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To determine how many constraint inequalities are present in this problem, we need to identify the constraints related to time, resources, and production.

### Problem Breakdown:
1. **Variables**: Let $ x $ represent the number of loaves of bread, and $ y $ represent the number of cakes.
2. **Constraints**:
   - **Time Constraint**: You have $ 6 \frac{2}{3} $ hours, which is equivalent to $ 6 \times 60 + 40 = 400 $ minutes.
     - Bread takes 5 minutes: $ 5x $.
     - Cake takes 4 minutes: $ 4y $.
     - Time constraint inequality: $ 5x + 4y \leq 400 $.
   - **Dough Constraint**: You have 43.2 pounds of flour dough.
     - Bread uses 0.15 pounds: $ 0.15x $.
     - Cake uses 0.6 pounds: $ 0.6y $.
     - Dough constraint inequality: $ 0.15x + 0.6y \leq 43.2 $.
   - **Non-negativity Constraints**: 
     - You cannot have a negative number of items produced: $ x \geq 0 $ and $ y \geq 0 $.

### Total Number of Constraints:
1. Time constraint: $ 5x + 4y \leq 400 $.
2. Dough constraint: $ 0.15x + 0.6y \leq 43.2 $.
3. Non-negativity for bread: $ x \geq 0 $.
4. Non-negativity for cake: $ y \geq 0 $.

**Total**: 4 constraints.

The correct answer is **4**.

Would you like further details or have any questions?

Here are some related questions:
1. How do you convert hours into minutes in constraint problems?
2. What role do non-negativity constraints play in linear programming?
3. How would the constraints change if the amount of flour dough increased?
4. What happens to the constraints if the baking times for bread and cake are swapped?
5. How would you set up the objective function for maximizing profit in this scenario?

**Tip**: When setting up constraints, always list each resource limitation separately to ensure you cover all restrictions!

How many constraint inequalities do you have for the given problem? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pounds of flower dough and cake takes 0.6 pounds. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.

Solve constraint inequalities for a baking business with bread and cake production. Learn how to set up time and dough constraints for bread and cake, and apply non-negativity conditions in this linear programming example. Suitable for Grades 9-12.

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