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Is k^2 +k -2 > 0 ?

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Is k^2 +k -2 > 0 ? [#permalink] New post 12 Feb 2013, 14:52
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Is k^2 + k - 2 > 0 ?

(1) k < 1
(2) k < -2
[Reveal] Spoiler: OA

Last edited by Bunuel on 12 Feb 2013, 14:57, edited 1 time in total.
Edited the question.
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Re: Is k^2 +k -2 > 0 ? [#permalink] New post 12 Feb 2013, 15:11
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Is k^2 + k - 2 > 0 ?

First let's see for which ranges of k the given inequality holds true: \(k^2 + k - 2 > 0\) --> \((k+2)(k-1)>0\) --> roots are -2 and 1 --> ">" sign indicates that the solution lies to the left of a smaller root and to the right of a larger root (check here for this technique: if-x-is-an-integer-what-is-the-value-of-x-1-x-2-4x-94661.html#p731476). Thus the given inequality holds true for: \(k<-2\) and \(k>1\).

So, the question asks whether \(k<-2\) or \(k>1\).

(1) k < 1. Not sufficient.
(2) k < -2. Sufficient.

Answer: B.
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Re: Is k^2 +k -2 > 0 ? [#permalink] New post 12 Feb 2013, 15:11
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rephrasing the statement we have

\((x + 2) (x - 1) > 0\) ====> \(x > -2\) AND \(x > 1\)

on a number line we have

-------------------0 ----- \(1\) +++++++

----- \(- 2\) +++++0++++++++++++

So from 1) we have \(x > 1\) so 2 intervals : one positive and one negative. two values. not suff

from2) \(x < - 2\) the only values are positive. so is suff.

B is the answer

Bunuel ?? correct my approach, pretty straightforward :)
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Re: Is k^2 +k -2 > 0 ? [#permalink] New post 14 Jan 2015, 08:58
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Re: Is k^2 +k -2 > 0 ? [#permalink] New post 14 Jan 2015, 16:54
Expert's post
Hi All,

If you spot the Quadratic expression in the prompt, then you can use an Algebra approach to get to the correct answer. This question can also be solved with a combination of TESTing VALUES and Number Properties.

We're asked if K^2 + K - 2 > 0. This is a YES/NO question

Fact 1: K < 1

IF....
K = 0
0^2 + 0 - 2 = -2 and the answer to the question is NO.

IF...
K = -3
(-3)^2 -3 - 2 = 4 and the answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: K < -2

With this Fact, we have an interesting "limit" issue.

Even though it's not permitted....IF K = -2, then
(-2)^2 - 2 + 2 = 0 which is NOT > 0

As K becomes "more negative" (re. -2.1, -3, -100, etc.)....
K^2 creates a "bigger positive" than (+K - 2) creates a "negative"

eg.
K = -3
(-3)^2 = +9 vs. (-3 - 2)

K = -2.1
(-2.1)^2 = +4.41 vs. (-2.1 - 2)
Etc.

Thus, the result of the calculation will ALWAYS be greater than 0 and the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT.

Final Answer:
[Reveal] Spoiler:
B


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Re: Is k^2 +k -2 > 0 ?   [#permalink] 14 Jan 2015, 16:54
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