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Re: M15-37 [#permalink]
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Mechmeera wrote:
My doubt may sound silly but this is confusing me a lot.
what is the difference between

1. \(|a| = |b|\).
2. \(a = |b|\).
3. \(|a| = b\).

please explain the concept.


1. \(|a| = |b|\) means that the distance from a to 0 is the same as the distance from b to 0. Or that the magnitudes of a and b are the same.

2. \(a = |b|\) means that the distance from b to 0 is a.

3. \(|a| = b\) means that the distance from a to 0 is b.
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Re: M15-37 [#permalink]
I think this is a high-quality question and I agree with explanation.
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Re: M15-37 [#permalink]
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Bunuel wrote:
\(a^2 - b^2 = b^2 - c^2\). Is \(a = |b|\)?


(1) \(b = |c|\)

(2) \(b = |a|\)


Target question: Is a = |b|?

Given: a² - b² = b² - c²

Statement 1: b = |c|
This tells us a few things, but with regard to this question, it tells us that b and c have the same magnitude
This also means that b² = c²
With this information, let's test some values that satisfy both statement 1 and the given information:
Case a: a = 0, b = 0 and c = 0. In this case, the answer to the target question is YES, a does EQUAL |b|
Case b: a = -1, b = 1 and c = -1. In this case, the answer to the target question is NO, a does NOT equal |b|
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b = |a|
Let's test values again.
There are several values of x and y that satisfy statement 2. Here are two:
Case a: a = 0, b = 0 and c = 0. In this case, the answer to the target question is YES, a does EQUAL |b|
Case b: a = -1, b = 1 and c = -1. In this case, the answer to the target question is NO, a does NOT equal |b|
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient.
So, the same counter-examples will satisfy the two statements COMBINED.

In other words,
Case a: a = 0, b = 0 and c = 0. In this case, the answer to the target question is YES, a does EQUAL |b|
Case b: a = -1, b = 1 and c = -1. In this case, the answer to the target question is NO, a does NOT equal |b|
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

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Re: M15-37 [#permalink]
I think this is a high-quality question and I agree with explanation.
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Re: M15-37 [#permalink]
Hi,

Please help me to find where have I gone wrong:

a^2−b^2=b^2−c^2
a^2= 2b^2-c^2
a=sqrt (2b^2-c^2)

Statement 1: b=mod(c)
so squaring both sides, b^2=c^2

Using this in the original equation, a=sqrt(b^2), so a=mod(b)

Thanks
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Re: M15-37 [#permalink]
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VardanShines wrote:
Hi,

Please help me to find where have I gone wrong:

a^2−b^2=b^2−c^2
a^2= 2b^2-c^2
a=sqrt (2b^2-c^2)

Statement 1: b=mod(c)
so squaring both sides, b^2=c^2

Using this in the original equation, a=sqrt(b^2), so a=mod(b)

Thanks


From \(a^2= 2b^2-c^2\) it follows that \(|a|=\sqrt{2b^2-c^2}\)
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Re: M15-37 [#permalink]
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VardanShines wrote:
Hi,

Please help me to find where have I gone wrong:

a^2−b^2=b^2−c^2
a^2= 2b^2-c^2
a=sqrt (2b^2-c^2)

Statement 1: b=mod(c)
so squaring both sides, b^2=c^2

Using this in the original equation, a=sqrt(b^2), so a=mod(b)

Thanks


Lets take an example: \(a^2 = 4\)
To get value of a, you take a square root.
so, \( a= \sqrt{4}\)
Now, \(\sqrt{4} = 2\).

But how many roots does a have? Is it just \(a = 2\)? or is it \(a = \pm 2\)?
When you take a square root, it is always to be interpreted as \(a = \pm\sqrt{ 4} = \pm2\)

It is recommended to add "\(\pm\)" before the square root.

Coming back to the question...

In the highlighted part, lets follow this protocol of adding "\(\pm\)" before the square root.

\(a=\pm \sqrt{b^2}\) = \(\pm |b|\)

Thus, \(a = \pm |b|\).... But the question asks if \(a = |b|..\)
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Re: M15-37 [#permalink]
I think this is a high-quality question and I agree with explanation. Top Question!
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Re: M15-37 [#permalink]
Bunuel wrote:
\(a^2 - b^2 = b^2 - c^2\). Is \(a = |b|\)?


(1) \(b = |c|\)

(2) \(b = |a|\)


VeritasKarishma Bunuel

When I arrived at \(a^2=b^2\)
If I take square root on both sides
\(\sqrt{a^2}=\sqrt{b^2}\)

I will get \(|a| = |b|\)
Thus
\(|a|=b\) or \(|a|=-b\)
Thus,
\(a=b\) or
\(-a=b\) or
\(a=-b\) or
\(-a=-b\)

Which ultimately is \(a=b\) or \(a=-b\)
Which is the definition of \(a=|b|\)

Please tell me what is wrong in this!
TIA!
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Re: M15-37 [#permalink]
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ProfChaos wrote:
Bunuel wrote:
\(a^2 - b^2 = b^2 - c^2\). Is \(a = |b|\)?


(1) \(b = |c|\)

(2) \(b = |a|\)


VeritasKarishma Bunuel

When I arrived at \(a^2=b^2\)
If I take square root on both sides
\(\sqrt{a^2}=\sqrt{b^2}\)

I will get \(|a| = |b|\)
Thus
\(|a|=b\) or \(|a|=-b\)
Thus,
\(a=b\) or
\(-a=b\) or
\(a=-b\) or
\(-a=-b\)

Which ultimately is \(a=b\) or \(a=-b\)
Which is the definition of \(a=|b|\)

Please tell me what is wrong in this!
TIA!


The question asks whether \(a = |b|\). Notice here that a, since it equals to an absolute value, cannot be negative. |a| = |b| doe snot guarantee that a is non-negative. For example, it's possible a^2 to be equal to b^2 and a not be equal to |b|. For example, a=-1 and b=1.
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Re: M15-37 [#permalink]
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Re: M15-37 [#permalink]
As we are getting a^2-b^2=0 means

(a+b)(a-b)= 0 so

a=b and a=-b ------- equation no 1

and value of |b| is = ' b' or '-b'

so we can write equation no 1 as - a=|b|

so dont u think 'A' shld be ans?????
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Re: M15-37 [#permalink]
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Mohit1994 wrote:
As we are getting a^2-b^2=0 means

(a+b)(a-b)= 0 so

a=b and a=-b ------- equation no 1

and value of |b| is = ' b' or '-b'

so we can write equation no 1 as - a=|b|

so dont u think 'A' shld be ans?????


a^2 = b^2 does not necessarily mean that a = |b|. For example, consider \(a = -1\) and \(b = 1\).
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