key term - Feasible Region

Definition

The feasible region is the set of all possible solutions that satisfy a system of linear inequalities. It represents the area or space where the constraints defined by the inequalities intersect and overlap, creating a region that contains the viable solutions to the problem.

5 Must Know Facts For Your Next Test

1. The feasible region is the intersection of all the half-planes defined by the linear inequalities in a system.
2. The feasible region is always a convex polygon, which means it has no indentations or holes.
3. The vertices of the feasible region are the points where the boundaries of the linear inequalities intersect.
4. The feasible region represents the set of all possible solutions that satisfy the constraints of the problem.
5. Identifying the feasible region is a crucial step in solving optimization problems using linear programming.

Review Questions

• Explain how the feasible region is determined from a system of linear inequalities.
  • The feasible region is determined by the intersection of all the half-planes defined by the linear inequalities in the system. Each linear inequality represents a constraint, and the feasible region is the area where all the constraints are satisfied simultaneously. The boundaries of the feasible region are the lines or line segments that represent the equality portions of the linear inequalities.
• Describe the properties of the feasible region and how they relate to the characteristics of linear inequalities.
  • The feasible region is always a convex polygon, meaning it has no indentations or holes. This is because linear inequalities define half-planes, and the intersection of these half-planes results in a convex set. The vertices of the feasible region are the points where the boundaries of the linear inequalities intersect. These vertices represent the extreme points of the feasible region, which are important in optimization problems.
• Analyze the significance of the feasible region in the context of solving optimization problems using linear programming.
  • The feasible region is a crucial component in solving optimization problems using linear programming. It represents the set of all possible solutions that satisfy the constraints of the problem. The objective function, which is the expression to be maximized or minimized, is then evaluated within the feasible region to determine the optimal solution. Identifying the feasible region and its characteristics, such as the vertices, is essential for effectively solving linear programming problems and finding the best solution that meets all the given constraints.

Related terms

Linear Inequality:

A linear inequality is a mathematical statement that compares a linear expression to a constant value using an inequality symbol (>, <, ≥, ≤).

System of Linear Inequalities:

A system of linear inequalities is a set of two or more linear inequalities that must be satisfied simultaneously.

Constraint:

A constraint is a restriction or limitation that must be met in order for a solution to be considered valid or feasible.