NCERT Solutions for Class 11 Maths Chapter 6 – Linear Inequalities
Equations tell you exactly where something is equal. Inequalities tell you about the range of values where something holds true — and real life is almost entirely about ranges, not exact values. A budget constraint is an inequality. A speed limit is an inequality. The number of items you can carry is bounded by an inequality. Chapter 6 of Class 11 Mathematics formalises this very natural idea.
This NCERT solutions chapter extends the algebra of linear equations into the territory of linear inequalities — both in one variable and in two variables. The one-variable portion (Exercise 6.1 and 6.2) is mostly algebraic: solving inequalities and representing the solution on a number line. The two-variable portion (Exercise 6.3) is where it gets visually interesting — you graph inequalities on the coordinate plane, shade half-planes, and find the feasible region formed by a system of linear inequalities. This feasible region concept becomes critically important in the Linear Programming unit in Class 12. For all Chapters must, read NCERT Solutions for Class 11 Maths and subject-wise NCERT Solutions for Class 11.
Students sometimes make the costly mistake of applying equation-solving rules blindly to inequalities. The single most important difference: when you multiply or divide both sides of an inequality by a negative number, the inequality sign flips. Miss this rule once and your entire solution is wrong. These solutions highlight every such step explicitly so you build the habit of checking for sign changes automatically.
Download PDF – All Exercises of NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequalities
| Exercise | Topic Covered | Number of Questions |
|---|---|---|
| Exercise 6.1 | Solving Linear Inequalities in One Variable (Number Line) | 26 Questions |
| Exercise 6.2 | Solving Systems of Inequalities in One Variable | 10 Questions |
| Exercise 6.3 | Graphical Solution of Linear Inequalities in Two Variables | 15 Questions |
| Miscellaneous | Mixed Inequality Problems | 14 Questions |
Chapter 6 – Linear Inequalities: Concepts, Explanation and Key Tables
Core Rules of Solving Inequalities
| Rule | Statement | Example |
|---|---|---|
| Addition/Subtraction Rule | Adding or subtracting the same number from both sides does not change the inequality | x – 3 > 5 → x > 8 |
| Multiplication/Division (Positive) | Multiplying/dividing by a positive number preserves the sign | 2x < 10 → x < 5 |
| Multiplication/Division (Negative) | Multiplying/dividing by a negative number REVERSES the sign | –2x < 10 → x > –5 |
| Transitive Property | a < b and b < c implies a < c | Used to combine solution ranges |
| Double Inequality | a < x < b means x lies strictly between a and b | –3 < 2x + 1 < 7 |
Types of Solution Sets and Their Notation
| Type | When It Occurs | Number Line Representation | Interval Notation |
|---|---|---|---|
| x > a (strict) | Strict greater than | Open circle at a, ray to the right | (a, ∞) |
| x ≥ a (non-strict) | Greater than or equal | Closed circle at a, ray to the right | [a, ∞) |
| x < a (strict) | Strict less than | Open circle at a, ray to the left | (–∞, a) |
| x ≤ a (non-strict) | Less than or equal | Closed circle at a, ray to the left | (–∞, a] |
| a < x < b | Open interval between two values | Open circles at both a and b | (a, b) |
| a ≤ x ≤ b | Closed interval between two values | Closed circles at both a and b | [a, b] |
Graphical Solution of Two-Variable Inequalities
For an inequality like 2x + 3y ≤ 12, the approach in Exercise 6.3 is:
- Treat it as the equation 2x + 3y = 12 and draw the line (find x and y intercepts).
- The line divides the plane into two half-planes. Pick a test point not on the line (origin is usually easiest).
- Substitute the test point into the inequality. If it satisfies it, shade the half-plane containing the test point. If not, shade the other half-plane.
- The boundary line is solid for ≤ or ≥ (point on line is included) and dashed for < or > (not included).
Solving Systems of Inequalities
When a problem has two or more inequalities in one variable, solve each separately, represent both on the same number line, and find the intersection (the values that satisfy ALL conditions simultaneously).
| System | Solution Set Principle |
|---|---|
| x > a AND x > b | x > max(a, b) |
| x > a AND x < b | a < x < b (only possible if a < b) |
| x > a OR x < b (where a > b) | All real numbers (union covers everything) |
| x > a AND x < b (where a > b) | No solution (empty set) |
Feasible Region in Two Variables
When Exercise 6.3 asks you to find the region satisfying a system of linear inequalities in x and y, the answer is the feasible region — the shaded area on the coordinate plane that satisfies all inequalities simultaneously. This region is a convex polygon (or unbounded region) bounded by the boundary lines of each inequality.
Study Tips for Chapter 6
- For Exercise 6.1, practise the sign-flip rule until it is reflexive — anytime you divide by a negative, check the inequality direction immediately before moving on.
- In Exercise 6.3, always draw clean, clearly labelled graphs — CBSE awards 1 mark for the graph itself.
- When finding the feasible region for a system, check two or three corner regions with the original inequalities to verify your shading before drawing the final answer.
- The miscellaneous exercise includes word problems (salary constraints, mixture problems) — translate each condition into a mathematical inequality systematically before solving.
FAQs for NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequalities
Linear inequalities are mathematical expressions that compare two quantities using inequality symbols such as >, <, ≥, or ≤. Unlike linear equations, which have a single solution or a fixed set of solutions, inequalities often have a range of values that satisfy the condition. In Class 11 Maths, students learn how to solve and represent these inequalities graphically. NCERT Solutions explain the concepts through detailed examples and solved exercises. Understanding linear inequalities is important because they are used in optimization problems, business calculations, economics, and higher mathematics. Regular practice helps students develop confidence in solving inequality-based questions accurately.
Solving a linear inequality in one variable is similar to solving a linear equation. Students simplify the inequality by performing the same operations on both sides. However, when multiplying or dividing by a negative number, the inequality sign must be reversed. This is one of the most important rules in the chapter. NCERT Solutions provide step-by-step explanations and examples that help students understand the correct procedure. After solving the inequality, the solution is usually represented on a number line. Practicing various types of questions helps students avoid common mistakes and strengthens their understanding of inequality concepts.
Graphical representation helps students visualize the solution set of a linear inequality. In inequalities involving two variables, the solutions are represented as a region on a coordinate plane rather than a single point. This visual approach makes it easier to identify all possible solutions that satisfy the inequality. NCERT Solutions explain how to draw boundary lines, determine the correct region, and interpret graphical results accurately. Understanding graphical representation is important because it is widely used in optimization, economics, and higher-level mathematics. Regular practice of graph-based questions improves analytical thinking and problem-solving abilities.
NCERT Solutions provide accurate and detailed answers to all textbook questions, helping students understand every concept thoroughly. They explain important topics such as solving inequalities, representing solutions on number lines, graphing inequalities, and interpreting solution regions. Step-by-step methods make complex questions easier to solve and improve conceptual clarity. Students can use these solutions for homework, revision, and exam preparation. Regular practice enhances logical reasoning, accuracy, and confidence. Since many school examination questions are based directly on NCERT exercises, studying these solutions helps students score better marks and develop a strong foundation for advanced mathematical topics.