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Is b < 0 ?

by Vincen » Fri Nov 10, 2017 6:15 am
Is b < 0 ?

(1) b^3 < b
(2) b^2 > b

The OA is C .

I don't have this DS question clear. Experts, may you clarify this to me?

Thanks.

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by Brent@GMATPrepNow » Fri Nov 10, 2017 7:56 am
Vincen wrote:Is b < 0 ?

(1) b³ < b
(2) b² > b
Target question: Is b < 0 ?

Statement 1: b³ < b
There are several values of b that satisfy statement 1. Here are two:
Case a: b = -2 (which means b³ = -8, and -8 < -2). In this case, the answer to the target question is YES, b IS less than 0
Case b: b = 1/2 (which means b³ = 1/8, and 1/8 < 1/2). In this case, the answer to the target question is NO, b is NOT less than 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b² > b
There are several values of b that satisfy statement 2. Here are two:
Case a: b = -1 (which means b² = 1, and 1 > -1). In this case, the answer to the target question is YES, b IS less than 0
Case b: b = 2 (which means b² = 4, and 4 > 2). In this case, the answer to the target question is NO, b is NOT less than 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that b³ < b
Statement 2 tells us that b² > b
Combine the inequalities to get: b³ < b < b²
This tells us that b³ < b²
Subtract b² from both sides to get: b³ - b² < 0
Factor to get: b(b² - b) < 0
If b(b² - b) < 0, then there are TWO POSSIBLE CASES:
Case a: b < 0 and (b² - b) > 0
OR
Case b: b > 0 and (b² - b) < 0
If we take statement 2 (b² > b) and subtract b from both sides, we get: b² - b > 0
This RULES OUT case b (above), which means case a must be true.
If case a is true, then it must be the case that b < 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

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by Rich.C@EMPOWERgmat.com » Fri Nov 10, 2017 10:47 am
Hi Vincen,

We're asked if B is less than 0. This is a YES/NO question. You can solve it by TESTing VALUES and/or by using Number Property Rules.

1) B^3 < B

There are two types of numbers that fit the inequality we're given in Fact 1:
-Negative numbers that are LESS than -1 (using any of these values would get us a YES answer).
-Positive fractions (using any of these values would get us a NO answer).
Fact 1 is INSUFFICIENT

2) B^2 > B

There are two types of numbers that fit the inequality we're given in Fact 2:
-Positive numbers that are GREATER than 1 (using any of these values would get us a NO answer).
-ANY Negative numbers (using any of these values would get us a YES answer).
Fact 2 is INSUFFICIENT

Combined, we have to look for 'overlap' between the two Facts (what groups of numbers fit BOTH Facts?). There's only one group that fits both:
-Negative numbers that are LESS than -1.
Since we're limited to just these values, the answer to the question is ALWAYS YES).
Combined, SUFFICIENT

Final Answer: C

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Rich
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