Step 1: Use the line constraint to express $ \beta $ in terms of $ \alpha $.
The point $ (\alpha, \beta) $ lies on $ 3x + y = 0 $. Substituting: $ 3\alpha + \beta = 0 $, so $ \beta = -3\alpha $. The point is $ (\alpha, -3\alpha) $.
Step 2: Evaluate the separating line at $ (3, 4) $.
For line $ L: 3x - 4y - 8 = 0 $, substitute $ (3,4) $: \[ L(3,4) = 3(3) - 4(4) - 8 = 9 - 16 - 8 = -15 < 0 \]
Step 3: Evaluate the separating line at $ (\alpha, -3\alpha) $.
\[ L(\alpha, -3\alpha) = 3\alpha - 4(-3\alpha) - 8 = 3\alpha + 12\alpha - 8 = 15\alpha - 8 \]
Step 4: Apply the opposite sides condition.
Two points lie on opposite sides of a line if and only if substituting them into the line equation gives values of opposite sign. So we need: \[ L(3,4) \cdot L(\alpha, -3\alpha) < 0 \] \[ (-15)(15\alpha - 8) < 0 \]
Step 5: Solve the inequality.
Dividing both sides by $ -15 $ (negative, so inequality flips): \[ 15\alpha - 8 > 0 \]
Step 6: State the answer.
\[ \boxed{15\alpha - 8 > 0} \]