SOLUTION: Hello, my question is: If x>y>z then prove xy + yz > (x+y)(y+z) / 2 Thanks in advance for your help!

Algebra ->  Proofs -> SOLUTION: Hello, my question is: If x>y>z then prove xy + yz > (x+y)(y+z) / 2 Thanks in advance for your help!      Log On


   

Question 1140916: Hello, my question is:
If x>y>z then prove xy + yz > (x+y)(y+z) / 2
Thanks in advance for your help!
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
If x%3Ey%3Ez , then x-y%3E0 and y-z%3E0 ,
and then
%28x-y%29%28y-z%29%3E0
xy-xz-y%5E2%2Byz%3E0
xy%2Byz-xz-y%5E2%3E0
2xy%2B2yz-xy-yz-xz-y%5E2%3E0
2xy%2B2yz-%28xy%2Byz%2Bxz%2By%5E2%29%3E0
2%28xy%2Byz%29-%28x%2By%29%28y%2Bz%29%3E0
%28xy%2Byz%29-%28x%2By%29%28y%2Bz%29%2F2%3E0
highlight%28xy%2Byz%3E%28x%2By%29%28y%2Bz%29%2F2%29

How did I come up with that?
Starting from the end and working backwards to the beginning
from one equivalent equation to another:
If and only if xy+%2B+yz+-+%28x%2By%29%28y%2Bz%29+%2F+2+%3E0 , then
+xy+%2B+yz+%3E+%28x%2By%29%28y%2Bz%29+%2F+2 .
The two equations are equivalent;
it is the same statement said "with other words",
so we just have to prove that
xy+%2B+yz+-+%28x%2By%29%28y%2Bz%29+%2F+2+%3E+0 or "in other words" that
%282xy+%2B2yz%29%2F2+-+%28x%2By%29%28y%2Bz%29+%2F+2+%3E+0 or "in other words" that
%282xy+%2B2yz-+%28x%2By%29%28y%2Bz%29%29%2F+2+%3E+0 or "in other words" that
%282xy+%2B2yz-+%28xy%2Bxz%2By%5E2%2Byz%29%29%2F+2+%3E+0 or "in other words" that
%282xy+%2B2yz-+xy-xz-y%5E2-yz%29%2F+2+%3E+0 or "in other words" that
%28xy+%2Byz-xz-y%5E2%29%2F+2+%3E+0 or "in other words" that
xy+%2Byz-xz-y%5E2%3E+0 or "in other words" that
%28z-y%29%28y-x%29%3E0