Shaded Region

Definition

The shaded region refers to the area on a graph that represents the solution set for a linear inequality or a system of linear inequalities. It is a visual representation of the values that satisfy the given inequality or set of inequalities.

5 Must Know Facts For Your Next Test

1. The shaded region on a graph represents the set of all points that satisfy the given linear inequality or system of linear inequalities.
2. The shape of the shaded region is determined by the type of inequality (strict or non-strict) and the number of inequalities in the system.
3. For a single linear inequality in two variables, the shaded region is a half-plane.
4. For a system of linear inequalities in two variables, the shaded region is the intersection of the individual half-planes.
5. The shaded region can be used to determine the feasible solutions to optimization problems, such as finding the maximum or minimum value of a linear function subject to a set of constraints.

Review Questions

• Explain how the shaded region on a graph represents the solution set for a single linear inequality in two variables.
  • The shaded region on a graph represents the solution set for a single linear inequality in two variables. The inequality divides the coordinate plane into two half-planes, and the shaded region is the half-plane that contains all the points that satisfy the inequality. The boundary of the shaded region is the line that represents the inequality, and the points inside the shaded region are the solutions to the inequality.
• Describe how the shaded region changes when graphing a system of linear inequalities in two variables.
  • When graphing a system of linear inequalities in two variables, the shaded region on the graph represents the intersection of the individual half-planes corresponding to each inequality. The shaded region is the area where all the inequalities are satisfied simultaneously. The shape of the shaded region can be more complex, such as a polygon or an irregular shape, depending on the number and orientation of the inequalities in the system.
• Discuss the importance of the shaded region in the context of optimization problems involving linear inequalities.
  • The shaded region on a graph is crucial in the context of optimization problems involving linear inequalities. The shaded region represents the feasible solutions, which are the set of values for the variables that satisfy all the constraints (linear inequalities). By identifying the shaded region, you can then determine the maximum or minimum value of a linear objective function within the feasible region, which is the optimal solution to the optimization problem. The shaded region provides a visual representation of the constraints and helps in the decision-making process for these types of problems.

Related terms

Linear Inequality: A linear inequality is a mathematical expression that represents a relationship between two or more variables, where the variables are connected by an inequality symbol (such as <, >, ≤, or ≥).

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

Half-Plane: A half-plane is the region on a graph that is bounded by a line and includes all the points on one side of the line.