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# What does |2b| equal?

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25 Jun 2012, 12:23
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What does |2b| equal?

(1) b^2 - |b| - 20 = 0

(2) |2b| = 3b + 25
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Re: What does |2b| equal? [#permalink]

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25 Jun 2012, 12:55
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What does |2b| equal?

(1) b^2-|b|-20=0. Solve quadratics for $$|b|$$: $$(|b|)^2-|b|-20=0$$ --> $$|b|=-4$$ or $$|b|=5$$. Since absolute value cannot be negative then we have that $$|b|=5$$ and $$|2b|=10$$. Sufficient.

(2) |2b|=3b+25. Two cases:

If $$b\leq{0}$$ then we would have that $$-2b=3b+25$$ --> $$b=-5$$.
If $$b>0$$ then we would have that $$2b=3b+25$$ --> $$b=-25$$, but since we are considering the rangeo when $$b>{0}$$ then discard this solution.

So, we have that $$b=-5$$, hence $$|2b|=10$$. Sufficient.

Hope it's clear.
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Re: What does |2b| equal? [#permalink]

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26 Jun 2012, 08:09
Hey Bunuel, the original question contained b^2-|b|-20=0 but you considered it to |b|^2-|b|-20=0
I know it does not matter that if it is |b|^2 or b^2
But it does matter especially when it is not mentioned that b is positive number...
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Re: What does |2b| equal? [#permalink]

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26 Jun 2012, 08:13
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manimani wrote:
Hey Bunuel, the original question contained b^2-|b|-20=0 but you considered it to |b|^2-|b|-20=0
I know it does not matter that if it is |b|^2 or b^2
But it does matter especially when it is not mentioned that b is positive number...

I'm not sure I understand your question. Anyway:

We are asked to find the value of |2b|, so we don't really care for (1) whether b is positive or negative.
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Re: What does |2b| equal? [#permalink]

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26 Jun 2012, 09:03
Bunuel wrote:
What does |2b| equal?

(1) b^2-|b|-20=0. Solve quadratics for $$|b|$$: $$(|b|)^2-|b|-20=0$$ --> $$|b|=-4$$ or $$|b|=5$$. Since absolute value cannot be negative then we have that $$|b|=5$$ and $$|2b|=10$$. Sufficient.

(2) |2b|=3b+25. Two cases:
If $$b\leq{0}$$ then we would have that $$-2b=3b+25$$ --> $$b=-5$$.
If $$b>0$$ then we would have that $$2b=3b+25$$ --> $$b=-25$$, but since we are considering the rangeo when $$b>{0}$$ then discard this solution.

So, we have that $$b=-5$$, hence $$|2b|=10$$. Sufficient.

Hope it's clear.

Hi, can you make the solution more clear?in case of (1) why you didn't consider the negative value of b? When b is positive, b^2 -b = 20 and when b is negative b^2 + b =20..then we will find 4 values of b...I am confused ..Can you explain? and in case of (2) how you discard the 2nd value? Thanks
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Re: What does |2b| equal? [#permalink]

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26 Jun 2012, 09:21
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farukqmul wrote:
Bunuel wrote:
What does |2b| equal?

(1) b^2-|b|-20=0. Solve quadratics for $$|b|$$: $$(|b|)^2-|b|-20=0$$ --> $$|b|=-4$$ or $$|b|=5$$. Since absolute value cannot be negative then we have that $$|b|=5$$ and $$|2b|=10$$. Sufficient.

(2) |2b|=3b+25. Two cases:
If $$b\leq{0}$$ then we would have that $$-2b=3b+25$$ --> $$b=-5$$.
If $$b>0$$ then we would have that $$2b=3b+25$$ --> $$b=-25$$, but since we are considering the rangeo when $$b>{0}$$ then discard this solution.

So, we have that $$b=-5$$, hence $$|2b|=10$$. Sufficient.

Hope it's clear.

Hi, can you make the solution more clear?in case of (1) why you didn't consider the negative value of b? When b is positive, b^2 -b = 20 and when b is negative b^2 + b =20..then we will find 4 values of b...I am confused ..Can you explain? and in case of (2) how you discard the 2nd value? Thanks

OK.

Say $$x=|b|$$, then we have that $$x^2-x-20=0$$ --> $$x=-4$$ or $$x=5$$. Now, $$x=|b|=-4$$ is not possible, since an absolute value of a number ($$|b|$$) cannot be negative. So, we have that $$|b|=5$$ and $$|2b|=10$$.

Now, you can solve this statement considering two ranges: $$b\leq{0}$$ and $$b>0$$, which will lead you to the same.

If $$b\leq{0}$$ then we'll have $$b^2+b-20=0$$ --> $$b=-5$$ or $$b=4$$ (not a valid solution since we are considering the range when $$b\leq{0}$$);
If $$b>{0}$$ then we'll have $$b^2-b-20=0$$ --> $$b=-4$$ (not a valid solution since we are considering the range when $$b>{0}$$) or $$b=5$$;

So, only two valid solutions: $$b=-5$$ or $$b=5$$ --> $$|2b|=10$$.

As for (2), it's explained in the post:
If $$b>0$$ then we would have that $$2b=3b+25$$ --> $$b=-25$$, but since we are considering the rangeo when $$b>{0}$$ then discard this solution.

Hope it's clear.
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Re: What does |2b| equal? [#permalink]

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26 Jun 2012, 09:22
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farukqmul wrote:
Bunuel wrote:
What does |2b| equal?

(1) b^2-|b|-20=0. Solve quadratics for $$|b|$$: $$(|b|)^2-|b|-20=0$$ --> $$|b|=-4$$ or $$|b|=5$$. Since absolute value cannot be negative then we have that $$|b|=5$$ and $$|2b|=10$$. Sufficient.

(2) |2b|=3b+25. Two cases:
If $$b\leq{0}$$ then we would have that $$-2b=3b+25$$ --> $$b=-5$$.
If $$b>0$$ then we would have that $$2b=3b+25$$ --> $$b=-25$$, but since we are considering the rangeo when $$b>{0}$$ then discard this solution.

So, we have that $$b=-5$$, hence $$|2b|=10$$. Sufficient.

Hope it's clear.

Hi, can you make the solution more clear?in case of (1) why you didn't consider the negative value of b? When b is positive, b^2 -b = 20 and when b is negative b^2 + b =20..then we will find 4 values of b...I am confused ..Can you explain? and in case of (2) how you discard the 2nd value? Thanks

hey farukqmul I also had the same doubt but then i tried to solve it properly and i found out that there is no need to do it the long way but i will post here to clear your doubt
b^2-|b|-20=0
Considering b>0 eq. becomes b^2-b-20=0 which gives us solution b=-4 and 5 since we assumed b>0 so b=-4 is rejected
Considering b<0 eq. becomes b^2+b-20=0 which gives us solution b=4 and -5 since we assumed b<0 so b=4 is rejected
In both the cases |2b|=10
Now for the 2nd part (copying Buenel's explanation)
|2b|=3b+25. Two cases:
If b>0 then we would have that -2b=3b+25 --> b=-5.
If b>0 then we would have that 2b=3b+25 --> b=-25, but since we are considering b>0 then discard this solution.
Hope it's all clear now.
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Re: What does |2b| equal? [#permalink]

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27 Jul 2012, 03:32
Hi, I am wondering why |b| cannot be other numbers greater than 5. What if |b| was to equal 16? This would also allow us to solve the quadratics right? in this case (1) would have multiple answers, 5 and 6, therefore causing (1) to be insufficient?

Please let me know if I misunderstand something. Thank you
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Re: What does |2b| equal? [#permalink]

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27 Jul 2012, 04:20
Solving the equation x^2 - x - 20 = 0 gives you two possible solution which are -4 and +5
In our case x = |b| so x cannot be negative.
That let x = 5 the only possible solution
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Re: What does |2b| equal? [#permalink]

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27 Jul 2012, 07:43
hi Arthur, I understand your point. However, I do not understand why "16" cannot be another possible answer. Since if we allow |b| to be 16, we can also solve the quadratics.
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Re: What does |2b| equal? [#permalink]

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27 Jul 2012, 08:20
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naruphanp wrote:
hi Arthur, I understand your point. However, I do not understand why "16" cannot be another possible answer. Since if we allow |b| to be 16, we can also solve the quadratics.

Frankly the above does not make ANY sense. How is |b|=16 satisfy any of the statements?
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Re: What does |2b| equal? [#permalink]

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28 Jul 2012, 18:51
Bunuel wrote:
What does |2b| equal?

(1) b^2-|b|-20=0. Solve quadratics for $$|b|$$: $$(|b|)^2-|b|-20=0$$ --> $$|b|=-4$$ or $$|b|=5$$. Since absolute value cannot be negative then we have that $$|b|=5$$ and $$|2b|=10$$. Sufficient.

(2) |2b|=3b+25. Two cases:
If $$b\leq{0}$$ then we would have that $$-2b=3b+25$$ --> $$b=-5$$.
If $$b>0$$ then we would have that $$2b=3b+25$$ --> $$b=-25$$, but since we are considering the rangeo when $$b>{0}$$ then discard this solution.

So, we have that $$b=-5$$, hence $$|2b|=10$$. Sufficient.

Hope it's clear.

Hey Bunuel,

I have always been meaning to ask you. I know that
(1) $$|x|=x$$ if $$x>0$$ and
(2) $$|x|=-x$$ if $$x <0$$.

But the "=" of the (less than or equal to) or (greater than or equal to), where does it go? I noticed that you put it on equation 2, while others place it on equation (1). Its Driving me nuts!!
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Re: What does |2b| equal? [#permalink]

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29 Jul 2012, 01:50
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alphabeta1234 wrote:
Bunuel wrote:
What does |2b| equal?

(1) b^2-|b|-20=0. Solve quadratics for $$|b|$$: $$(|b|)^2-|b|-20=0$$ --> $$|b|=-4$$ or $$|b|=5$$. Since absolute value cannot be negative then we have that $$|b|=5$$ and $$|2b|=10$$. Sufficient.

(2) |2b|=3b+25. Two cases:
If $$b\leq{0}$$ then we would have that $$-2b=3b+25$$ --> $$b=-5$$.
If $$b>0$$ then we would have that $$2b=3b+25$$ --> $$b=-25$$, but since we are considering the rangeo when $$b>{0}$$ then discard this solution.

So, we have that $$b=-5$$, hence $$|2b|=10$$. Sufficient.

Hope it's clear.

Hey Bunuel,

I have always been meaning to ask you. I know that
(1) $$|x|=x$$ if $$x>0$$ and
(2) $$|x|=-x$$ if $$x <0$$.

But the "=" of the (less than or equal to) or (greater than or equal to), where does it go? I noticed that you put it on equation 2, while others place it on equation (1). Its Driving me nuts!!

First of all:
$$|x|=-x$$ when $$x\leq{0}$$;

$$|x|=x$$ when $$x\geq{0}$$.

Next, as for "=" sign in the solution: you could include it either in the first case or in the second, it doesn't matter at all.
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Re: What does |2b| equal? [#permalink]

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04 Jul 2013, 01:24
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Re: What does |2b| equal? [#permalink]

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04 Jul 2013, 21:49
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(1) If b >0, eqn reduces to (b-5)(b+4)=0 , since we have assumed b >0, this gives us a soln of b=5. If b <0, eqn reduces to (b+5)(b-4) =0 giving us a soln of b=-5 (since b <0). Since we are asked for the value of |2b| it doesnt matter if b is 5 or -5. Sufficient.

(2) If b >0, it gives us 2b=3b+25 or b =-25 but tht goes against our assumption. So b <0 and -2b=3b+25 or b=-5. Sufficient.

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Re: What does |2b| equal? [#permalink]

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07 Jul 2013, 18:24
What does |2b| equal?

(1) b^2-|b|-20=0
use quadratic formula to try and get a value for b.
(|b|-5) (|b|+4) = 0
|b|=5 OR |b|=-4
Well, an absolute value cannot be equal to a negative number so |b|=5. Therefore, |2b|=10
SUFFICIENT

(2) |2b|=3b+25

Find positive/negative cases to try and isolate b.
|2b|=3b+25
If x>= 0
2b=3b+25
-b=25
b=-25
Not valid as -2b is not greater than or equal to zero.
x<0
-2b=3b+25
-5b=25
b=-5
Valid as -5 falls within the range of <0
SUFFICIENT
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Re: What does |2b| equal? [#permalink]

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11 Aug 2014, 00:39
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11 Aug 2014, 01:44
This is how I solved it:

1. b^2 - |b| = 20
Since both b^2 and absolute value of b cannot be negative,

b ( b - 1 ) = 20 and b= 5

2. I got two solutions following the same method as Bunuel did, but the part I don't understand is why we have to discard b = -25, where does it say that we are considering b>0 ??

Can someone please explain what exactly I missed or overlooked! Thank you
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12 Aug 2014, 08:14
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suhaschan wrote:
This is how I solved it:

1. b^2 - |b| = 20
Since both b^2 and absolute value of b cannot be negative,

b ( b - 1 ) = 20 and b= 5

2. I got two solutions following the same method as Bunuel did, but the part I don't understand is why we have to discard b = -25, where does it say that we are considering b>0 ??

Can someone please explain what exactly I missed or overlooked! Thank you

For (1): b(b - 1) = b^2 - b not b^2 - |b|. Also, b(b - 1) = 20 has two roots: b = -4 and b = 5.

For (2): plug b = -25. Does it satisfy the equation?
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Re: What does |2b| equal? [#permalink]

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29 Aug 2015, 13:01
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Re: What does |2b| equal?   [#permalink] 29 Aug 2015, 13:01
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