if \(x^2>y^2\), then \(|x|>|y|\)

if x>0 y>0, then x>y since \(|x|>|y|\);

if x>0 y<0, then x>y. Because \(x>-|y|\) and \(-|y|=-(-y)=y\);

x<0 y>0 is not possible. Because \(x>-|y|\) and \(-|y|=-y\), so \(x>-y\), which means \(|x|<|y|\), but we already know \(|x|>|y|\), so this is a contradiction;

x<0 y<0 is also not possbile, Because \(x>-|y|\) and \(-|y|=-(-y)=y\) therefore x>y but then \(|x|<|y|\), and we already know \(|x|>|y|\), so this is a contradiction;

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Yuan Liu, Creator of Gregory. Dartmouth Grad.

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