Q. 5

# Are there numbers x, y such that |x + y| < |x| + |y|?

To prove : |x + y| < |x| + |y|

We know that, |x| and |y|

Therefore, 2|x||y|

Adding x2 + y2 to both sides,

We have, x2 + y2 + 2|x||y| x2 + y2 + 2xy

|x|2 + |y|2 + 2|x||y| x2 + y2 + 2xy

(|x| + |y|)2 (x + y)2

(|x| + |y|)2 (|x + y|)2

|x| + |y| |x + y|

We can also say that |x| + |y| > |x + y|

Therefore, this inequality holds true for all x and y.

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