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Challenge question: Is |1 - 4k| < k?

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Challenge question: Is |1 - 4k| < k?

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Is |1 - 4k| < k?

(1) k > 4x³
(2) k < 2x - x² - 2

Answer: B
Difficulty level: 700+
Source: www.gmatprepnow.com

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Brent@GMATPrepNow wrote:
Is |1 - 4k| < k?

(1) k > 4x³
(2) k < 2x - x² - 2
Target question: Is |1 - 4k| < k?

Statement 1: k > 4x³
This pretty much tells is that k can have ANY value.
For example, notice that, if x = -100, then 4x³ = -4,000,000
So, for this value of x, k can be any number greater than -4,000,000

Let's TEST some values.
There are several values of k that satisfy statement 1. Here are two:
Case a: k = 0. Here, |1 - 4k| = |1 - 4(0)| = |1| = 1. In this case, the answer to the target question is NO, it is NOT the case that |1 - 4k| < k
Case b: k = 0.25. Here, |1 - 4k| = |1 - 4(0.25)| = |0| = 0. In this case, the answer to the target question is YES, it IS the case that |1 - 4k| < k
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: k < 2x - x² - 2
Rewrite as: k < -x² + 2x - 1 - 1
Rewrite as: k < -(x² - 2x + 1) - 1
Factor to get: k < -(x - 1)² - 1
Notice that (x - 1)² is always greater than or equal to zero
So, -(x - 1)² is always LESS THAN or equal to zero
So -(x - 1)² - 1 must be NEGATIVE
So, statement 2 essentially tells us that k < some NEGATIVE number
This means k must be NEGATIVE
Since |1 - 4k| is always greater than or equal to zero, the answer to the target question is NO, it is NOT the case that |1 - 4k| < k
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

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Brent Hanneson – Creator of GMATPrepNow.com
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