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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]

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28 Oct 2012, 12:06
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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 29 Oct 2012, 00:58, edited 1 time in total.
Renamed the topic and edited the question.

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28 Oct 2012, 12:27
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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28 Oct 2012, 23:00
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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
$$-5 \neq 5$$

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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17771 [15], given: 235 Intern Joined: 29 Aug 2012 Posts: 26 Kudos [?]: 35 [0], given: 56 Schools: Babson '14 GMAT Date: 02-28-2013 Re: Modulus Ques. [#permalink] ### Show Tags 29 Oct 2012, 00:21 1 This post was BOOKMARKED 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 Kudos [?]: 35 [0], given: 56 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7736 Kudos [?]: 17771 [0], given: 235 Location: Pune, India Re: Modulus Ques. [#permalink] ### Show Tags 29 Oct 2012, 02:03 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 Get started with Veritas Prep GMAT On Demand for$199

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29 Oct 2012, 02:08
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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29 Oct 2012, 10:49
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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29 Oct 2012, 11:03
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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]

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05 Dec 2012, 01:52
|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]

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04 Jul 2013, 01:45
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: |a|=|b|, which of the following must be true : [#permalink]

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04 Jul 2013, 18:33
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]

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16 Sep 2014, 20:44
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Re: |a|=|b|, which of the following must be true : [#permalink]

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20 Sep 2014, 22:57
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]

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21 Sep 2014, 00:09
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]

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21 Sep 2015, 01:38
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Re: |a|=|b|, which of the following must be true : [#permalink]

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30 Mar 2017, 10:40
Hello from the GMAT Club BumpBot!

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Re: |a|=|b|, which of the following must be true : [#permalink]

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04 Apr 2017, 16:18
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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

Since |a| = |b|, we see that a = b, -a = b, a = -b, or -a = -b.

Since we have all four of those options as possibilities, none of the Roman numerals MUST BE TRUE.

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Re: |a|=|b|, which of the following must be true :   [#permalink] 04 Apr 2017, 16:18
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