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

Question

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

KarishmaB · Accepted Answer

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| 
eq -5

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

Answer (E)

clarkkent0610 · Answer

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

himanshuhpr · Answer

^^ 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

KarishmaB · Answer

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.

himanshuhpr · Answer

Ok. thanks very much for the clarification... your blogs and posts are very informative

prep · Answer

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.

Bunuel · Answer

|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.

mbaiseasy · Answer

|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

Bunuel · Answer

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SVaidyaraman · Answer

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.

prashd · Answer

Where am i going wrong ??

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

Bunuel · Answer

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.

ScottTargetTestPrep · Answer

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.

Answer: E

bv8562 · Answer

I solved it this way:

|a|=|b| squaring on both sides
a^2=b^2
a^2-b^2=0
(a-b)(a+b)=0
either a=b or a=-b

since, either of them could be true and both of them will not be true at the same time. So, the answer is NONE (E)

Am I doing it right?  Bunuel   VeritasKarishma

KarishmaB · Answer

bv8562

Your method is correct but not your inference.

(a-b)(a+b)=0 
means
Either a=b or a=-b

But what does either a = b or a = -b mean? 
It means that one of the following 3 possibilities will occur:
1. a = b and a is not equal to -b
2. a = -b and a is not equal to b
3. a = b and a = -b (for example when a = 0, b = 0)

But which one of these 3 will occur, we cannot say. So is it necessary that a = b? No. In case a = -b, a may not be equal to b. 
Is it necessary that a = -b? No. In case a = b, a may not be equal to -b etc. 
It is not true that both cannot occur at the same time. They may. It is one of the possibilities.

Note that "either X or Y" means "at least one of X and Y".

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

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