1. **State the problem:** Solve the inequality $x^2 - 14x > 0$. 2. **Rewrite the inequality:** Factor the left side: $$x^2 - 14x = x(x - 14)$$ So the inequality becomes: $$x(x - 14) > 0$$ 3. **Analyze the inequality:** The product of two factors is greater than zero when both factors are positive or both are negative. 4. **Case 1: Both factors positive** $$x > 0 \quad \text{and} \quad x - 14 > 0$$ $$x > 0 \quad \text{and} \quad x > 14$$ Combined: $$x > 14$$ 5. **Case 2: Both factors negative** $$x < 0 \quad \text{and} \quad x - 14 < 0$$ $$x < 0 \quad \text{and} \quad x < 14$$ Combined: $$x < 0$$ 6. **Combine solutions:** $$x < 0 \quad \text{or} \quad x > 14$$ 7. **Final answer:** $$\boxed{x < 0 \text{ or } x > 14}$$