System of inequalities

Definition

A system of inequalities is a collection of two or more inequalities that involve the same variables. It is used to describe a range of possible solutions that satisfy all the inequalities simultaneously, which can be represented graphically in a coordinate plane. The intersections and boundaries created by these inequalities are crucial for determining feasible regions, especially in real-world applications like optimization problems.

5 Must Know Facts For Your Next Test

1. Each inequality in a system can define a half-plane on a graph, and the solution to the system is where these half-planes overlap.
2. When graphing a system of inequalities, solid lines are used for '≤' or '≥' inequalities, while dashed lines are used for '<' or '>' inequalities to indicate whether points on the line are included in the solution set.
3. Systems of inequalities can model real-world scenarios, such as budgeting constraints, resource allocation, or any situation where multiple conditions must be satisfied simultaneously.
4. The solution to a system of inequalities is often represented as a shaded region on a graph, which contains all the possible solutions that meet the criteria established by the inequalities.
5. In optimization problems, systems of inequalities can help identify the maximum or minimum values of a function within the feasible region defined by the inequalities.

Review Questions

• How do you determine the feasible region for a system of inequalities on a graph?
  • To determine the feasible region for a system of inequalities, first graph each inequality individually on the same coordinate plane. Use solid lines for '≤' or '≥' and dashed lines for '<' or '>'. Shade the appropriate half-plane for each inequality. The overlapping shaded area represents the feasible region, where all conditions are satisfied simultaneously.
• What role do boundary lines play in understanding a system of inequalities?
  • Boundary lines are essential in visualizing systems of inequalities because they represent the points where the expressions are equal. These lines divide the graph into different regions, helping identify which areas satisfy each inequality. The nature of these boundary lines—whether solid or dashed—also indicates whether points on the line are part of the solution set, impacting how we interpret the overall solution.
• Evaluate how systems of inequalities can be applied in real-life scenarios like resource management and provide an example.
  • Systems of inequalities are valuable in resource management because they allow for modeling multiple constraints that must be adhered to simultaneously. For instance, consider a factory that produces two products with limited resources: labor hours and raw materials. By setting up inequalities based on available hours and materials needed per product, one can create a system of inequalities. This can help determine how many units of each product can be produced without exceeding available resources while optimizing profit within those constraints.

Related terms

Inequality: A mathematical statement that shows the relationship between two expressions that are not equal, often using symbols such as '>', '<', '≥', or '≤'.

Feasible Region: The set of all possible points that satisfy all the constraints represented by the inequalities in a system, usually illustrated as a shaded area on a graph.

Boundary Line: A line that represents the equality part of an inequality; it divides the coordinate plane into different regions, with one side representing solutions to the inequality.