is a high-quality question and I agree with explanation.
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Joined: 15 Jul 2016
Posts: 7

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10 Oct 2016, 17:19
Bunuel : How do we know x<y when x^3 < y^3 ------ this is only true if X and Y are not fractions
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Joined: 02 Sep 2009
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11 Oct 2016, 05:20
2
1
utkarsh240884 wrote:
Bunuel : How do we know x<y when x^3 < y^3 ------ this is only true if X and Y are not fractions

You could simply test your theory with fractions to test whether it's true.

We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).

Solving Quadratic Inequalities - Graphic Approach
Inequality tips
Wavy Line Method Application - Complex Algebraic Inequalities

DS Inequalities Problems
PS Inequalities Problems

700+ Inequalities problems

inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html
everything-is-less-than-zero-108884.html
graphic-approach-to-problems-with-inequalities-68037.html

Hope it helps.
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Joined: 06 Jun 2017
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16 Nov 2017, 04:49
For statement 1 to be true, cant all three be fractions ?

[quote="Bunuel"]

You could simply test your theory with fractions to test whether it's true.

We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).
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16 Nov 2017, 04:54
HariharanIyeer0 wrote:
For statement 1 to be true, cant all three be fractions ?

From (1) all we can deduce is that a is negative (and that not both from x and y are 0). Other than that, the variables could be integers, fractions, or irrational numbers, we don't know that.
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Re: M31-46 &nbs [#permalink] 16 Nov 2017, 04:54
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# M31-46

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