To solve the equation (X-3)/(X-2) + (X-2)/(X-3) = 2 1/2, we need to first find a common denominator for the fractions. The common denominator in this case is (X-2)(X-3).

Rewriting the equation with the common denominator:

[(X-3)^2 + (X-2)^2] / [(X-2)(X-3)] = 5/2

Expanding the numerators:

[X^2 - 6X + 9 + X^2 - 4X + 4] / [(X-2)(X-3)] = 5/2

Simplifying the numerators:

[2X^2 - 10X + 13] / [(X-2)(X-3)] = 5/2

Multiplying both sides by 2:

2(2X^2 - 10X + 13) = 5(X^2 - 5X + 6)

Expanding both sides:

4X^2 - 20X + 26 = 5X^2 - 25X + 30

Rearranging terms:

X^2 + 5X - 4 = 0

Factoring the quadratic equation:

(X + 4)(X - 1) = 0

Setting each factor to zero:

X + 4 = 0 or X - 1 = 0

Therefore, the solutions are X = -4 and X = 1.