Question

解分式方程1/x(x-1) + 1/x(x+1) +1/(x+1)(x+2) +``` +1/(x+9)(x+10)=11/x-1

Answer 1

1/x(x-1) + 1/x(x+1) +1/(x+1)(x+2) +…+1/(x+9)(x+10)
=1/(x-1)-1/x+1/x-1/(x+1)+1/(x+1)-1/(x+2)+…+1/(x+9)-1/(x+10)
=1/(x-1)-1/(x+10)
=11/[(x-1)(x+10)]
原方程为：11/[(x-1)(x+10)]=11/(x-1)、1/(x+10)=1、x=-9。
经检验，x=-9是增根。
所以，原方程无根。

Answer 2

1/x(x-1) + 1/x(x+1) +1/(x+1)(x+2) +``` +1/(x+9)(x+10)=11/(x-1)
1/(x-1)-1/x+1/x-1/(x+1)+1/(x+1)-1/(x+2)+..........+1/(x+9)-1/(x+10)=11/(x-1)
1/(x-1)-1/(x+10)=11/(x-1)
[(x+10)-(x-1)]/(x-1)(x+10)=11/(x-1)
(x+10-x+1)/(x-1)(x+10)=11/(x-1)
11/(x-1)(x+10)=11/(x-1)
x+10=1
x=-9
经检验 x=-9是增根，所以方程无解

Answer 3

X=-9
1/x(x-1) + 1/x(x+1) +1/(x+1)(x+2) +…+1/(x+9)(x+10)
=1/(x-1)-1/x+1/x-1/(x+1)+1/(x+1)-1/(x+2)+…+1/(x+9)-1/(x+10)
=1/(x-1)-1/(x+10)
=11/[(x-1)(x+10)]
原式可写成
11/[(x-1)(x+10)]=11/(x-1)
x=-9 为增根，愿方程无解

Answer 4

1/(x-1)-1/x+1/x-1/(x+1) +1/(x+1)-1/(x+2)+.....+1/(x+9)-1/(x+10)=1/(x-1)-1/(x+10)=11/(x-1)(x+10)