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# Is |a-b| < |a| +|b| ?

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Manager
Joined: 15 Dec 2015
Posts: 114

Kudos [?]: 111 [0], given: 70

GMAT 1: 660 Q46 V35
GPA: 4
WE: Information Technology (Computer Software)
Is |a-b| < |a| +|b| ? [#permalink]

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03 Aug 2017, 09:35
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Difficulty:

65% (hard)

Question Stats:

51% (01:10) correct 49% (01:15) wrong based on 102 sessions

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Is |a-b| < |a| +|b| ?

Statement 1: $$\frac{a}{b}$$<0
Statement 2: $$a^2b$$<0
[Reveal] Spoiler: OA

Kudos [?]: 111 [0], given: 70

Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4426

Kudos [?]: 8453 [1], given: 102

Re: Is |a-b| < |a| +|b| ? [#permalink]

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03 Aug 2017, 10:52
1
KUDOS
Expert's post
DH99 wrote:
Is |a-b| < |a| +|b| ?

Statement 1: $$\frac{a}{b}$$<0
Statement 2: $$a^2b$$<0

Dear DH99,

I'm happy to respond.

This is a very clever question. Let's think about the prompt.

If a & b have the same sign, then subtracting produces a difference of a smaller absolute value.
$$5 - 3 = 2 (-5) - (-3) = -2$$
These are examples of choices that would produce "yes" answers for the prompt question.

If a & b have opposite signs, the subtracting produces a difference that has the same absolute value as the sum of the absolute values of a & b separately.
$$5 - (-3) = 8$$ and $$|8| = 8 = |5| + |-3|$$
$$(-5) - 3 = -8$$ and $$|-8| = 8 = |-5| + |3|$$
These are examples of choices that would produce "no" answers for the prompt question.

So really, the prompt question is: do a and b have the same sign?

Statement #1: $$\frac{a}{b}$$<0
When is a fraction negative? It's negative when the numerator and denominator have opposite signs. Thus, a & b have opposite signs. We can give a definitive "no" to the prompt question. Because we can give a definitive answer, we know that Statement #1, alone and by itself, must be sufficient.

Statement #2: $$a^2b$$<0
I assume that this was entered correctly, and that DH99 meant $$a^2b$$ and not $$a^{2b}$$.

Taking this statement at face value, we know $$a^2$$ is always positive, regardless of the sign of a, so we know that b just be negative. Thus, we know the sign of b, but the sign of a could be anything. We are not able to give a definitive answer to the prompt. Thus, we know that Statement #2, alone and by itself, must be insufficient.

OA = (A)

Does all this make sense?
Mike
_________________

Mike McGarry
Magoosh Test Prep

Kudos [?]: 8453 [1], given: 102

Manager
Joined: 15 Dec 2015
Posts: 114

Kudos [?]: 111 [1], given: 70

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GPA: 4
WE: Information Technology (Computer Software)
Re: Is |a-b| < |a| +|b| ? [#permalink]

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03 Aug 2017, 10:59
1
KUDOS
mikemcgarry wrote:
DH99 wrote:
Is |a-b| < |a| +|b| ?

Statement 1: $$\frac{a}{b}$$<0
Statement 2: $$a^2b$$<0

Dear DH99,

I'm happy to respond.

This is a very clever question. Let's think about the prompt.

If a & b have the same sign, then subtracting produces a difference of a smaller absolute value.
$$5 - 3 = 2 (-5) - (-3) = -2$$
These are examples of choices that would produce "yes" answers for the prompt question.

If a & b have opposite signs, the subtracting produces a difference that has the same absolute value as the sum of the absolute values of a & b separately.
$$5 - (-3) = 8$$ and $$|8| = 8 = |5| + |-3|$$
$$(-5) - 3 = -8$$ and $$|-8| = 8 = |-5| + |3|$$
These are examples of choices that would produce "no" answers for the prompt question.

So really, the prompt question is: do a and b have the same sign?

Statement #1: $$\frac{a}{b}$$<0
When is a fraction negative? It's negative when the numerator and denominator have opposite signs. Thus, a & b have opposite signs. We can give a definitive "no" to the prompt question. Because we can give a definitive answer, we know that Statement #1, alone and by itself, must be sufficient.

Statement #2: $$a^2b$$<0
I assume that this was entered correctly, and that DH99 meant $$a^2b$$ and not $$a^{2b}$$.

Taking this statement at face value, we know $$a^2$$ is always positive, regardless of the sign of a, so we know that b just be negative. Thus, we know the sign of b, but the sign of a could be anything. We are not able to give a definitive answer to the prompt. Thus, we know that Statement #2, alone and by itself, must be insufficient.

OA = (A)

Does all this make sense?
Mike

Yes,thanks mike

Kudos [?]: 111 [1], given: 70

Senior Manager
Joined: 29 Jun 2017
Posts: 345

Kudos [?]: 66 [0], given: 64

WE: Engineering (Transportation)
Re: Is |a-b| < |a| +|b| ? [#permalink]

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24 Aug 2017, 11:03
Clearly from A we are getting definite NO.
See the attached pic.
Attachments

IMG_2982.JPG [ 905.22 KiB | Viewed 467 times ]

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Give Kudos for correct answer and/or if you like the solution.

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Is |a-b| < |a| +|b| ? [#permalink]

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24 Aug 2017, 12:06
1
KUDOS
DH99 wrote:
Is |a-b| < |a| +|b| ?

Statement 1: $$\frac{a}{b}$$<0
Statement 2: $$a^2b$$<0

Statement 1: implies that either $$a<0$$ or $$b<0$$
Putting the values of $$a$$ & $$b$$ in the inequality as per the scenario
if $$a<0$$, then $$|a-b| = |-a-b| = |a+b| = |a| +|b|$$. so we get a $$NO$$ for the question stem
if $$b<0$$, then $$|a-b| = |a-(-b)| = |a+b| = |a| + |b|$$. so we get a $$NO$$ for the question stem
Hence $$Sufficient$$

[b]Statement 2:/b] implies that $$b<0$$ but $$a<0$$ or $$a>0$$
Putting the values of $$a$$ & $$b$$ in the inequality as per the scenario
if both $$a<0$$ and $$b<0$$, then $$|a-b| = |-a-(-b)| = |-a+b|<|a| + |b|$$. so we get a $$YES$$ for the question stem
but if $$a>0$$ and $$b<0$$, then $$|a-b| = |a-(-b)| = |a+b| = |a| + |b|$$. so we get a $$NO$$ for the question stem
Hence $$Insufficient$$

Option $$A$$

Kudos [?]: 186 [1], given: 31

Is |a-b| < |a| +|b| ?   [#permalink] 24 Aug 2017, 12:06
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