Use linear programming and graph four inequalities

Here is the graph representing the four inequalities:

1. \( x + 2y \leq 10 \) (blue line with shading below it),
2. \( x - y \geq 3 \) (green line with shading above it),
3. \( x \geq 0 \) (shading in the right half-plane),
4. \( y \geq 0 \) (shading in the upper half-plane).

The shaded region where all the constraints overlap is the feasible region, which represents all possible solutions to the system of inequalities.

Would you like more details on how to interpret the feasible region, or do you have any questions about the inequalities? 

Here are five related questions to deepen your understanding:
1. What is the significance of the feasible region in linear programming?
2. How can we determine the optimal solution within the feasible region?
3. What would happen if there were no feasible region?
4. How do we interpret unbounded feasible regions in linear programming?
5. How would the graph change if we modified one of the inequalities?

**Tip**: In linear programming, optimal solutions always occur at the vertices (corner points) of the feasible region.

Explore how to solve and graph four linear inequalities using linear programming. This problem covers graphing techniques, determining the feasible region, and understanding key theorems like the Fundamental Theorem of Linear Programming. Ideal for students in Grades 9-12.

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