Log0.5(x^2-x-2)+2>=0

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To solve this inequality, we will first simplify the expression:

log0.5(x^2-x-2) + 2 >= 0

We can rewrite the left side using the properties of logarithms:

log0.5(x^2-x-2) + log0.5(2) >= 0

Using the properties of logarithms, we can combine the logs:

log0.5(2(x^2-x-2)) >= 0

Now, we can rewrite the inequality without using logarithms:

0.5(2(x^2-x-2)) >= 1

Simplify the expression:

x^2 - x - 2 >= 1

Now we have a quadratic inequality. We can solve it by first setting it equal to 0:

x^2 - x - 2 - 1 >= 0

x^2 - x - 3 >= 0

Now, we can factor the quadratic:

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

Solve for x by setting each factor equal to 0:

x-2 = 0 --> x = 2
x+1 = 0 --> x = -1

Now we have the critical points x = -1 and x = 2. We can test the inequality at different intervals:

When x < -1:
Choose x = -2:
(-2 - 2)(-2 + 1) = (-4)(-1) = 4 > 0. True

When -1 < x < 2:
Choose x = 0:
(0 - 2)(0 + 1) = (-2)(1) = -2 < 0. False

When x > 2:
Choose x = 3:
(3 - 2)(3 + 1) = (1)(4) = 4 > 0. True

Therefore, the solution to the inequality is x <= -1 or x >= 2.