Problem: The arithmetic mean between x+ax\dfrac{x+a}{x}xx+aβ and xβax\dfrac{x-a}{x}xxβaβ, when xβ 0x \neq 0xξ =0, is (the symbol β \neqξ = means "not equal to"):
Answer Choices:
A. 2 , if a β 0\neq 0ξ =0
B. 111
C. 1 , if a=0a=0a=0 only
D. a/xa / xa/x
E. xxx Solution:
12(x+ax+xβax)=12(2xx)=1\dfrac{1}{2}\left(\dfrac{x+a}{x}+\dfrac{x-a}{x}\right)=\dfrac{1}{2}\left(\dfrac{2 x}{x}\right)=121β(xx+aβ+xxβaβ)=21β(x2xβ)=1