(x+1)/(x+6)+1/x=(2x+1)/(x+6) Solve the equation by using the LCD. Check for extraneous solutions.

(x+1)/(x+6)+1/x=(2x+1)/(x+6)
LCD is x(x+6)
Multiply all the terms of the equation by LCD and simplify,
x(x+6)((x+1)/(x+6))+x(x+6)(1/x)=x(x+6)((2x+1)/(x+6))
x(x+1)+(x+6)=x(2x+1)
x^2+x+x+6=2x^2+x
x^2+2x+6=2x^2+x
x^2+2x+6-2x^2-x=0
-x^2+x+6=0
Factorize the above equation,
-1(x^2-x-6)=0
(x^2+2x-3x-6)=0
(x(x+2)-3(x+2))=0
(x+2)(x-3)=0
use the zero product property,
x+2=0  or x-3=0
x=-2   or x=3
Let's check the solutions by plugging them in the original equation,
For x=-2,
(-2+1)/(-2+6)+1/(-2)=((2(-2)+1))/(-2+6)
(-1)/4-1/2=(-3)/4
-3/4=-3/4
It's true.
For x=3,
(3+1)/(3+6)+1/3=(2(3)+1)/(3+6)
4/9+1/3=7/9
7/9=7/9
It's true.
So, the solutions of the equation are 3 and -2.