https://file.aiquickdraw.com/mathbot/file/17259907622138ynmzb62.jpg

Let's break down the problem and the system of inequalities based on the given image.

We have two constraints to model the garden:

1. **Length constraint**: The length of the garden must be at least 110 feet.
   - This translates to the inequality: \( y \geq 110 \), where \( y \) represents the length of the garden.

2. **Perimeter constraint**: The total distance around the garden should be no more than 380 feet. The formula for the perimeter of a rectangle is \( 2x + 2y \), where \( x \) is the width, and \( y \) is the length.
   - This gives the inequality: \( 2x + 2y \leq 380 \), or simplified as \( x + y \leq 190 \).

Thus, the system of inequalities is:

\[
\begin{cases}
y \geq 110 \\
x + y \leq 190
\end{cases}
\]

Now, to analyze which of the graphs is correct, let's look at the conditions for the solution:

- The shaded region should represent the area where both inequalities hold true.
- The line \( x + y = 190 \) should appear in the graph as the boundary, and the region below this line should be shaded (since \( x + y \leq 190 \)).
- The line \( y = 110 \) should also be included, and the area above this line should be shaded (since \( y \geq 110 \)).

### Analyzing the graphs:
- **Graph a**: The shaded region lies below \( x + y = 190 \), and the length constraint \( y \geq 110 \) is satisfied. This is likely correct.
- **Graph b**: The shading extends far above \( y = 110 \), which violates the length constraint, making this incorrect.
- **Graph c**: The shaded region does not properly satisfy the \( y \geq 110 \) condition, as the shading is too low. This is incorrect.
- **Graph d**: The shading violates both the perimeter constraint \( x + y \leq 190 \) and the length constraint \( y \geq 110 \), so this is incorrect.

### Conclusion:
- The correct graph is **a** because it accurately represents the solution set for both inequalities.

Would you like further clarification or details on any part of the solution?

### Related questions:
1. How do you graph a system of inequalities on a coordinate plane?
2. Can a system of inequalities have no solution? If so, how would that look graphically?
3. How does changing the length or perimeter affect the shaded region in the graph?
4. What is the importance of testing boundary conditions in inequality graphs?
5. How do you solve systems of inequalities algebraically without graphing?

**Tip:** When graphing inequalities, always remember to test points in and out of the boundary to ensure proper shading of the solution set.

A local citizen wants to fence a rectangular community garden. The length of the garden should be at least 110 ft, and the distance around should be no more than 380 ft. Write a system of inequalities that models the possible dimensions of the garden. Graph the system to show all possible solutions.

Learn how to graph a system of inequalities for a community garden problem. The garden's length must be at least 110 feet, and the perimeter must be no more than 380 feet. This solution involves graphing the inequalities y ≥ 110 and 2x + 2y ≤ 380, showcasing the possible dimensions for the garden. Suitable for students in Grades 8-10.

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