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