By Cauchy-Schwarz's inequality we have:

\(VT=\dfrac{x}{2x+y+z}+\dfrac{y}{2y+x+z}+\dfrac{z}{2z+x+y}\)

\(=\dfrac{x}{\left(x+y\right)+\left(x+z\right)}+\dfrac{y}{\left(y+z\right)+\left(y+x\right)}+\dfrac{z}{\left(z+x\right)+\left(y+z\right)}\)

\(\le\dfrac{1}{4}\left(\dfrac{x}{x+y}+\dfrac{x}{x+z}\right)+\dfrac{1}{4}\left(\dfrac{y}{y+z}+\dfrac{y}{x+y}\right)+\dfrac{1}{4}\left(\dfrac{z}{x+z}+\dfrac{z}{z+y}\right)\)

\(=\dfrac{1}{4}\left(\dfrac{x+y}{x+y}+\dfrac{y+z}{y+z}+\dfrac{z+x}{z+x}\right)=\dfrac{3}{4}\)

When \(x=y=z\)

I never have studied inequality Co-si yet. So can you give me different cases? 

P/s: I am grade 7