To solve this equation, we first need to simplify the left side of the equation by factoring the quadratic in the numerator and denominator.

(x^2 - 7x + 3) = (x-6)(x-1)
(x^2 - 36) = (x-6)(x+6)

So, we have:

(x-5.5)((x-6)(x-1))/(x-6)(x+6) = -3(x-5.5)

Now, we can cancel out the (x-6) terms in the numerator and denominator:

(x - 5.5)(x - 1)/(x + 6) = -3(x - 5.5)

Now, we can expand the left side of the equation:

(x^2 - 6x - x + 6)/(x + 6) = -3x + 16.5

(x^2 - 7x + 6)/(x + 6) = -3x + 16.5

((x-1)(x-6))/(x + 6) = -3x + 16.5

Now, we can multiply both sides of the equation by (x + 6) to get rid of the fraction:

(x - 1)(x - 6) = -3x^2 + 16.5x + 6x + 36

Expanding both sides gives us:

x^2 - 7x + 6 = -3x^2 + 22.5x + 36

Rearranging and combining like terms:

4x^2 - 29.5x - 30 = 0

Now, we have a quadratic equation that can be solved by factoring or using the quadratic formula.