To solve the inequality log0.5(x^2 - 3x + 2) > -1, we need to first rewrite the inequality in exponential form.

0.5^(log0.5(x^2 - 3x + 2)) > 0.5^(-1)

This simplifies to:

x^2 - 3x + 2 > 1

Now we have a quadratic inequality that we need to solve. We can rewrite this as:

x^2 - 3x + 1 > 0

Next, we can factor the quadratic to find its roots:

(x - 1)(x - 2) > 0

The roots of the quadratic are x = 1 and x = 2. We need to determine when the inequality is greater than 0, which occurs when x is between the two roots. So the solution to the inequality is:

1 < x < 2

Therefore, the solution to the original inequality log0.5(x^2 - 3x + 2) > -1 is 1 < x < 2.