Solve the equation `x((3-x)/(x+1))+(x+(3-x)/(x+1))=2`

Explanation

Hence `x+1!=0`
and let `x((3-x)/(x+1))=u` and `x+(3-x)/(x+1)=v`
`:.uv=2`……….i
and `u+v=x((3-x)/(x+1))+x+((3-x)/(x+1))`
`=(x+1)((3-x)/(x+1))+x=3-x+x=3`
`:.u+v=3` and `uv=2`
Then `u=2,v=1` or `u=1,v=2`
Givnen equation is equivalent to the collection
`:.{(x((3-x)/(x+1))=2),(x+(3-x)/(x+1)=1):}` or `{(x((3-x)/(x+1)))=1),(x+(3-x)/(x+1)=2):}`
`implies{(x^(2)-x+2=0),(x^(2)-x+2=0):}` or `{(x^(2)-2x+1=0),(x^(2)-2x+1=0):}`
`implies{(x^(2)-x+2=0),(x^(2)-2x+1=0):}implies{((x-1/2)^(2)+7/4!=0),((x-1)^(2)=0):}`
`:.(x-1)^(2)=0`
`impliesx=1` is a unique solution of the original equation.