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Tricky Absolute Value Question

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Tricky Absolute Value Question [#permalink] New post 06 Feb 2013, 04:20
Not sure if this type of question has been posted, but I could not find. The problem says expression |6-4Pi+3\sqrt{2}| + |8-6Pi+4 \sqrt{2}| is in the form of a,bPi,c \sqrt{2}, what is a+b+c?

I do not have the answers, nor the source, however I am confused as to the action to take here. How would you recommend solving this please? I am totally confused as to how I could even start here, and the absolute values complicate this in my mind even more...Thanks much in advance.
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Re: Tricky Absolute Value Question [#permalink] New post 06 Feb 2013, 11:54
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Hi There,

This a strange one and I don't think that this is super close to a real GMAT question but just for fun let's take a look at it.

My interpretation is that the question is asking you to simplify the expression and then add all of the coefficients. Let's start by breaking down absolute value a bit.

|-3| = 3. So we just take away the negative sign. By the same logic you may make the mistake of thinking that |-3+5| is 8 (by taking away the negative in front of -3 and adding the numbers) but it is in fact 2 because we have to combine the values before we mess with the signs. All pretty basic stuff but important to think about while we do this.

If all of the numbers in this expression were positive this would be a no brainer. If all of the numbers were rational this would be a no brainer. Unfortunately we have irrational numbers that we can't combine with our rational numbers.

Well, it's almost as if those irrational numbers are variables so let's treat them as variables and make Pi = x and Root 2 = y. So in the first expression we have |6-4x+y|. We know that 4x is greater than 6+y so that the entire expression will end up being less than 0 and therefore the absolute value will have the opposite sign. To represent that - negate the whole expressions. So, |6-4x+y| = -6+4x-y (remember we are taking into account that we know that 4x is greater than 6+y).

This may be confusing so let's explain a bit with real numbers: |4-10+2| = -4+10-2 = 4 Whenever you have a doubt about rules that you are applying to variables, try the rules out with real numbers and see if the rules work.

We can do the same thing for the second expression. |8-6x+4| = -8+6x-4

Now we can sum the expressions: (-8+6x-4y)+(-6+4x-3y) = -14+10x-7y

a=-14 b=10 c=-7

a+b+c= -11

This question turned out to be a nice exploration into absolute value. I hope that all of this helps. Let me know if you need more advice on the subject.

HG.
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Re: Tricky Absolute Value Question [#permalink] New post 06 Feb 2013, 12:08
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obs23 wrote:
Not sure if this type of question has been posted, but I could not find. The problem says expression |6-4Pi+3\sqrt{2}| + |8-6Pi+4 \sqrt{2}| is in the form of a,bPi,c \sqrt{2}, what is a+b+c?

I do not have the answers, nor the source, however I am confused as to the action to take here. How would you recommend solving this please? I am totally confused as to how I could even start here, and the absolute values complicate this in my mind even more...Thanks much in advance.

I'm happy to help with this. :-)

First of all, let me say --- as it stands, this question is not likely at all to appear on the GMAT. Not only is it not in the proper test-question format (multiple choice or DS), but more importantly, I have never seen the GMAT ask anything like this. This is, in essence, an estimation problem.

Basically, we have to figure out whether each expression is positive or negative.

Think about 6-4Pi+3\sqrt{2} first. Is this positive or negative? Well, pi is a little more than 3, so 4*(pi) is little more than 12. The square root of two is about 1.4 ---- this is a very handy approximation to know ---- so 3*sqrt(2) is about 4.2
Well, the positive terms, 6 + 3*sqrt(2) are around 10.2, which is less than the negative term, 4*(pi). Taking the absolute value will multiply this expression by an "opposite" sign, making it all the opposite sign

|6-4Pi+3\sqrt{2}|= 4Pi-6-3\sqrt{2}

Now, think about the second expression, 8-6Pi+4 \sqrt{2}. Here, the only negative term, 6*(pi), is a little larger than 18. 4*sqrt(2) is around 4*1.4 = 5.6, so the sum of the positive terms, 8 + 4*sqrt(2), is around 13.6, and this is clearly less than 18. Once again, this expression is negative, so all terms get multiplied by the opposite sign.

|8-6Pi+4 \sqrt{2}|=6Pi-8-4\sqrt{2}

Now that these are both out of the absolute values, we can add them

Sum = 10Pi-14-7\sqrt{2}

I don't know the exact form in which the a/b/c stuff is stated, but if the sum is supposed to be in the form .....

Sum = a+bPi+c\sqrt{2}

Then a = -14, b = +10, and c = -7, so a+b+c = -11

Does all this make sense?
Mike :-)
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Re: Tricky Absolute Value Question [#permalink] New post 06 Feb 2013, 21:01
Guys, this is extremely helpful and I get the idea - I was so lost here. Humble kudos to both!
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Re: Tricky Absolute Value Question   [#permalink] 06 Feb 2013, 21:01
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