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# |a|=|b|, which of the following must be true :

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|a|=|b|, which of the following must be true : [#permalink]  28 Oct 2012, 11:06
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|a|=|b|, which of the following must be true :

I. a=b
II. |a|=-b
III. -a=-b

A. I only
B. II only.
C. III only.
D. I and III only.
E. None
[Reveal] Spoiler: OA

Last edited by Bunuel on 28 Oct 2012, 23:58, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Modulus Ques. [#permalink]  28 Oct 2012, 11:27
E

Let's say l a l = 1 and l b l = 1

For l a l = 1 ; a can be 1 or -1
Similarly b can be 1 or -1

This reasoning is used to get the answer
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Re: Modulus Ques. [#permalink]  28 Oct 2012, 22:00
10
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Expert's post
himanshuhpr wrote:
|a|=|b| , which of the following must be true :

1. a=b 2.|a|=-b 3.-a=-b

a. 1 only b. 2 only. C. 3 only. D. 1 and 3 only. E.none

Responding to a pm:

Neither method needs to be used here. Just think of the definition of mod we use to remove the mod sign.

|x| = x if x >= 0 and |x| = -x if x < 0

We don't know whether a and b are positive or negative. |a|=|b| when absolute values of both a and b are the same. The signs can be different or same. There are 4 cases: a and b are positive, a is positive b is negative, a is negative b is positive, a and b are negative.
For a must be true question, the relation should hold in every case.

1. a=b
Doesn't hold when a and b have opposite signs. e.g. a = 5, b= -5

2.|a|=-b
Doesn't hold when b is positive because -b will become negative while left hand side is always non negative. e.g. a = 5, b = 5
|5| \neq -5

3.-a=-b
Doesn't hold when a and b have opposite signs. e.g. a = 5, b = -5
-5 \neq 5

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Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Intern Joined: 29 Aug 2012 Posts: 26 Schools: Babson '14 GMAT Date: 02-28-2013 Followers: 0 Kudos [?]: 10 [0], given: 56 Re: Modulus Ques. [#permalink] 28 Oct 2012, 23:21 VeritasPrepKarishma wrote: himanshuhpr wrote: |a|=|b| , which of the following must be true : 1. a=b 2.|a|=-b 3.-a=-b a. 1 only b. 2 only. C. 3 only. D. 1 and 3 only. E.none Responding to a pm: Neither method needs to be used here. Just think of the definition of mod we use to remove the mod sign. |x| = x if x >= 0 and |x| = -x if x < 0 We don't know whether a and b are positive or negative. |a|=|b| when absolute values of both a and b are the same. The signs can be different or same. There are 4 cases: a and b are positive, a is positive b is negative, a is negative b is positive, a and b are negative. For a must be true question, the relation should hold in every case. 1. a=b Doesn't hold when a and b have opposite signs. e.g. a = 5, b= -5 2.|a|=-b Doesn't hold when b is positive because -b will become negative while left hand side is always non negative. e.g. a = 5, b = 5 |5| \neq -5 3.-a=-b Doesn't hold when a and b have opposite signs. e.g. a = 5, b = -5 -5 \neq 5 Answer (E) ^^ by the highlighted statement above you mean that all the four cases you listed out should hold true for every stmt. 1. 2. 3. individually. If yes then the only possible solution the to the question would be |a|=|b| , pl. re confirm ... thanks Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5153 Location: Pune, India Followers: 1250 Kudos [?]: 6058 [0], given: 172 Re: Modulus Ques. [#permalink] 29 Oct 2012, 01:03 Expert's post himanshuhpr wrote: ^^ by the highlighted statement above you mean that all the four cases you listed out should hold true for every stmt. 1. 2. 3. individually. If yes then the only possible solution the to the question would be |a|=|b| , pl. re confirm ... thanks What I mean is that if we say any statement 'must be true' then it must hold for all 4 cases i.e. both a and b are positive, a is positive b is negative, a is negative b is positive and a and b are negative. i.e. if statement 1 i.e. a = b must be true, then it should be true in all 4 cases. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting
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Re: Modulus Ques. [#permalink]  29 Oct 2012, 01:08
VeritasPrepKarishma wrote:
himanshuhpr wrote:
^^ by the highlighted statement above you mean that all the four cases you listed out should hold true for every stmt. 1. 2. 3. individually.

If yes then the only possible solution the to the question would be |a|=|b| , pl. re confirm ... thanks

What I mean is that if we say any statement 'must be true' then it must hold for all 4 cases i.e. both a and b are positive, a is positive b is negative, a is negative b is positive and a and b are negative.

i.e. if statement 1 i.e. a = b must be true, then it should be true in all 4 cases.

Ok. thanks very much for the clarification... your blogs and posts are very informative
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Re: Modulus Ques. [#permalink]  29 Oct 2012, 09:49
Thanks for the explanation.
Had a query on this one. Suppose if numbers weren't chosen to evaluate this.

Consider: |a|= |b|
this can be evaluated as: a,b have same signs or a,b have opposite signs

thus, a =b (same signs) and (a = -b or -a = b) for opposite signs.

|a| = -b would have two cases: a +ve , a -ve
thus, a = -b or -a = -b => a = b.
Thus, a = -b or -a=b AND a = b. which is what |a| = |b| boils down to.

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Re: Modulus Ques. [#permalink]  29 Oct 2012, 10:03
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prep wrote:
Thanks for the explanation.
Had a query on this one. Suppose if numbers weren't chosen to evaluate this.

Consider: |a|= |b|
this can be evaluated as: a,b have same signs or a,b have opposite signs

thus, a =b (same signs) and (a = -b or -a = b) for opposite signs.

|a| = -b would have two cases: a +ve , a -ve
thus, a = -b or -a = -b => a = b.
Thus, a = -b or -a=b AND a = b. which is what |a| = |b| boils down to.

|a|= |b| basically means that the distance between a and zero on the number line is the same as the distance between b and zero on the number line.

Thus either a=b (notice that it's the same as -a=-b) or a=-b (notice that it's the same as -a=b).

Hope it helps.
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Re: |a|=|b|, which of the following must be true : [#permalink]  05 Dec 2012, 00:52
|a|=|b|

The equation doesn't tell us anything about the sign of a and b. All we know is that their absolute values are equal.

Possibilities:
|-5| = |5|
|5| = |5|
|5| = |-5|

I. a=b ==> When a=5 and b=-5, this is false!
II. |a|=-b ==> When a=-5 and b=5, this is false!
III. -a=-b ==> When a=-5 and b=5, this is false!

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Re: |a|=|b|, which of the following must be true : [#permalink]  04 Jul 2013, 00:45
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Re: |a|=|b|, which of the following must be true : [#permalink]  04 Jul 2013, 17:33
himanshuhpr wrote:
|a|=|b|, which of the following must be true :

I. a=b
II. |a|=-b
III. -a=-b

A. I only
B. II only.
C. III only.
D. I and III only.
E. None

Replace mod with its equivalent

We have one of these 4 equivalents for |a|=|b|:

-(a) = -(b)
-(a) = b
a = -(b)
a=b

In the answer choices we can see that,

(i) is not the only possibility because we see there are other possibilities as seen above
(ii) is equivalent to -(a) = -b or a = -b. Again these are not the only possibilities as we see there are other possibilities as seen above
(iii) again is not the only possibility as there are other possibilities as seen above

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Re: |a|=|b|, which of the following must be true : [#permalink]  16 Sep 2014, 19:44
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Re: |a|=|b|, which of the following must be true : [#permalink]  20 Sep 2014, 21:57
Where am i going wrong ??

|a| = |b|
\sqrt{a^2} = \sqrt{b^2}
a^2 = b^2
a=b
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Re: |a|=|b|, which of the following must be true : [#permalink]  20 Sep 2014, 23:09
Expert's post
prashd wrote:
Where am i going wrong ??

|a| = |b|
\sqrt{a^2} = \sqrt{b^2}
a^2 = b^2
a=b

Have you checked this: a-b-which-of-the-following-must-be-true-141468.html#p1137162

a^2 = b^2 does not necessarily means that a = b. Consider a = 1 and b = -1. a^2 = b^2 means a = b or a = -b.
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Re: |a|=|b|, which of the following must be true :   [#permalink] 20 Sep 2014, 23:09
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