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To solve the linear programming problem shown in your image, we need to graph the inequalities and then find the feasible region by plotting the boundary lines and shading the appropriate areas. Let's break it down step by step:

### System of inequalities:
1. \( x + 4y \leq 24 \)
2. \( 5x + 3y \leq 35 \)
3. \( x \geq 0 \) (this ensures the region lies to the right of the y-axis)
4. \( y \geq 0 \) (this ensures the region lies above the x-axis)

### Step 1: Graph each inequality's boundary
We first graph each inequality as an equality, finding the intercepts for each.

- For \( x + 4y = 24 \):
  - \( x \)-intercept: set \( y = 0 \), so \( x = 24 \) (point is \( (24, 0) \)).
  - \( y \)-intercept: set \( x = 0 \), so \( 4y = 24 \), which gives \( y = 6 \) (point is \( (0, 6) \)).
  - Plot and connect these points to get the line.

- For \( 5x + 3y = 35 \):
  - \( x \)-intercept: set \( y = 0 \), so \( 5x = 35 \), which gives \( x = 7 \) (point is \( (7, 0) \)).
  - \( y \)-intercept: set \( x = 0 \), so \( 3y = 35 \), which gives \( y \approx 11.67 \) (point is \( (0, 11.67) \)).
  - Plot and connect these points to get the second line.

### Step 2: Shade the region
- For \( x + 4y \leq 24 \), shade below the line.
- For \( 5x + 3y \leq 35 \), shade below this line.
- The constraints \( x \geq 0 \) and \( y \geq 0 \) mean we are only considering the first quadrant (the area where both \( x \) and \( y \) are positive).

### Step 3: Find the corner points
The feasible region will be a polygon formed by the intersection of the lines and the axes. We find the corner points where the boundary lines intersect, and where they intersect the axes.

- Intersection of \( x + 4y = 24 \) and \( 5x + 3y = 35 \): Solve this system of equations to find the point of intersection. We can use substitution or elimination methods to find this.

- The other corner points will be on the axes.

Once the polygon is formed by these points, we can check the region it encloses. Would you like help calculating the intersection point or additional clarification on graphing the inequalities?

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Here are five questions to deepen understanding:
1. How do we solve the system of equations to find the intersection of the two boundary lines?
2. How can you check if a point lies in the feasible region?
3. What are the properties of the feasible region in linear programming problems?
4. Why are \( x \geq 0 \) and \( y \geq 0 \) essential in defining the feasible region?
5. How does the graphical method help in solving linear programming problems?

**Tip:** Always remember to shade the region that satisfies all inequalities. This is key to finding the feasible region!

Graph the feasible region for the following system of inequalities by drawing a polygon around the feasible region. Click to set the corner points.

Learn how to graph the feasible region for a system of linear inequalities using boundary lines and shading. The problem involves inequalities like x + 4y ≤ 24 and 5x + 3y ≤ 35, finding intercepts, and solving for corner points in a linear programming context.

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