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# Is 2a < a^2 + b ?

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Senior Manager
Joined: 22 Feb 2018
Posts: 396
Is 2a < a^2 + b ?  [#permalink]

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26 Oct 2018, 09:33
2
00:00

Difficulty:

35% (medium)

Question Stats:

74% (01:06) correct 26% (01:19) wrong based on 27 sessions

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Is $$2a < a^2 + b$$ ?

(1) $$0 < a < 2$$
(2) $$-2 < b < 0$$

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Joined: 11 Sep 2015
Posts: 3125
Re: Is 2a < a^2 + b ?  [#permalink]

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26 Oct 2018, 10:35
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Top Contributor
Princ wrote:
Is $$2a < a^2 + b$$ ?

(1) $$0 < a < 2$$
(2) $$-2 < b < 0$$

Target question: Is 2a < a² + b

Statement 1: 0 < a < 2
No information about b, so statement 1 is NOT SUFFICIENT

Statement 2: -2 < b < 0
No information about a, so statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
It may help to REPHRASE the target question by getting the a's on one side of the equation
Take: Is 2a < a² + b?
Subtract a² from both sides to get: Is 2a - a² < b?
Factor left side to get: Is a(2 - a) < b?

Since 0 < a < 2, we know that a is POSITIVE, and we know that (2 - a) is POSITIVE
Also, if -2 < b < 0, then we know that b is NEGATIVE

So, our target question becomes: Is (POSITIVE)(POSITIVE) < NEGATIVE?
Simplify left side to get: Is POSITIVE < NEGATIVE?
The answer to this REPHRASED question is "NO, it is definitely NOT the case that POSITIVE < NEGATIVE"
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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Re: Is 2a < a^2 + b ? &nbs [#permalink] 26 Oct 2018, 10:35
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