SOLUTION: Prove that |x + y| ≦ |x| + |y|

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Question 643819: Prove that |x + y| ≦ |x| + |y|
Answer by AnlytcPhil(1806)   (Show Source): You can put this solution on YOUR website!

           xy ≦ |xy|   If either is 0, or both have the same sign, equality
                       holds.  If one is negative and the other positive the
                       left side is negative and the right side positive.  
                       Thus this inequality holds in all cases
                       
                       Multiply both sides by 2

          2xy ≦ 2|xy|        
                       Add x² + y² to both sides:

x² + 2xy + y² ≦ x² + 2|xy| + y²

                      Factor both sides: 

    (x + y)² ≦ (|x| + |y|)²

                      Take non-negative square roots of both sides:

    |x + y| ≦ |x| + |y|

Edwin
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