解分式方程x+1⼀x+2+x+6⼀x+7=x+5⼀x+6+x+2⼀x+3

解分式方程 (x+1)/(x+2)+(x+6/(x+7)=(x+5)/(x+6)+(x+2)/(x+3)
解：1--1/(x+2)+1--1/(x+7)=1--1/(x+6)+1--1/(x+3)
1/(x+6)--1/(x+7)=1/(x+2)--1/(x+3)
[(x+7)--(x+6)]/[(x+6)(x+7)]=[(x+3)--(x+2)]/[(x+2)(x+3)]
1/[(x+6)(x+7)]=1/[(x+2)(x+3)]
x^2+13x+42=x^2+5x+6
13x--5x=6--42
8x=--36
x=--9/2
经检验：x=--9/2是原分式方程的解。

(x+1)/(x+2)+(x+6)/(x+7)=(x+5)/(x+6)+(x+2)/(x+3)
化为：1-1/(x+2)+1-1/(x+7)=1-1/(x+6)+1-1/(x+3)
即：1/(x+2)-1/(x+3)=1/(x+6)-1/(x+7)
1/[(x+2)(x+3)]=1/[(x+6)(x+7)]
(x+2)(x+3)=(x+6)(x+7)
展开为：x�0�5+5x+6=x�0�5+13x+42
解得：x=-9/2=4.5

x+1/x+2+x+6/x+7=x+5/x+6+x+2/x+3
[(x+2)-1]/(x+2)+[(x+7)-1]/(x+6)=[(x+6)-1]/(x+5)+[(x+3)-1]/(x+3)
1-1/(x+2)+1-1/(x+7)=1-1/(x+6)+1-1/(x+3)
-1/(x+2)-1/(x+7)=-1/(x+6)-1/(x+3)
1/(x+3)-1/(x+2)=1/(x+7)-1/(x+6)
(x+2-x-3)/(x+2)(x+3)=(x+6-x-7)/(x+6)(x+7)
(x+6)(x+7)=(x+2)(x+3)
x²+13x+42=x²+5x+6
8x=-36
x=-9/2
检验：