- #1
jimmyly
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Homework Statement
|x + y| ≥ |x| - |y| [Hint: write out x = x + y - y, and apply Theorem 3, together with the fact that |-y| = |y|]
Homework Equations
Theorem 3: |a + b| ≤ |a| + |b|
x = x + y - y
|-y| = |y|
The Attempt at a Solution
|x + y| ≥ |x| - |y|
x = x + y - y (don't know where to use it)
xy ≤ |xy| = |x| |y| ( I really don't know why I am taking these steps, I am pretty much following the proof of theorem 3 in this book)
2xy ≤ 2|x||y| ( does the |-y| = |y| come into play here to flip the inequality? )
(x+y)^2 = x^2 + 2xy + y^2 ≤ x^2 - 2|x||y| + y^2 = (|x| - |y|)^2
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