To solve the given equation:

1/(x-1) + 11/(x-11) = 9/(x-9) + 10/(x-10)

First, find a common denominator for all the terms. The common denominator here would be (x-1)(x-11)(x-9)(x-10).

Multiplying through by the common denominator, we get:

(x-11)(x-9)(x-10) + 11(x-9)(x-10) = 9(x-1)(x-10) + 10(x-1)(x-11)

Expanding and simplifying:

(x^2 - 20x + 99) + 11(x^2 - 19x + 90) = 9(x^2 - 11x - 10) + 10(x^2 - 12x + 11)

(x^2 - 20x + 99) + (11x^2 - 209x + 990) = (9x^2 - 99x - 90) + (10x^2 - 120x + 110)

Combining like terms:

12x^2 - 229x + 1089 = 19x^2 - 219x + 20

Rearranging the terms:

7x^2 - 10x - 1069 = 0

Now you can solve for x using the quadratic formula or by factoring the equation.