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

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Manager
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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03 Aug 2017, 09:38
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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

Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4488

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

Is |a| - |b| < |a-b| ? [#permalink]

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03 Aug 2017, 10:35
1
KUDOS
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.

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

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
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]

### Show Tags

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.

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

Manager
Joined: 12 Sep 2016
Posts: 71

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

Location: India
Schools: Kellogg '20
GMAT 1: 700 Q50 V34
GPA: 3.33
Re: Is |a| - |b| < |a-b| ? [#permalink]

### Show Tags

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.

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

Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4488

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

Is |a| - |b| < |a-b| ? [#permalink]

### Show Tags

03 Aug 2017, 14:30
1
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

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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