anonymous
  • anonymous
Prove |x+y| ≥ |x| - |y|. It gives a hint to turn x=x+y-y and to use |a + b| ≤ |a| + |b| (triangle inequality) with the fact |-y| = |y|, but I'm not sure where to start.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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dan815
  • dan815
lemme rewrite this, in a more simple way
dan815
  • dan815
|dw:1370078585243:dw|
dan815
  • dan815
|dw:1370078646046:dw|

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dan815
  • dan815
|dw:1370078847393:dw|
dan815
  • dan815
since its -2 * magnitude of x and y then its always a negative number or 0 when x and y = 0 so this equality holds true 2YX can only be greater or equal to -2}y}}x}
reemii
  • reemii
\(|x| = |(x-y)+y|\le |(x-y)| + |y|\). therefore, removing |y| on both sides: \(|x|-|y|\le |x-y| \). Then you just have to be able to explain why you can replace \(y\) by \(-y\). and you're done.
anonymous
  • anonymous
Would this work? Let a = x + y, b = -y i) |a + b| ≤ |a| + |b| (triangle inequality) ii) |(x+y) - y| ≤ |x+y| + |-y| (substitution) iii) |x| - |-y| ≤ |x+y| (subtract |-y| from both sides) iv) |x| - |y| ≤ |x+y| (|-y| = |y| definition of absolute value)
reemii
  • reemii
exactly how you should do it. well done.
anonymous
  • anonymous
Thanks to both of you. What reemii wrote gave me a flash of insight. However, the way the hint is written in the book was confusing as I was literally substituting a in the triangle inequality with a = x + y - y which was confusing me for a long time.

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