Feasible region
Identify the feasible region in linear programming: graph all constraint inequalities, shade each valid side, and locate the overlapping zone that contains every possible solution.
Key Concepts
The feasible region is the shaded area on a graph that contains all the possible solutions satisfying the constraints. This is your playground! It’s the area on the graph containing all the valid moves that do not break any rules. Any point inside this zone is a possible, 'legal' solution to your problem.
Example 1: For the constraints x ≥ 0, y ≥ 0, and x + y ≤ 8, the feasible region is a triangle with vertices at (0,0), (8,0), and (0,8).
Example 2: For constraints x ≤ 10, y ≤ 15, and x + y ≥ 20, the feasible region is the area bounded by those three lines.
Common Questions
What is a feasible region in linear programming?
The feasible region is the shaded area on a coordinate plane where every point satisfies all the given constraints simultaneously. Any coordinate pair inside this zone is a valid solution to the optimization problem.
How do you find the vertices of a feasible region?
Graph each constraint inequality as a boundary line, then find where pairs of lines intersect. Each intersection point on the boundary of the overlapping shaded area is a vertex. For x≥0, y≥0, x+y≤5, the vertices are (0,0), (5,0), and (0,5).
How do you know which side of a line to shade for an inequality?
Test a convenient point, typically (0,0), by substituting it into the inequality. If the inequality holds true, shade the side containing (0,0). If it is false, shade the opposite side.