To solve the inequality, we need to rewrite it in exponential form first.

2log2(x-2) + log0.5(x-3) > 2
log2(x-2)^2 + log0.5(x-3) > 2
log2(x-2)^2 0.5(x-3) > 2
log((x-2)^2) log((x-3)^0.5) > 2
log((x-2)^2 (x-3)^0.5) > 2
(x-2)^2 (x-3)^0.5 > 10^2
(x-2)^2 * (x-3)^0.5 > 100

Now we need to solve for x by using properties of logarithms:

(x-2)(x-2)(x-3)^0.5 > 100
(x^2 - 4x + 4)*(x-3)^0.5 > 100
x^3 - 7x^2 + 16x - 12 > 100
x^3 - 7x^2 + 16x - 112 > 0

Now, we can find the roots of this polynomial or use a graphing calculator to determine the values of x that satisfy the inequality.