1. **Problem statement:** We are given the equation of a stone archway as $$10x^2 + 9y - 90 = 0$$ and need to write an inequality representing the region below the archway. 2. **Rewrite the equation:** Solve for $y$ to express the archway curve explicitly: $$10x^2 + 9y - 90 = 0 \implies 9y = 90 - 10x^2 \implies y = \frac{90 - 10x^2}{9} = 10 - \frac{10}{9}x^2$$ 3. **Understanding the region below the archway:** The archway curve is given by $y = 10 - \frac{10}{9}x^2$. The region below this curve consists of all points $(x,y)$ where $y$ is less than or equal to the curve's $y$-value. 4. **Write the inequality:** $$y \leq 10 - \frac{10}{9}x^2$$ 5. **Rewrite inequality in standard form:** Multiply both sides by 9 to clear the denominator: $$9y \leq 90 - 10x^2$$ 6. **Bring all terms to one side:** $$10x^2 + 9y - 90 \leq 0$$ **Final answer:** The inequality representing the region below the archway is $$10x^2 + 9y - 90 \leq 0$$