The equation given is lg(x+1) - lg 9 = 1 - lg x
I assume the base of the logarithm is 10. Use the property that lg a - lg b = lg a/b and 1 = lg 10
lg(x+1) - lg 9 = 1 - lg x
=> lg(x+1) - lg 9 = lg 10 - lg x
=> lg [(x+1)/9] = lg (10/x)
(x + 1)/9 = 10/x
=> x^2 + x = 90
=> x^2 + 10x - 9x -90 = 0
=> x(x + 10) - 9(x + 10) = 0
=> (x -9)(x + 10) = 0
x = 9 and x = -10
As log of negative numbers is not defined we eliminate x = -10
The solution of the equation is x = 9