Solve the system graphically and indicate whether the solution region is bounded or unbounded. Find the coordinates of each corner point.
nothingxplusnothingyless than or equals3
nothingxplus2yless than or equals4
xgreater than or equals0
ygreater than or equals0
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Part 1
Use the graphing tool to graph the system of inequalities. Graph the region that represents the correct solution only once.



To solve the system of inequalities graphically, we first rewrite the system in a form suitable for graphing. Here are the inequalities provided:

1. \( 0x + 0y \leq 3 \) — This inequality doesn't affect the region since it is trivially satisfied.
2. \( 0x + 2y \leq 4 \), which simplifies to \( y \leq 2 \).
3. \( x \geq 0 \) — This inequality restricts the solution to the region where \( x \) is non-negative (right side of the y-axis).
4. \( y \geq 0 \) — This restricts the solution to the region where \( y \) is non-negative (above the x-axis).

### Steps to Graph the System of Inequalities:

1. **Inequality 1: \( y \leq 2 \)**  
   - This is a horizontal line at \( y = 2 \). The region below or on this line (where \( y \) is less than or equal to 2) is part of the solution.

2. **Inequality 2: \( x \geq 0 \)**  
   - This restricts the solution to the region where \( x \geq 0 \), which is the right half-plane (to the right of the y-axis).

3. **Inequality 3: \( y \geq 0 \)**  
   - This restricts the solution to the region where \( y \geq 0 \), which is the upper half-plane (above the x-axis).

### Graphical Interpretation:
The solution region will be a triangular area bounded by the following:
- The horizontal line \( y = 2 \),
- The y-axis \( x = 0 \),
- The x-axis \( y = 0 \).

### Corner Points:
The corner points of the solution region are:
1. \( (0, 0) \) — where the x-axis and y-axis meet.
2. \( (2, 0) \) — where the line \( y = 2 \) intersects the x-axis.
3. \( (0, 2) \) — where the line \( y = 2 \) intersects the y-axis.

### Bounded or Unbounded:
The region is **bounded** because it is enclosed on all sides by the x-axis, the y-axis, and the line \( y = 2 \).

### Graphing Summary:
- The region of interest is a triangle with vertices at \( (0, 0) \), \( (2, 0) \), and \( (0, 2) \).
- This region is bounded.

Let me know if you'd like a detailed graphical representation.

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Do you want any more details or have questions?  
Here are 5 related questions to expand on the topic:

1. How do we identify if a region is bounded or unbounded?
2. How do corner points help in solving systems of inequalities graphically?
3. What would change if the inequality were \( y \geq 2 \) instead of \( y \leq 2 \)?
4. Can the graphical method be applied to non-linear inequalities?
5. How do we find the feasible region in a system with more than two variables?

**Tip:** Always label the axes and the inequality lines when graphing, as it helps in visualizing the solution region easily.

Solve the system graphically and indicate whether the solution region is bounded or unbounded. Find the coordinates of each corner point. nothingxplusnothingyless than or equals3 nothingxplus2yless than or equals4 xgreater than or equals0 ygreater than or equals0

Solve the system of linear inequalities graphically, including x ≥ 0, y ≥ 0, and y ≤ 2. Learn how to find the corner points (0, 0), (2, 0), and (0, 2) and determine if the solution region is bounded. Suitable for students in Grades 9-10.

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