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The problem you're dealing with involves setting up a system of inequalities based on the information provided. Let’s break it down:

### Given:
- **Price of a bag of popcorn** = $7
- **Price of a pretzel** = $5
- **Corey’s total budget** = $85
- **Corey must buy a minimum of 13 items (popcorn + pretzels)**

Let **x** represent the number of bags of popcorn, and **y** represent the number of pretzels purchased.

### Constraints:
1. **Cost constraint**: The total cost of popcorn and pretzels cannot exceed $85. This gives the inequality:
   \[
   7x + 5y \leq 85
   \]

2. **Minimum item constraint**: Corey must buy at least 13 items (popcorn + pretzels), which leads to the inequality:
   \[
   x + y \geq 13
   \]

So, the system of inequalities is:
\[
\begin{cases}
7x + 5y \leq 85 \\
x + y \geq 13
\end{cases}
\]

### One possible solution:
You can graph these inequalities, but one possible solution can be found by solving this system manually or graphically. For example:
- Suppose Corey buys **7 bags of popcorn** (i.e., **x = 7**), then we can substitute into the second inequality $x + y \geq 13$, which gives $7 + y \geq 13$. Therefore, Corey must buy at least $y = 6$ pretzels.

Now, check if this satisfies the cost constraint:
\[
7(7) + 5(6) = 49 + 30 = 79
\]
This is within the $85 budget, so $x = 7$ and $y = 6$ is one possible solution.

Would you like more details on solving or graphing the system? Here are 5 additional questions related to this topic:

1. How can you solve this system graphically?
2. How do we determine if other combinations of popcorn and pretzels work?
3. What happens if Corey has a different budget?
4. Can you create a general method to solve this type of word problem?
5. How would the solution change if the minimum number of items increased?

**Tip:** When solving inequalities, always check the solution by substituting values into both inequalities to ensure they are valid.

Corey and his children went into a movie theater that sells bags of popcorn for $7 each and pretzels for $5 each. Corey has $85 to spend and must buy a minimum of 13 bags of popcorn and pretzels altogether. If x represents the number of bags of popcorn purchased and y represents the number of pretzels purchased, write and solve a system of inequalities graphically and determine one possible solution.

Learn how to solve a system of inequalities involving budgeting for popcorn and pretzels, with a $85 limit and a minimum purchase of 13 items. This problem explores linear inequalities and provides a step-by-step graphical solution. Suitable for Grades 8-10.

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