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# Is (a − 5)^2 < (b + 5)^2 ?

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Math Expert
Joined: 02 Sep 2009
Posts: 47110
Is (a − 5)^2 < (b + 5)^2 ? [#permalink]

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24 Apr 2017, 04:27
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Difficulty:

85% (hard)

Question Stats:

51% (01:45) correct 49% (01:27) wrong based on 118 sessions

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Is (a − 5)^2 < (b + 5)^2 ?

(1) a < b
(2) a + b > 5

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Joined: 26 Feb 2016
Posts: 2946
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Re: Is (a − 5)^2 < (b + 5)^2 ? [#permalink]

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24 Apr 2017, 06:14
Is (a − 5)^2 < (b + 5)^2 ?
Using trial and error method.
(1) a < b
Lets consider the value a=0,b=1. These values, satisfy the condition
But if we take a=-2,b=-1, the condition does not hold.
Hence, not sufficient.

(2) a + b > 5
Lets consider a=4,b=2, the condition holds good.
However, if a = 15,b = -2, the condition doesn't hold good.
Hence, not sufficient.

On combining both, since one of the number definitely has to be positive,
we can mark the solution sufficient(Option C)
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Re: Is (a − 5)^2 < (b + 5)^2 ? [#permalink]

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03 Sep 2017, 10:59
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Is (a − 5)^2 < (b + 5)^2 ? [#permalink]

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03 Sep 2017, 11:48
Bunuel wrote:
Is (a − 5)^2 < (b + 5)^2 ?

(1) a < b
(2) a + b > 5

Simplifying the question stem:
$$(a-5)^2-(b+5)^2<0$$ or $$(a-5+b+5)(a-5-b-5)<0$$ or $$(a+b)(a-b-10)<0$$.
Hence the question becomes IS $$a+b<0$$ & $$(a-b-10)>0$$ or $$(a+b)>0$$ & $$(a-b-10)<0$$

Statement 1: $$a<b$$ or $$a-b<0$$ so we can say that $$a-b-10<0$$ (as it is summation of two negative terms)
but we know nothing about $$a$$ & $$b$$ so whether $$a+b<0$$ or $$a+b>0$$ can't be arrived at. For example if $$a=-3$$ & $$b=-1$$, then $$a+b<0$$ as well as $$a-b<0$$ but if $$a=1, b=2$$, then $$a+b>0$$ but $$a-b<0$$. Hence the statement is not sufficient.

Statement 2: this implies that $$a+b>0$$ but we know nothing about $$(a-b)$$, hence can't say that whether $$a-b-10<0$$ or $$a-b-10>0$$. For ex. if $$a=-1, b=7$$ then $$a-b-10<0$$ but if $$a=-1, b=13$$ then $$a-b-10>0$$. Hence insufficient

Combining 1 & 2
we know that $$a-b<0$$, hence $$a-b-10<0$$ and $$a+b>5$$ hence $$a+b>0$$. therefore $$(a+b)(a-b-10)<0$$. Hence sufficient

Option C
Is (a − 5)^2 < (b + 5)^2 ?   [#permalink] 03 Sep 2017, 11:48
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