Solve the equation lg( x + 1 ) - lg 9 = 1 - lg( x ).

Let's recall the logarithmic properties before:

1 = lg10

The quotient law: lgx - lgy = lg(x/y)

x > 0

Now, let's solve the equation:

lg(x + 1) - lg9 = lg[(x + 1)/9]

1 - lgx = lg10 - lgx = lg(10/x)

lg[(x + 1)/9] = lg(10/x)

From one to one  property of logarithmic functions, we'll get:

(x + 1)/9 = 10/x

We'll use the cross multiplying:

x*(x + 1) = 9*10

x^2 + x - 90 = 0

x1 = [-1+ sq root(1 + 4*90)]/2 = (-1 + 19)/2 = 9

x2 = (-1 -19)/2 = -10

From the existence condition of the logarithm, x > 0, so the only accepted solution of the equation is x1 = 9.