Q:

simple algebraic inequality question

For any  $a,b,c$ such that $0 < a < b < c$ ,  prove/disprove: 
$\frac{b}{a+b}  + \frac{c}{b+c}  - \frac{c}{a+c} - \frac{1}{2}  > 0$

A:

http://www.wolframalpha.com/input/?i=simplify+b%2F%28a%2Bb%29+%2B+c%2F%28b%2Bc%29+-+c%2F%28a%2Bc%29+-+1%2F2
The denominator is positive, and two of the terms in the numerator are negative, while the remaining term is positive.

