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Let's break down the problem step by step.

### Problem Summary:
- Ham costs **$4.59 per pound**.
- There is a **limit of 0.78 pounds** per customer.
- The variable **x** represents the pounds of ham, and **y** represents the total cost of buying **x** pounds.

We are asked to determine which compound inequality represents the **domain** (possible values of **x**) and the **range** (possible values of **y**).

### Step 1: Determine the Domain of x
The domain represents the possible values of **x** (pounds of ham), which is the number of pounds a customer can purchase.

- Since there is a limit of **0.78 pounds** per customer, **x** must lie between **0** (no ham bought) and **0.78** pounds (maximum ham bought).
  
Thus, the domain is:
\[ 0 \leq x \leq 0.78 \]

### Step 2: Determine the Range of y
The range represents the possible values of **y**, which is the total cost of buying **x** pounds of ham.

- The cost per pound is **$4.59**. The minimum cost (when **x = 0**) is **$0**, and the maximum cost (when **x = 0.78**) can be calculated as:

\[
y = 4.59 \times 0.78 = 3.5802 \approx 3.58
\]

Thus, the range is:
\[ 0 \leq y \leq 3.58 \]

### Step 3: Compound Inequality
We now need to match these inequalities with the options provided.

The correct compound inequality for the domain and range is:
\[
0 \leq x \leq 0.78 \text{ and } 0 \leq y \leq 3.58
\]

This corresponds to the **first option**.

### Answer: The correct choice is the first option:
\[ 0 \leq x \leq 0.78 \text{ and } 0 \leq y \leq 3.58 \]

Would you like further clarification, or do you have additional questions?

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Here are 5 related questions to consider:
1. How do we define domain and range in real-world applications?
2. What would change if the price per pound were different?
3. How can we graph the relationship between **x** and **y**?
4. What happens to the range if the limit on pounds per customer is removed?
5. Can we use this method for other linear pricing problems?

**Tip**: Always double-check the relationship between quantities and costs when forming inequalities to ensure you're capturing both the domain and the range correctly.

At a local grocery store, the deli counter has ham on sale for $4.59 per pound with a limit of 0.78 pounds per customer. If y represents the total cost for buying x pounds of ham, then which set of compound inequalities below represents the domain and range of this situation?

Learn how to solve a real-world problem involving ham prices and weight limits using compound inequalities. Understand the domain and range for the number of pounds of ham (x) and the total cost (y). This problem is suitable for students in Grades 6-9.

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