方程化为:1/(x+1)+2k/(x-4)=(3k+1)/[(x-4)(x+1)]
两边同时乘以(x-4)(x+1)得:
x-4+2k(x+1)=(3k+1)
x(2k+1)=k+5
若2k+1=0, 即k=-1/2时,方程无解,原方程也无解
若2k+1≠0, 则x=(k+5)/(2k+1), 若原方程无解,则此根为增根,有2种情况:
为增根-1, 则有-1=(k+5)/(2k+1), 解得: k=-2
为增根4,则有4=(k+5)/(2k+1),解得:k=1/7
综合得:k=-1/2 或 -2 或 1/7