Data Sufficiency

ziyuenlau wrote:If b<0, is 5b(a+b)> (−a^2 − b^2) ? 

1) a+2b>0
2) a+3b>0

Source : Math Revolution
Official Answer : B

Hi ziyuenlau,

The inequality 5b(a+b) > (−a^2 − b^2) can be written as:

5ab + 5b^2 > −a^2 − b^2

=> -5ab - 5b^2 < a^2 + b^2; multiplying the inequality by '-' and reversing the sign of the inequality

=> 0 < a^2 + 5ab + 6b^2

=> 0 < a^2 + 3ab +2ab + 6b^2

=> a(a+3b) + 2b(a+3b)

=> 0 < (a+3b)(a+2b)

Since b < 0, we can write,

0 < (a-3|b|)(a-2|b|)

So, we have to see if (a-3|b|)(a-2|b|) > 0.

Let's take each statement one by one.

S1: a+2b>0

=> a > -2b
=> a > 2|b|

Case 1: Say a = 2.5|b|

Then (a-3|b|)(a-2|b|) = (2.5|b|-3|b|)(2.5|b|-2|b|) = (-0.5|b|)*(0.5|b|) = -0.25b^2 = a negative number. The answer is NO.

Case 2: Say a = 4|b|

Then (a-3|b|)(a-2|b|) = (4|b|-3|b|)(4|b|-2|b|) = (|b|)*(2|b|) = 2b^2 = a positive number. The answer is Yes.

No unique answer. Insufficient.

S2: S1: a+3b>0

=> a > -3b
=> a > 3|b|

We see that Case 2 discussed above applies here too. Even if you increase the value of a more than 4|b|, the answer is always Yes.

Sufficient.

The correct answer: B

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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