Roots. Solve and check

1/4 -1/x+6= 1/x

1/4 -1/x+6= 1/x ?

I have this feeling you meant
1/4 - 1/(x+6) = 1/x

multiply by 4x(x+6) , the LCD

x(x+6) - 4x = 4(x+6)
x^2 + 6x - 4x = 4x + 24
x^2 - 2x - 24 = 0
(x-6)(x+4) = 0

x = 6 or x = -4