Absolute value of ( x-y)

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Absolute value of ( x-y)

Postby Ddanndt » Mon Apr 05, 2010 9:29 am UTC

Hi everybody. I'm working on a proof and I want to get a relation between |x-y| and |x|-|y| but in many books I see the relation ||x|-|y||<=|x-y|<=|x|+|y| and although I understand why there is ||x|-|y|| I dont know what i should do to get a relation between |x-y| and |x|-|y|. Thanks for your help.
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Re: Absolute value of ( x-y)

Postby jestingrabbit » Mon Apr 05, 2010 9:48 am UTC

If |y|<|x| is it cool to drop the absolute value? If |y|>|x| what then?

edit: Totally misread sorry.
Last edited by jestingrabbit on Mon Apr 05, 2010 2:24 pm UTC, edited 1 time in total.
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Re: Absolute value of ( x-y)

Postby Natty » Mon Apr 05, 2010 1:12 pm UTC

One of the best ways to work with absolute values in proofs is to break your proof down into cases. For example, if there is an |x| involved in your problem, split your proof into two cases, the first in which |x|=x and the second in which |x|=-x. Then go through each proof and try to arrive at a similar result.

In your example you would need more than just two cases.

One pitfall for these problems is that students don't always consider every single case. For example, if your problem has both x and y in the, they break it down into case 1: |x|=x and |y|=y, case 2: |x|=-x and |y|=y, case 3: |x|=x and |y|=-y, case 4: |x|=-x and |y|=-y, but they fail to consider all values in the problem and will forget that in the case of |x-y| for certain cases the outcome is unknown and needs to be separated into even more cases of |x-y|=-(x-y) and |x-y|=(x-y).

Hope this helps.
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Re: Absolute value of ( x-y)

Postby Yakk » Mon Apr 05, 2010 3:45 pm UTC

What you are reading about is known as the triangle inequality.
| a + b | <= |a| + |b|
There are lots of equations that are similar to the above.

Using ~ as "unknown relation", we can play:
|x-y| ~ |x| - |y|
|x-y| + |y| ~ x
now, you'll note that (x-y)+y = x...
Define a = (x-y), and b = y. Then x = a+b.

Does anything pop out?
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

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