The roots of $x/(x+1)+(x+1)/x=5/2$ are:

Put $t=x/(x+1)$, so the equation becomes $t+1/t=5/2$, i.e. $2t^2-5t+2=0$ and $t=2$ or $t=1/2$. From $x/(x+1)=2$ we get $x=2x+2$, so $x=-2$; from $x/(x+1)=1/2$ we get $2x=x+1$, so $x=1$. Check: at $x=1$, 1/2 + 2/1 = 5/2; at $x=-2,2+1/2=5/2$.