1. **State the problem:** Solve the inequality $$x^2 - 10x > -24$$ and graph the solution on the number line. 2. **Rewrite the inequality:** Move all terms to one side to set the inequality to zero: $$x^2 - 10x + 24 > 0$$ 3. **Factor the quadratic:** Find two numbers that multiply to 24 and add to -10. These are -6 and -4. $$x^2 - 10x + 24 = (x - 6)(x - 4)$$ 4. **Set the inequality:** $$(x - 6)(x - 4) > 0$$ 5. **Analyze the sign of the product:** The product of two factors is greater than zero when both factors are positive or both are negative. - Both positive: $$x - 6 > 0 \Rightarrow x > 6$$ and $$x - 4 > 0 \Rightarrow x > 4$$ so combined $$x > 6$$ - Both negative: $$x - 6 < 0 \Rightarrow x < 6$$ and $$x - 4 < 0 \Rightarrow x < 4$$ so combined $$x < 4$$ 6. **Solution set:** $$x < 4 \quad \text{or} \quad x > 6$$ 7. **Graph on the number line:** Shade all values less than 4 and all values greater than 6, leaving the interval between 4 and 6 unshaded. **Final answer:** $$x < 4 \text{ or } x > 6$$