1. **Stating the problem:** Solve the system of inequalities: $$x^2 - 2x \leq 0$$ $$x^2 - 1 > 0$$ 2. **Solve the first inequality:** $$x^2 - 2x \leq 0$$ Factor the left side: $$x(x - 2) \leq 0$$ 3. **Analyze the sign of the product:** The product $x(x-2)$ is less than or equal to zero when $x$ is between the roots 0 and 2, including the endpoints: $$0 \leq x \leq 2$$ 4. **Solve the second inequality:** $$x^2 - 1 > 0$$ Factor the left side: $$(x - 1)(x + 1) > 0$$ 5. **Analyze the sign of the product:** The product $(x-1)(x+1)$ is greater than zero when $x < -1$ or $x > 1$. 6. **Combine the two inequalities:** From the first inequality: $0 \leq x \leq 2$ From the second inequality: $x < -1$ or $x > 1$ The intersection is where both conditions hold: $$1 < x \leq 2$$ 7. **Interpret the third expression:** The expression $3 - x$ is given but not an inequality or equation to solve, so it does not affect the solution set. **Final answer:** $$\boxed{(1, 2]}$$