\(\left\{{}\begin{matrix}\dfrac{x}{x+y}>\dfrac{x}{x+y+z}\\\dfrac{y}{y+z}>\dfrac{y}{x+y+z}\\\dfrac{z}{x+z}>\dfrac{z}{x+y+z}\end{matrix}\right.\)
\(\Rightarrow\dfrac{x}{x+y}+\dfrac{y}{y+z}+\dfrac{z}{z+x}>\dfrac{x}{x+y+z}+\dfrac{y}{x+y+z}+\dfrac{z}{x+y+z}\)
\(\Rightarrow\dfrac{x}{x+y}+\dfrac{y}{y+z}+\dfrac{z}{z+x}>1\)
\(\left\{{}\begin{matrix}\dfrac{x}{x+y}< \dfrac{x+z}{x+y+z}\\\dfrac{y}{y+z}< \dfrac{y+x}{x+y+z}\\\dfrac{z}{x+z}< \dfrac{z+y}{x+y+z}\end{matrix}\right.\)
\(\Rightarrow\dfrac{x}{x+y}+\dfrac{y}{y+z}+\dfrac{z}{z+x}< \dfrac{x+z}{x+y+z}+\dfrac{y+x}{x+y+z}+\dfrac{z+y}{x+y+z}\)
\(\Rightarrow\dfrac{x}{x+y}+\dfrac{y}{y+z}+\dfrac{z}{z+x}< \dfrac{x+z+y+x+z+y}{x+y+z}\)
\(\Rightarrow\dfrac{x}{x+y}+\dfrac{y}{y+z}+\dfrac{z}{z+x}< \dfrac{2\left(x+y+z\right)}{x+y+z}=2\)
\(\Rightarrow1< \dfrac{x}{x+y}+\dfrac{y}{y+z}+\dfrac{z}{z+x}< 2\)
