1. **State the problem:** Use the distributive property of multiplication over subtraction to simplify expressions like $$4 \frac{1}{2} \left( \frac{2}{3} - 1 \frac{3}{5} \right)$$ and verify the property. 2. **Recall the distributive property formula:** $$a(b - c) = ab - ac$$ This means multiplying a number by a difference is the same as multiplying the number by each term and then subtracting. 3. **Convert mixed numbers to improper fractions:** $$4 \frac{1}{2} = \frac{9}{2}, \quad 1 \frac{3}{5} = \frac{8}{5}$$ 4. **Apply the property to the first expression:** $$4 \frac{1}{2} \left( \frac{2}{3} - 1 \frac{3}{5} \right) = \frac{9}{2} \left( \frac{2}{3} - \frac{8}{5} \right) = \frac{9}{2} \cdot \frac{2}{3} - \frac{9}{2} \cdot \frac{8}{5}$$ 5. **Calculate each product:** $$\frac{9}{2} \cdot \frac{2}{3} = \frac{\cancel{9}^3}{\cancel{2}^1} \cdot \frac{\cancel{2}^1}{\cancel{3}^1} = \frac{3}{1} = 3$$ $$\frac{9}{2} \cdot \frac{8}{5} = \frac{9 \cdot 8}{2 \cdot 5} = \frac{72}{10} = \frac{36}{5} = 7 \frac{1}{5}$$ 6. **Subtract the products:** $$3 - 7 \frac{1}{5} = -4 \frac{1}{5}$$ 7. **Verify by direct subtraction inside parentheses first:** $$\frac{2}{3} - \frac{8}{5} = \frac{10}{15} - \frac{24}{15} = -\frac{14}{15}$$ 8. **Multiply by $$\frac{9}{2}$$:** $$\frac{9}{2} \cdot -\frac{14}{15} = -\frac{126}{30} = -\frac{21}{5} = -4 \frac{1}{5}$$ 9. **Conclusion:** Both methods give the same result, confirming the distributive property. **Final answer:** $$4 \frac{1}{2} \left( \frac{2}{3} - 1 \frac{3}{5} \right) = -4 \frac{1}{5}$$