Rose is packing small and large bags of peppers from her garden. A small bag contains 3 peppers and a large bag contains 7 peppers. Rose plans to pack small bags, large bugs or both, according to the specifications below.
• She needs to pack at least 18 peppers total.
She needs to pack at least 3 bags.
Let x represent the number of small bags that Rose will pack and let y represent the number of large bags. Write a system of linear inequalities that represents this situation

To represent this situation, let's break down the conditions given in the problem.

### Variables
- Let \( x \) represent the number of small bags.
- Let \( y \) represent the number of large bags.

### Information Given
1. **Each small bag contains 3 peppers:** This means that \( 3x \) represents the total number of peppers in small bags.
2. **Each large bag contains 7 peppers:** This means that \( 7y \) represents the total number of peppers in large bags.
3. **She needs at least 18 peppers total.**
4. **She needs at least 3 bags in total.**

### Inequalities
1. To satisfy the requirement of at least 18 peppers:
   \[
   3x + 7y \geq 18
   \]

2. To meet the requirement of at least 3 bags in total:
   \[
   x + y \geq 3
   \]

3. Additionally, since the number of bags cannot be negative:
   \[
   x \geq 0
   \]
   \[
   y \geq 0
   \]

### System of Inequalities
The system of linear inequalities that represents this situation is:
\[
\begin{cases}
3x + 7y \geq 18 \\
x + y \geq 3 \\
x \geq 0 \\
y \geq 0
\end{cases}
\]

Would you like a graphical representation of these inequalities, or further clarification on any part?

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#### Related Questions:
1. How would the inequalities change if each small bag held 5 peppers instead of 3?
2. If Rose packs only large bags, what is the minimum number she must pack to meet the pepper requirement?
3. What would the inequalities look like if Rose wanted exactly 18 peppers instead of at least 18?
4. How would adding a third bag type with different pepper counts affect the inequalities?
5. Can you identify values for \( x \) and \( y \) that satisfy all the inequalities simultaneously?

#### Tip:
In systems of inequalities, always check if each inequality has a feasible solution within the context. This will help determine if the conditions are achievable.

Rose is packing small and large bags of peppers from her garden. A small bag contains 3 peppers and a large bag contains 7 peppers. Rose plans to pack small bags, large bags, or both, according to the specifications below. She needs to pack at least 18 peppers total. She needs to pack at least 3 bags. Let x represent the number of small bags that Rose will pack and let y represent the number of large bags. Write a system of linear inequalities that represents this situation.

Learn to create a system of linear inequalities to solve a real-world problem involving packing small and large bags of peppers. This solution uses algebraic concepts and linear inequality properties to model requirements for the number of bags and peppers needed. Suitable for Grades 9-10.

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