1. **Problem Statement:** Solve the inequality $$x^2 - 9x + 14 \leq 0$$ and represent the solution on a number line and graph. 2. **Formula and Rules:** This is a quadratic inequality. To solve it, first find the roots of the quadratic equation $$x^2 - 9x + 14 = 0$$ by factoring or using the quadratic formula. The solution to the inequality depends on the sign of the quadratic expression between and outside the roots. 3. **Find the roots:** Factor the quadratic: $$x^2 - 9x + 14 = (x - 2)(x - 7) = 0$$ So, the roots are $$x = 2$$ and $$x = 7$$. 4. **Analyze intervals:** The parabola opens upwards (coefficient of $$x^2$$ is positive), so the quadratic expression is: - Positive when $$x < 2$$ or $$x > 7$$ - Negative or zero when $$2 \leq x \leq 7$$ 5. **Solution to inequality:** Since we want $$x^2 - 9x + 14 \leq 0$$, the solution is: $$2 \leq x \leq 7$$ 6. **Number line representation:** The solution interval is from 2 to 7, including both endpoints. 7. **Graph description:** The parabola touches the x-axis at $$x=2$$ and $$x=7$$ and lies below or on the x-axis between these points, consistent with the inequality. **Final answer:** $$\boxed{2 \leq x \leq 7}$$