Shaded Region

Definition

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

5 Must Know Facts For Your Next Test

1. The shaded region on a graph of a linear inequality represents the set of all points that satisfy the inequality.
2. The boundary of the shaded region is the line that represents the equality part of the inequality (e.g., $2x + 3y = 12$).
3. The shaded region is determined by the direction of the inequality symbol (e.g., \geq or \leq).
4. For a system of linear inequalities, the shaded region represents the intersection of the individual half-planes.
5. The shaded region can be used to solve optimization problems by identifying the feasible region and finding the optimal solution.

Review Questions

• Explain how the shaded region on a graph of a linear inequality represents the solutions to the inequality.
  • The shaded region on a graph of a linear inequality represents the set of all points that satisfy the inequality. The boundary of the shaded region is the line that represents the equality part of the inequality (e.g., $2x + 3y = 12$). The direction of the inequality symbol (e.g., \geq or \leq) determines which side of the boundary line is included in the shaded region. All points within the shaded region are solutions to the inequality, while points outside the shaded region do not satisfy the inequality.
• Describe the process of finding the shaded region for a system of linear inequalities.
  • To find the shaded region for a system of linear inequalities, you need to graph each individual inequality and identify the half-planes created by the boundary lines. The shaded region represents the intersection of all the individual half-planes. This means that the shaded region is the area where all the inequalities are satisfied simultaneously. The boundary of the shaded region is determined by the equality parts of the individual inequalities, and the direction of the inequality symbols determines which side of the boundary lines is included in the shaded region.
• Explain how the shaded region can be used to solve optimization problems involving linear inequalities.
  • The shaded region on a graph of linear inequalities represents the feasible region, which is the set of all points that satisfy the given constraints. To solve an optimization problem, such as finding the maximum or minimum value of a linear function subject to a system of linear inequalities, you can first identify the shaded region on the graph. The optimal solution will be a point within the shaded region that maximizes or minimizes the objective function. By analyzing the shaded region, you can determine the corner points of the feasible region and evaluate the objective function at those points to find the optimal solution.

Related terms

Linear Inequality: A linear inequality is an inequality that can be written in the form $ax + by \geq c$, where $a$, $b$, and $c$ are real numbers.

Half-plane: A half-plane is one of the two regions created by a line on a coordinate plane. The shaded region of a linear inequality is the intersection of one or more half-planes.

Feasible Region: The feasible region is the set of all points that satisfy a system of linear inequalities. The shaded region on the graph represents the feasible region.