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# M15-37

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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. $$|a| = b$$.

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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I think this is a high-quality question and I agree with explanation.
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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

Cheers,
Brent
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I think this is a high-quality question and I agree with explanation.
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Hi,

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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VardanShines wrote:
Hi,

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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VardanShines wrote:
Hi,

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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I think this is a high-quality question and I agree with explanation. Top Question!
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Bunuel wrote:
$$a^2 - b^2 = b^2 - c^2$$. Is $$a = |b|$$?

(1) $$b = |c|$$

(2) $$b = |a|$$

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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ProfChaos wrote:
Bunuel wrote:
$$a^2 - b^2 = b^2 - c^2$$. Is $$a = |b|$$?

(1) $$b = |c|$$

(2) $$b = |a|$$

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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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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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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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 $$b = 1$$.