1. **State the problem:** We need to find the minimum price for a dozen eggs if the total cost of four dozen eggs plus a half gallon of milk costing 2.97 exceeds 10. 2. **Define variables and write the inequality:** Let $x$ be the price of one dozen eggs. The cost of four dozen eggs is $4x$. The cost of the milk is 2.97. The total cost exceeds 10, so: $$4x + 2.97 > 10$$ 3. **Solve the inequality:** Subtract 2.97 from both sides: $$4x + 2.97 - 2.97 > 10 - 2.97$$ $$4x > 7.03$$ 4. **Divide both sides by 4 to isolate $x$:** $$\cancel{4}x > \cancel{4} \times \frac{7.03}{4}$$ $$x > \frac{7.03}{4}$$ 5. **Calculate the value:** $$x > 1.7575$$ 6. **Interpretation:** The minimum price for a dozen eggs must be greater than 1.7575 to make the total cost exceed 10. **Final answer:** $$\boxed{x > 1.76}$$ (rounded to two decimal places)