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The problem involves determining the possible dimensions of a rectangular garden based on the system of inequalities, which are visually represented in the graph. The relevant information includes the conditions for width and length, constrained by the amount of fencing Irina can use. Let's break down the problem:

### Given Conditions:
1. **Width Constraint**: \( w \geq 10 \) (the width must be at least 10 feet).
2. **Fence Constraint**: The total perimeter of the garden must not exceed 150 feet. The formula for the perimeter is given by:
   \[
   2l + 2w \leq 150
   \]
   This simplifies to:
   \[
   l + w \leq 75
   \]

### Step-by-Step Approach:
From the graph, we can observe:
- The region shaded blue represents the possible dimensions that satisfy both inequalities \( l + w \leq 75 \) and \( w \geq 10 \).
- The red region represents the area that is below \( w = 10 \), which is not allowed based on the problem statement.

Now, to check the given options:

### Option 1: \( l = 20 \, \text{ft}, w = 5 \, \text{ft} \)
- This option violates \( w \geq 10 \) since the width is less than 10 ft. **Not valid.**

### Option 2: \( l = 20 \, \text{ft}, w = 10 \, \text{ft} \)
- Check if the perimeter condition is met: 
  \[
  l + w = 20 + 10 = 30 \quad \text{(which is less than 75)}
  \]
  This satisfies both conditions. **Valid.**

### Option 3: \( l = 60 \, \text{ft}, w = 20 \, \text{ft} \)
- Check if the perimeter condition is met:
  \[
  l + w = 60 + 20 = 80 \quad \text{(which exceeds 75)}
  \]
  This does not satisfy the perimeter constraint. **Not valid.**

### Option 4: \( l = 55 \, \text{ft}, w = 30 \, \text{ft} \)
- Check if the perimeter condition is met:
  \[
  l + w = 55 + 30 = 85 \quad \text{(which exceeds 75)}
  \]
  This does not satisfy the perimeter constraint. **Not valid.**

### Conclusion:
The correct answer is **\( l = 20 \, \text{ft}, w = 10 \, \text{ft} \)**.

Would you like more details or have any questions?

Here are 5 relative questions to deepen understanding:
1. What would be the maximum possible width if the length is 40 feet?
2. How would the solution change if the available fencing was 200 feet?
3. How can we calculate the area of the garden given its dimensions?
4. What happens if the constraint for the width changes to \( w \geq 15 \)?
5. Can a garden with dimensions \( l = 30 \, \text{ft}, w = 40 \, \text{ft} \) be built with the given fencing?

**Tip**: When solving inequalities involving geometry, it's often helpful to graph the solution space as shown in the problem—this visual representation makes it easier to identify valid solutions.

Irina wants to build a fence around a rectangular vegetable garden so that it has a width of at least 10 feet. She can use a maximum of 150 feet of fencing. The system of inequalities that models the possible lengths, l, and widths, w, of her garden is shown. Which length and width are possible dimensions for the garden?

Explore how to solve a fencing problem involving inequalities and geometric constraints. Learn to find possible dimensions of a garden given maximum perimeter and width constraints. Perfect for students in Grades 7-10.

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