Systems of Inequalities and Feasible Regions

Active learning transforms graphing inequalities from a static exercise into a dynamic process where students physically shade regions and test points. When students move between stations and collaborate, they correct their own misconceptions about shading and feasible regions in real time, which is essential for mastering systems of inequalities.

• Year 11
• Mathematics
• 4 activities
• 20 min–45 min

Learning Objectives

1. Graph linear and non-linear inequalities on a Cartesian plane, accurately shading the solution region for each.
2. Analyze the intersection of shaded regions to determine the feasible region that satisfies a system of inequalities.
3. Evaluate the vertices of a feasible region to find the optimal (maximum or minimum) value of a given objective function.
4. Design a system of linear inequalities to model real-world resource constraints, such as time or budget limitations.
5. Explain how the graphical representation of inequalities aids in decision-making for optimization problems.

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: Graphing Inequalities

Prepare four stations with pairs of inequalities: linear-linear, linear-nonlinear, three inequalities, and a programming problem. Groups graph on mini whiteboards, shade feasible regions, and test points. Rotate every 10 minutes, then share one insight per group.

Prepare & details

Explain how the intersection of shaded regions represents the solution to a system of inequalities.

Facilitation Tip: During Station Rotation: Graphing Inequalities, circulate with a red pen to immediately correct shading errors by having students test (0,0) aloud while you watch them mark the graph.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 30 min·Pairs

Pairs Challenge: Design Constraints

Pairs receive a scenario like fencing a yard with limited materials. They write and graph a system of inequalities, identify the feasible region, and find optimal vertices. Switch partners to critique and refine.

Prepare & details

Evaluate the vertices of a feasible region to determine optimal solutions in linear programming.

Facilitation Tip: When students work in pairs on Design Constraints, require each pair to swap their system with another pair and graph the new system to verify the feasible region before presenting.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 35 min·Whole Class

Whole Class: Linear Programming Race

Project a resource problem. Students individually graph constraints, then vote on feasible region boundaries. Discuss vertices as a class and compute optima, racing to verify solutions.

Prepare & details

Design a system of inequalities to represent a given set of resource constraints.

Facilitation Tip: To run the Linear Programming Race, set a visible timer and provide a shared whiteboard for each team to post their vertices and objective function value as soon as they find them.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 20 min·Individual

Individual: Digital Feasible Explorer

Students use Desmos or GeoGebra to input teacher-provided inequalities, adjust sliders for non-linear ones, and screenshot feasible regions with vertices labelled for optimisation.

Prepare & details

Explain how the intersection of shaded regions represents the solution to a system of inequalities.

Facilitation Tip: Use the Digital Feasible Explorer to assign differentiated graphs so students practice both linear and non-linear inequalities at their level before moving to optimization.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Teaching This Topic

Teach graphing inequalities by starting with linear examples, then introduce non-linear boundaries gradually so students see the continuity of the concept. Avoid rushing to optimization; spend most of the time on accurate graphing and testing, as this foundation prevents errors later. Research shows students benefit from multiple representations—algebraic, graphical, and contextual—so connect each inequality to a real scenario before abstracting it to the coordinate plane.

What to Expect

Students will confidently graph linear and non-linear inequalities, identify overlapping feasible regions, and justify their shading choices using test points. They will also translate real constraints into systems of inequalities and explain why vertices matter in optimization problems.

Watch Out for These Misconceptions

Common Misconception

During Station Rotation: Graphing Inequalities, watch for students who stop shading after drawing boundary lines or who shade only the intersection point.

What to Teach Instead

Require students to test the point (0,0) on each inequality and physically shade the correct side before moving to the next station, using a shared whiteboard to compare their shading with peers.

Common Misconception

During Station Rotation: Graphing Inequalities, watch for students who shade the opposite side of the boundary line.

What to Teach Instead

Provide a set of cut-out graphs and inequality cards; students must flip the card to test (0,0) and confirm shading direction before marking the graph with a colored pencil.

Common Misconception

During Pairs Challenge: Design Constraints, watch for students who assume non-linear inequalities cannot form a feasible region.

What to Teach Instead

Give pairs a prompt with one linear and one quadratic constraint, then have them sketch both boundaries and shade overlaps to discover the feasible region together.

Assessment Ideas

Quick Check

After Station Rotation: Graphing Inequalities, give students a pre-printed graph with a feasible region and its vertices. Ask them to write the system of inequalities and identify one vertex coordinate to assess their ability to reverse-engineer the graph.

Exit Ticket

During Pairs Challenge: Design Constraints, ask each pair to submit their system of inequalities and a rough sketch of the feasible region before leaving class to assess their translation of context into mathematical models.

Discussion Prompt

After Linear Programming Race, pose the question: 'Why is it important to check the vertices of a feasible region when optimizing?' Facilitate a class discussion where students explain that the optimal solution for a linear objective function always occurs at one of the vertices.

Extensions & Scaffolding

• Challenge students to create a real-world scenario involving two non-linear constraints (e.g., circular garden and parabolic path) and find the feasible region by hand, then verify with graphing software.
• For students struggling with shading, provide cut-out inequality cards that they physically place on coordinate planes to test regions before drawing.
• Deeper exploration: Have students research and present a case where linear programming was used in industry, explaining how feasible regions and vertices guided decision-making.

Key Vocabulary

Inequality	A mathematical statement comparing two expressions using symbols like <, >, ≤, or ≥, indicating that one expression is not equal to the other.
Feasible Region	The area on a graph where the shaded regions of all inequalities in a system overlap, representing all possible solutions that satisfy every condition.
Vertex	A corner point of a feasible region, formed by the intersection of the boundary lines of two or more inequalities.
Objective Function	A mathematical expression, typically linear, that represents the quantity to be maximized or minimized within a feasible region, such as profit or cost.
Linear Programming	A mathematical method used to find the best possible outcome or solution in a given situation with linear relationships and constraints.