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

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S
Joined: 15 Dec 2015
Posts: 112

Kudos [?]: 116 [0], given: 71

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

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New post 03 Aug 2017, 09:38
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A
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E

Difficulty:

  75% (hard)

Question Stats:

41% (01:27) correct 59% (02:03) wrong based on 64 sessions

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

Statement 1: \(b^a\)<0
Statement 2: |b|>|a|
[Reveal] Spoiler: OA

Kudos [?]: 116 [0], given: 71

Expert Post
1 KUDOS received
Magoosh GMAT Instructor
User avatar
G
Joined: 28 Dec 2011
Posts: 4488

Kudos [?]: 8735 [1], given: 105

Is |a| - |b| < |a-b| ? [#permalink]

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New post 03 Aug 2017, 10:35
1
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Expert's post
DH99 wrote:
Is |a| - |b| < |a-b| ?

Statement 1: \(b^a\)<0
Statement 2: |b|>|a|

Dear DH99,

I'm happy to respond. :-)

This is an intriguing question. I don't agree with the answer: I wonder if you posted the correct OA.

Statement #1 is particularly interesting.
\(b^a\)<0
This means a few things. The number b must be negative and it doesn't have to be an integer. The number a cannot be zero, but it could be any positive or negative odd integer.
Example #1: \(a = 3\) and \(b = -2\)
This yields \(b^a\) = \((-2)^3\) = \(- 8\) < \(0\), so these choices are consistent with Statement #1
Then \(|a| - |b| = |3| - |-2| = 3 - 2 = 1\)
and \(|a - b| = |3 - (-2)| = |5| = 5\)
This choice gives a "yes" answer to the prompt question.
Example #2: \(a = -3\) and \(b = -2\)
This yields \(b^a\) = \((-2)^{-3}\) = \(- \tfrac{1}{8}\) < \(0\), so these choices are consistent with Statement #1
Then \(|a| - |b| = |-3| - |-2| = 3 - 2 = 1\)
and \(|a - b| = |-3 - (-2)| = |-1| = 1\)
This choice gives a "no" answer to the prompt question.

Two different choices consistent with the statement yield two different answer to the prompt question, so statement #1, alone and by itself, is insufficient.

We can do algebraic work with Statement #1
\(|b|>|a|\)
Subtract |b| from both sides
\(0 > |a| - |b|\)
So this difference now is always negative. Of course, a single absolute value is always positive (or zero of a = b, but that's excluded by this statement). Thus,
\(|a| - |b| < 0 < |a - b|\)
We absolutely have a "yes" answer to this statement.
Because we have arrived at a definitive answer, we know that statement #2, alone and by itself, is sufficient.

I find the answer (B).

Again, I am not sure whether something was copied incorrect, but my answer doesn't match the OA posted.

Does all this make sense?
Mike :-)
_________________

Mike McGarry
Magoosh Test Prep

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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Kudos [?]: 8735 [1], given: 105

Manager
Manager
User avatar
S
Joined: 15 Dec 2015
Posts: 112

Kudos [?]: 116 [0], given: 71

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

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New post 03 Aug 2017, 11:01
mikemcgarry wrote:
DH99 wrote:
Is |a| - |b| < |a-b| ?

Statement 1: \(b^a\)<0
Statement 2: |b|>|a|

Dear DH99,

I'm happy to respond. :-)

This is an intriguing question. I don't agree with the answer: I wonder if you posted the correct OA.

Statement #1 is particularly interesting.
\(b^a\)<0
This means a few things. The number b must be negative and it doesn't have to be an integer. The number a cannot be zero, but it could be any positive or negative odd integer.
Example #1: \(a = 3\) and \(b = -2\)
This yields \(b^a\) = \((-2)^3\) = \(- 8\) < \(0\), so these choices are consistent with Statement #1
Then \(|a| - |b| = |3| - |-2| = 3 - 2 = 1\)
and \(|a - b| = |3 - (-2)| = |5| = 5\)
This choice gives a "yes" answer to the prompt question.
Example #2: \(a = -3\) and \(b = -2\)
This yields \(b^a\) = \((-2)^{-3}\) = \(- \tfrac{1}{8}\) < \(0\), so these choices are consistent with Statement #1
Then \(|a| - |b| = |-3| - |-2| = 3 - 2 = 1\)
and \(|a - b| = |-3 - (-2)| = |-1| = 1\)
This choice gives a "no" answer to the prompt question.

Two different choices consistent with the statement yield two different answer to the prompt question, so statement #1, alone and by itself, is insufficient.

We can do algebraic work with Statement #1
\(|b|>|a|\)
Subtract |b| from both sides
\(0 > |a| - |b|\)
So this difference now is always negative. Of course, a single absolute value is always positive (or zero of a = b, but that's excluded by this statement). Thus,
\(|a| - |b| < 0 < |a - b|\)
We absolutely have a "yes" answer to this statement.
Because we have arrived at a definitive answer, we know that statement #2, alone and by itself, is sufficient.

I find the answer (B).

Again, I am not sure whether something was copied incorrect, but my answer doesn't match the OA posted.

Does all this make sense?
Mike :-)


Hi, I just checked there is no typo with this question and the OA is also given as E.It is possible that the OA is wrong.

Kudos [?]: 116 [0], given: 71

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Location: India
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GMAT 1: 700 Q50 V34
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Re: Is |a| - |b| < |a-b| ? [#permalink]

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New post 03 Aug 2017, 12:00
mikemcgarry wrote:
DH99 wrote:
Is |a| - |b| < |a-b| ?

Statement 1: \(b^a\)<0
Statement 2: |b|>|a|

Dear DH99,

I'm happy to respond. :-)

This is an intriguing question. I don't agree with the answer: I wonder if you posted the correct OA.

Statement #1 is particularly interesting.
\(b^a\)<0
This means a few things. The number b must be negative and it doesn't have to be an integer. The number a cannot be zero, but it could be any positive or negative odd integer.
Example #1: \(a = 3\) and \(b = -2\)
This yields \(b^a\) = \((-2)^3\) = \(- 8\) < \(0\), so these choices are consistent with Statement #1
Then \(|a| - |b| = |3| - |-2| = 3 - 2 = 1\)
and \(|a - b| = |3 - (-2)| = |5| = 5\)
This choice gives a "yes" answer to the prompt question.
Example #2: \(a = -3\) and \(b = -2\)
This yields \(b^a\) = \((-2)^{-3}\) = \(- \tfrac{1}{8}\) < \(0\), so these choices are consistent with Statement #1
Then \(|a| - |b| = |-3| - |-2| = 3 - 2 = 1\)
and \(|a - b| = |-3 - (-2)| = |-1| = 1\)
This choice gives a "no" answer to the prompt question.

Two different choices consistent with the statement yield two different answer to the prompt question, so statement #1, alone and by itself, is insufficient.

We can do algebraic work with Statement #1
\(|b|>|a|\)
Subtract |b| from both sides
\(0 > |a| - |b|\)
So this difference now is always negative. Of course, a single absolute value is always positive (or zero of a = b, but that's excluded by this statement). Thus,
\(|a| - |b| < 0 < |a - b|\)
We absolutely have a "yes" answer to this statement.
Because we have arrived at a definitive answer, we know that statement #2, alone and by itself, is sufficient.

I find the answer (B).

Again, I am not sure whether something was copied incorrect, but my answer doesn't match the OA posted.

Does all this make sense?
Mike :-)



Hi mikemcgarry
While I do understand from your method why statement 2 is sufficient, could you please tell me what I am doing wrong.

For |a| - |b| < |a-b| to hold true , a and b should have opposite signs.

Now in statement 2

|b|>|a|

|3| > |2| { b = 3 and a =2 }
|-3| > |2| {b =-3 and a =2 }

since a and b can have the same sign as well as the opposite sign

so by this logic, statement 2 is insufficient.

Kudos [?]: 9 [0], given: 796

Expert Post
1 KUDOS received
Magoosh GMAT Instructor
User avatar
G
Joined: 28 Dec 2011
Posts: 4488

Kudos [?]: 8735 [1], given: 105

Is |a| - |b| < |a-b| ? [#permalink]

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New post 03 Aug 2017, 14:30
1
This post received
KUDOS
Expert's post
darn wrote:
Hi mikemcgarry
While I do understand from your method why statement 2 is sufficient, could you please tell me what I am doing wrong.

For |a| - |b| < |a-b| to hold true , a and b should have opposite signs.

Now in statement 2

|b|>|a|

|3| > |2| { b = 3 and a =2 }
|-3| > |2| {b =-3 and a =2 }

since a and b can have the same sign as well as the opposite sign

so by this logic, statement 2 is insufficient.

Dear darn,

I'm happy to respond. :-)

The other problem similar to this could be deciphered according to whether a & b had the same or opposite signs, but that's not the case here. Here, the relative size of the absolute values of a & b is pertinent.

It's true both {a = 2, b = 3} and {a = 2, b = -3}. For both of these, the left side of the prompt inequality, |a| - |b|, will equal -1, and the right side will equal something positive. Something positive is always greater than -1. Thus, both pairs produce a "yes" answer to the prompt.

In this problem, regardless of the signs of a & b, if statement #2 is satisfied, then the answer to the prompt question is "yes."

Does all this make sense?
Mike :-)
_________________

Mike McGarry
Magoosh Test Prep

Image

Image

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Kudos [?]: 8735 [1], given: 105

Is |a| - |b| < |a-b| ?   [#permalink] 03 Aug 2017, 14:30
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