https://file.aiquickdraw.com/mathbot/file/17284249750458xdx8ekp.jpg

Which set of constraints below correctly bound the feasible region shown in the graph? 

To determine the correct set of inequalities that bound the feasible region shown in the graph, let's analyze each line and shading in the graph.

### Analysis of the Graph
1. **Green Line (Bottom Boundary):** This line seems to be the x-axis constraint, meaning \( y \geq 0 \).
2. **Blue Line (Left Boundary):** The line has a negative slope and appears to represent a linear inequality. It restricts the region to the left and below this line.
3. **Black Line (Diagonal Boundary):** This line appears to have a positive slope and seems to bound the region on the right side.

### Step-by-Step Solution
#### 1. Checking Inequality Conditions
   - **x-axis Constraint:** As this feasible region is in the upper part of the graph, \( y \geq 0 \) is likely one of the constraints.
   - **y-axis Constraint:** The region is to the right of the y-axis, so we expect \( x \geq 0 \) to also be a constraint.
   - **Diagonal Boundaries:** To determine which inequalities match the blue and black lines, we would test the points or compare with the options.

#### 2. Options Analysis
Let's match each option to see which correctly bounds the feasible region:

   - **Option (a)**: 
     - \( 2x - 3y \leq 12 \)
     - \( y \leq -2x + 4 \)
     - \( y \geq 0 \)
     - \( x \geq 0 \)

   - **Option (b)**: 
     - \( 2x + 3y \leq 12 \)
     - \( y \leq -2x + 4 \)
     - \( y \geq 0 \)
     - \( x \geq 0 \)

   - **Option (c)**: 
     - \( 2x + 3y \leq 12 \)
     - \( y \geq -2x + 4 \)
     - \( y \geq 0 \)
     - \( x \geq 0 \)

   - **Option (d)**:
     - \( 2x - 3y \leq 12 \)
     - \( y \geq -2x + 4 \)
     - \( y \geq 0 \)
     - \( x \geq 0 \)

### Conclusion
Based on the boundaries provided and the shading, **Option (b)** seems to be the correct answer as it matches the constraints on the feasible region's boundaries.

Would you like further details on how each boundary was matched to the inequalities?

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#### Related Questions
1. How do you determine the feasible region in a system of linear inequalities?
2. What methods can be used to solve systems of linear inequalities graphically?
3. How can you verify if a point is within the feasible region of a system of inequalities?
4. Why is it important to check boundary conditions when working with inequalities?
5. How do constraints affect the shape and location of the feasible region in a graph?

#### Tip:
Always start by identifying the basic boundaries, like the axes, when interpreting the feasible region on a graph.

Which set of constraints below correctly bound the feasible region shown in the graph?

Learn how to identify the correct set of constraints for a feasible region in a system of linear inequalities using graph analysis. This problem involves understanding algebraic inequalities, boundary conditions, and graphical representation, suitable for students in grades 9-11.

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