To solve this system of inequalities, we will first find the values of x that satisfy each inequality separately.

1) x^2 - x > 0

Factoring out an x, we get:
x(x-1) > 0

Now, we find the critical points where the expression equals 0:
x = 0 or x = 1

These critical points divide the real number line into three intervals: (-∞, 0), (0, 1), and (1, ∞).

We test a value in each interval:
For x = -1, we get: (-1)(-1-1) = 2 > 0, true for (-∞, 0)
For x = 0.5, we get: (0.5)(0.5-1) = -0.25 < 0, false for (0, 1)
For x = 2, we get: (2)(2-1) = 2 > 0, true for (1, ∞)

Therefore, the solution for x^2 - x > 0 is x ∈ (-∞, 0) U (1, ∞).

2) x^2 - x < 2

Rearranging the inequality, we get:
x^2 - x - 2 < 0

Factorizing the quadratic expression, we get:
(x-2)(x+1) < 0

The critical points where the expression equals 0 are x = -1 and x = 2.

Testing a value in each interval:
For x = -2, we get: (-2-2)(-2+1) = -4(-1) = 4 > 0, false for (-∞, -1)
For x = 0, we get: (-2)(1) = -2 < 0, true for (-1, 2)
For x = 3, we get: (1)(4) = 4 > 0, false for (2, ∞)

Therefore, the solution for x^2 - x < 2 is x ∈ (-1, 2).

So, the solution to the system of inequalities {x^2 - x > 0, x^2 - x < 2} is x ∈ (-1, 0) U (1, 2).