(3x+1)/(x+1)=k-2

3x+1=(x+1)(k-2)

3x+1=xk-2x+k-2

5x-xk=k-3

x(5-k)=k-3

x=(k-3)/(5-k)

(k-3)/(5-k)<0

(k-3)(5-k)<0

-(k-3)(k-5)<0

k∈(-∞,3)u(5,∞)