Solve for x lg(x+1)-lg9=1-lgx

We'll re-write the equation, moving the terms in x to the
left side and the terms without x, to the right
side.

lg(x+1) - lg9 = 1 -
lgx

lg(x+1) + lgx = 1 +
lg9

We'll re-write 1 as lg
10

lg(x+1) + lgx = lg 10 +
lg9

Since the bases are matching, we'll use the product
rule of logarithms both sides:

lg x(x+1) = lg
90

Since the bases are matching, we'll use one to one
rule:

x(x+1) = 90

We'll remove
the brackets:

x^2 + x - 90 =
0

We'll apply the quadratic
formula:

x1 =
[-1+sqrt(1+360)]/2

x1 =
(-1+19)/2

x1 = 9

x2 =
(-1-19)/2

x2 =
-10

We'll reject the second solution, because
it's negative. We'll keep the solution x = 9.