Solving Systems of Linear Inequalities

Active learning deepens understanding here because graphing systems of inequalities demands visual reasoning and collaborative correction. When students work together, they catch shading errors instantly, refine their boundary choices, and internalise how overlapping regions form solutions. Teamwork turns abstract graphs into tangible, shared knowledge.

• Class 11
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Graph the feasible region for a system of two linear inequalities in two variables.
2. Compare the solution sets of individual linear inequalities versus systems of linear inequalities.
3. Evaluate the effectiveness of graphical methods for identifying the intersection of multiple constraints.
4. Design a system of linear inequalities to model a simple real-world scenario with at least two constraints.

Project-Based Learning

⏱ 35 min·Small Groups

Small Groups: Shading Relay

Divide class into groups of four. Each member graphs and shades one inequality on shared graph paper, passes to next for verification. Groups present feasible regions and justify boundaries. Conclude with class vote on most accurate.

Prepare & details

Evaluate the effectiveness of graphical solutions for systems of inequalities.

Facilitation Tip: During Shading Relay, assign each group a unique inequality pair and rotate groups every 5 minutes so students experience varied systems and corrections.

Setup: Standard classroom of 40–50 students; printed task and role cards are recommended over digital display to allow simultaneous group work without device dependency.

Materials: Printed driving question and role cards, Chart paper and markers for group outputs, NCERT textbooks and supplementary board materials as base resources, Local data sources — newspapers, community interviews, government census data, Internal assessment rubric aligned to board project guidelines

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

Project-Based Learning

⏱ 40 min·Pairs

Pairs: Real-World Model Build

Pairs create a system for a scenario like fencing a garden with budget constraints. Graph on coordinate paper, shade feasible region, test points. Swap with another pair to verify and discuss adjustments.

Prepare & details

Differentiate between the solution of a single inequality and a system of inequalities.

Facilitation Tip: In Real-World Model Build, provide physical graph paper and coloured pencils so pairs can physically shade and erase without erasing mistakes.

Setup: Standard classroom of 40–50 students; printed task and role cards are recommended over digital display to allow simultaneous group work without device dependency.

Materials: Printed driving question and role cards, Chart paper and markers for group outputs, NCERT textbooks and supplementary board materials as base resources, Local data sources — newspapers, community interviews, government census data, Internal assessment rubric aligned to board project guidelines

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

Project-Based Learning

⏱ 30 min·Whole Class

Whole Class: Constraint Gallery Walk

Project systems on board, students use sticky notes to mark feasible regions on printed graphs around room. Discuss variations as class walks, vote on challenges. Tally for common patterns.

Prepare & details

Design a system of inequalities to represent a real-world problem with multiple constraints.

Facilitation Tip: For Constraint Gallery Walk, place large graph sheets on walls and have students mark corrections with sticky notes so peers can revisit and revise their work.

Setup: Standard classroom of 40–50 students; printed task and role cards are recommended over digital display to allow simultaneous group work without device dependency.

Materials: Printed driving question and role cards, Chart paper and markers for group outputs, NCERT textbooks and supplementary board materials as base resources, Local data sources — newspapers, community interviews, government census data, Internal assessment rubric aligned to board project guidelines

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

Project-Based Learning

⏱ 25 min·Individual

Individual: Digital Verification

Students use GeoGebra to input teacher-provided systems, shade regions, screenshot feasible areas. Submit with one test point explanation. Review in pairs next class.

Prepare & details

Evaluate the effectiveness of graphical solutions for systems of inequalities.

Facilitation Tip: Use Digital Verification to let students input their graphs into software and compare digital shading with their hand-drawn versions for immediate feedback.

Setup: Standard classroom of 40–50 students; printed task and role cards are recommended over digital display to allow simultaneous group work without device dependency.

Materials: Printed driving question and role cards, Chart paper and markers for group outputs, NCERT textbooks and supplementary board materials as base resources, Local data sources — newspapers, community interviews, government census data, Internal assessment rubric aligned to board project guidelines

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

Teaching This Topic

Start with whole-class demonstrations of single inequalities, then shift to pairs to solve simpler systems before tackling complex ones. Research shows that letting students test points aloud and justify their shading decisions reduces misconceptions faster than lecturing. Avoid rushing to the final graph; instead, emphasise the process of boundary selection and half-plane testing.

What to Expect

By the end of these activities, students should confidently sketch systems, choose solid or dashed lines accurately, and explain why only the overlapping shaded region meets all conditions. They should also articulate whether the feasible region is bounded or unbounded with clear reasoning.

Watch Out for These Misconceptions

Common Misconception

During Shading Relay, watch for students who shade the same side for every inequality without checking the inequality sign or testing a point.

What to Teach Instead

Hand each group a test point card and ask them to test it in both inequalities before shading, then compare results with another group to confirm overlaps.

Common Misconception

During Real-World Model Build, watch for students who assume the feasible region must always form a closed polygon.

What to Teach Instead

Provide examples with open regions and ask pairs to sketch two inequalities where one boundary extends infinitely, then discuss why the region remains valid.

Common Misconception

During Constraint Gallery Walk, watch for students who mark solid lines for all boundaries regardless of the inequality symbol.

What to Teach Instead

Give each pair cut-out boundary strips with different symbols; ask them to place the correct strip on each line and explain why the edge is included or excluded before posting their graphs.

Assessment Ideas

Quick Check

After Shading Relay, provide a graph with a shaded feasible region formed by two inequalities. Ask students to write the two inequalities, justifying their choice of solid or dashed lines and the shaded areas.

Exit Ticket

After Digital Verification, ask students to graph the system y > 2x - 1 and y ≤ -x + 3 on paper, label boundaries, shade the region, and answer whether (1,1) is a solution and why.

Discussion Prompt

During Constraint Gallery Walk, pose the bakery scenario and ask students to represent constraints as inequalities, sketch the feasible region, and explain what the region tells them about daily production limits.

Extensions & Scaffolding

• Challenge early finishers to create a system with an unbounded feasible region and explain why it remains valid despite extending infinitely.
• For struggling students, provide pre-drawn boundary lines and ask them to shade only one inequality at a time before combining.
• Use extra time to invite students to design their own real-world constraint scenario and graph it, then peer-review each other’s feasible regions.

Key Vocabulary

Linear Inequality	An inequality involving two variables where the highest power of each variable is one. Its graph is a half-plane.
System of Linear Inequalities	A collection of two or more linear inequalities that must be satisfied simultaneously.
Feasible Region	The region on a graph where the solution sets of all inequalities in a system overlap. This region represents all possible solutions to the system.
Boundary Line	The line associated with a linear inequality, which divides the coordinate plane into two half-planes. It is solid for '≤' or '≥' and dashed for '<' or '>'.