Mathematics · Class 11

Active learning ideas

Applications of Linear Inequalities

Take your students beyond simple equations and into the world of real-life decision making. This topic shows them how maths is used to solve complex problems with multiple constraints, like running a business or planning a diet.

• CBSE Learning Outcomes
• NCERT Class 11 Mathematics: Chapter 6 - Linear Inequalities

• 30–50 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Problem-Based Learning
• 45 min · Pairs

The Canteen Diet Plan

Students are given a list of food items from the school canteen with their prices and nutritional values (e.g., calories, protein). They must create a lunch plan that meets certain nutritional requirements (e.g., at least 500 calories, no more than ₹70) by formulating and graphing inequalities.

Analyse a word problem to formulate a system of linear inequalities that represents the given constraints.

Facilitation Tip

Provide a pre-made table of food items to save time and keep the focus on the maths.

What to look for

Give students a short word problem and ask them only to write the system of inequalities, not to solve it. This 'translation check' quickly assesses their understanding of problem formulation.

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

Activity 02

• Problem-Based Learning
• 50 min · Small Groups

Small Business Production Challenge

In small groups, students act as managers of a small business making two products (e.g., kurtas and pyjamas). They are given constraints on labour hours and cloth availability. Their task is to graph the inequalities and identify the feasible region of all possible production combinations.

Explain how a graphical solution to a system of inequalities can help a manufacturer decide on production levels.

Facilitation Tip

Encourage groups to present their feasible region and explain what a point on the boundary represents versus a point inside.

What to look for

A mini-project where students create their own real-world problem, formulate the inequalities, solve it graphically, and write a paragraph explaining what any two points in the feasible region mean in the context of their problem.

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

Activity 03

• Problem-Based Learning
• 30 min · Whole Class

Event Planning Budget

As a whole class, plan a hypothetical farewell party. Brainstorm constraints like total budget, minimum number of snacks per person, and maximum venue capacity. The class then works together to formulate the system of inequalities representing these constraints.

Evaluate the constraints in a given problem to determine if a solution is feasible.

Facilitation Tip

Use a projector or smartboard so everyone can contribute to building the graphical solution together.

What to look for

Provide a checklist for students to review their own or a peer's graphical solution. The checklist can include points like: 'Are the lines solid/dashed correctly?', 'Is the shading for each inequality correct?', 'Is the final feasible region clearly marked?'

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

A few notes on teaching this unit

Begin with a very simple, relatable scenario like 'I have ₹100 to spend on pens (₹10) and notebooks (₹20)'. First model how to write the inequality. Then, introduce a second constraint and show how to graph both on the same axes. Emphasise the 'test point' method using (0,0) to determine shading, and clearly explain that the overlapping region is the 'feasible region' containing all valid answers.

Students will learn to translate complex word problems into a system of mathematical inequalities and use graphs to visualise all the possible solutions that satisfy the given conditions.

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Watch Out for These Misconceptions

• Students often mix up the inequality signs, for example, using '<' for 'at most' or '>' for 'at least'.

  The phrase 'at most' means that value or less, so it corresponds to the 'less than or equal to' sign (≤). Similarly, 'at least' means that value or more, corresponding to the 'greater than or equal to' sign (≥). Create a chart with these keywords and their symbols.

• When graphing, students shade the wrong side of the line.

  Always use a test point that is not on the line, typically the origin (0,0) for simplicity. Substitute the coordinates of the test point into the inequality. If the resulting statement is true, shade the region containing the test point; if it is false, shade the other region.

• Students forget to include non-negativity constraints (like x ≥ 0 and y ≥ 0) in real-world problems.

  In most application problems, the variables represent physical quantities like the number of items, time, or weight. These quantities cannot be negative. Therefore, we must almost always include constraints restricting the solution to the first quadrant of the graph.

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Methods used in this brief

• Problem-Based Learning