What is x for lg(x)+lg(x+1)=(1+lg9) to exist?

We'll impose the constraints of existence of
logarithms:

x > 0

x + 1
> 0

x > -1

The
interval of admissible values for x is (o ;
+infinite).

We'll re-write the equation and we'll put lg 10
instead of value of 1:

lg(x+1) + lgx = 1 + lg9 is changing
into 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
property:

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 valid solution, x =
9.