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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] New post 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] New post 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] New post 28 Oct 2012, 22:00
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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)
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Re: Modulus Ques. [#permalink] New post 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
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Re: Modulus Ques. [#permalink] New post 29 Oct 2012, 01:03
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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.
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Re: Modulus Ques. [#permalink] New post 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] New post 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.

Please help me understand if I'm missing anything.
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Re: Modulus Ques. [#permalink] New post 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.

Please help me understand if I'm missing anything.


|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] New post 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!

Answer: NONE or E
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Re: |a|=|b|, which of the following must be true : [#permalink] New post 04 Jul 2013, 00:45
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Re: |a|=|b|, which of the following must be true : [#permalink] New post 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

So the answer is E.
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Re: |a|=|b|, which of the following must be true :   [#permalink] 04 Jul 2013, 17:33
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