# Inequality- absolute value function

• Feb 18th 2013, 04:44 AM
sachinrajsharma
Inequality- absolute value function
Prove that if the numbers x,y are of one sign, then $|\frac{x+y}{2}-\sqrt{xy}|+|\frac{x+y}{2}+\sqrt{xy}|=|x|+|y|$
• Feb 18th 2013, 04:57 AM
Plato
Re: Inequality- absolute value function
Quote:

Originally Posted by sachinrajsharma
Prove that if the numbers x,y are of one sign, then $|\frac{x+y}{2}-\sqrt{xy}|+|\frac{x+y}{2}+\sqrt{xy}|=|x|+|y|$

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• Feb 18th 2013, 09:49 PM
sachinrajsharma
Re: Inequality- absolute value function
I am sorry as I am new to this forum I didnt know that I have to show my work as well.. moreover this is not my homework what I have done

$|\frac{x+y}{2}-\sqrt{xy}|+|\frac{x+y}{2}+\sqrt{xy}|=|x|+|y|$

Solving R.HS. i.e. $|\frac{x+y}{2}-\sqrt{xy}|= |\frac{x+y -\sqrt{xy}}{2}| = \frac{(\sqrt{x}-\sqrt{y})^2}{2}$

$|\frac{x+y}{2}+\sqrt{xy}| = |\frac{ (\sqrt{x}+\sqrt{y})^2}{2} |$