Solve for real x equation lgx+lg(9-2x)=1.

We'll impose the constraints of existence of the
logarithms:

x>0

9-2x>0
=> -2x>-9 => x < 9/2

The common
interval of admissible values is (0 ; 9/2).

We'll solve the
equation using the product rule of logarithms:

lg x*(9-2x)
= 1

We'll take
antilogarithm:

x*(9-2x) =
10

We'll remove the
brackets:

9x - 2x^2 = 10

We'll
move all terms to the left:

-2x^2 + 9x - 10 =
0

2x^2 - 9x + 10 = 0

We'll
apply quadratic formula:

x1 = [9+sqrt(81 -
80)]/4

x1 = (9+1)/4

x1 =
5/2

x2 = 8/4

x2 =
2

Since both x values belong to the interval
(0 ; 9/2), we'll accept them as solutions: {2 ;
5/2}.