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.