Quadratic Inequality 0C6D5F

1. **State the problem:** Solve the inequality $$x^2 - 11x + 24 \leq 0$$ algebraically. 2. **Identify the type of inequality:** This is a quadratic inequality of the form $$ax^2 + bx + c \leq 0$$ where $$a=1$$, $$b=-11$$, and $$c=24$$. 3. **Find the roots of the quadratic equation:** Solve $$x^2 - 11x + 24 = 0$$. Use the factoring method: $$x^2 - 11x + 24 = (x - 8)(x - 3) = 0$$ So, the roots are $$x=8$$ and $$x=3$$. 4. **Determine the intervals:** The roots divide the number line into three intervals: - $$(-\infty, 3)$$ - $$[3, 8]$$ - $$(8, +\infty)$$ 5. **Test the sign of the quadratic in each interval:** - For $$x < 3$$, pick $$x=0$$: $$0^2 - 11(0) + 24 = 24 > 0$$ - For $$3 < x < 8$$, pick $$x=5$$: $$5^2 - 11(5) + 24 = 25 - 55 + 24 = -6 < 0$$ - For $$x > 8$$, pick $$x=9$$: $$9^2 - 11(9) + 24 = 81 - 99 + 24 = 6 > 0$$ 6. **Interpret the inequality:** We want where $$x^2 - 11x + 24 \leq 0$$, so where the quadratic is less than or equal to zero. From the tests, this is true on the interval $$[3, 8]$$. 7. **Final answer:** $$\boxed{3 \leq x \leq 8}$$