A system of linear inequalities is a set of two or more linear inequalities with the same variables.
The solution to a system of linear inequalities is the set of all ordered pairs $(x, y)$ that satisfy all the inequalities in the system.
NoteThe solution to a system of linear inequalities is a regionin the coordinate plane, not just a single point or line.
Graphing a System of Linear Inequalities
To graph a system of linear inequalities:
- Graph each inequality in the system on the same coordinate plane.
- Shade the region that satisfies each inequality.
- The intersection of the shaded regions is the solution to the system.
How does the concept of intersectionin systems of inequalities relate to the idea of consensusor agreementin other areas of knowledge?
Applications of Systems of Linear Inequalities
Systems of linear inequalities are used to model real-world situations where multiple constraints must be satisfied simultaneously.
Self reviewCreate your own real-world scenario that can be modeled using a system of linear inequalities. Write the inequalities and graph the solution region.
NoteThe solution to a system of linear inequalities is often called the feasible region. In optimization problems, the goal is to find the optimal solutionwithin this region.
Theory of KnowledgeHow do systems of inequalities help us make decisions in situations with multiple constraints? Can you think of examples in other areas of knowledge where similar reasoning is used?
Solving Systems of Linear Inequalities with Technology
Graphing calculators and computer software can be used to graph systems of linear inequalities and find the solution region.
NoteDifferent calculators may have different methods for entering inequalities. Refer to the user manual or online resources for specific instructions.
Self reviewTry graphing a system of linear inequalities using a graphing calculator or online tool. Compare the solution region with your hand-drawn graph.
NoteLinear programming is a powerful tool used in fields such as economics, engineering, and logistics. It will be covered in more detail in later sections.
Theory of KnowledgeHow do mathematical models like systems of inequalities help us make decisions in complex situations? What are the limitations of these models?
TipPractice graphing systems of inequalities by hand to build intuition, but also explore technology tools to handle more complex systems.
Note1. How do systems of inequalities help us understand the relationship between multiple constraints? 2. Can you think of real-world situations where systems of inequalities are used to make decisions? 3. What are the challenges of working with systems of inequalities, and how can they be overcome?
