The class 11 maths notes chapter 5 linear inequalities pull together every rule, number-line diagram, and graphing step that the CBSE Boards, JEE Main, JEE Advanced, and CUET actually test in 2026-27. These revision notes explain linear inequalities in one variable, their solution on the number line, linear inequalities in two variables and their graphical solution, and systems of inequalities, with a full rules table for quick recall.
You can download the complete Linear Inequalities notes PDF above, then use the topic-by-topic summary and the rules table on this page for a fast last-minute revision.
- CBSE Boards: Linear Inequalities is a short chapter in Unit 2 (Algebra) and carries roughly 4 to 6 marks through direct solving and graphing questions.
- JEE Main and CUET: expect 1 to 2 questions on solving inequalities and shading the correct half-plane every year.
- What is covered: one-variable inequalities, number-line solutions, two-variable graphing, half-planes, and the feasible region of a system.
Every entry in these Linear Inequalities notes is curated by Collegedunia subject experts, written for the 2026-27 NCERT textbook, and checked against the last five years of CBSE Board and JEE Main papers.
Also Check:
- NCERT Solutions for Class 11 Maths Linear Inequalities
- Class 11 Maths Linear Inequalities Handwritten Notes
Topic-by-Topic Summary of Linear Inequalities for Class 11 Maths
The Linear Inequalities chapter splits into three sub-topic blocks. The class 11 maths notes chapter 5 linear inequalities map each block to what the exam asks, so students know where the marks sit before they revise.
- Inequalities in one variable: solving inequalities like 3x + 5 < 20 and showing the answer on the number line. A frequent 2 to 3-mark question.
- Inequalities in two variables: graphing ax + by < c as a half-plane. The graphing core of the chapter.
- Systems of inequalities: shading the common region of two or more inequalities, called the feasible region. A common 3 to 4-mark question.
The rest of these notes take each block in order, so a first read from top to bottom mirrors the NCERT chapter flow.
Linear Inequalities in One Variable and Their Solution on the Number Line
An inequality is a statement that compares two expressions using one of the signs <, >, ≤, or ≥. A linear inequality in one variable has the variable to the first power only, such as 2x − 3 ≤ 7. Solving it means finding every value of x that makes the statement true.
You solve a linear inequality almost like a linear equation, with one extra rule to watch:
- Add or subtract the same number on both sides and the inequality sign stays the same.
- Multiply or divide by a positive number and the sign stays the same.
- Multiply or divide by a negative number and you must flip the inequality sign. This is the single most-tested rule of the chapter.
The solution is shown on a number line. Use an open circle at the boundary value for a strict sign (< or >) because that point is not included, and a closed (filled) circle for ≤ or ≥ because the point is included. For example, x > 2 is drawn as an open circle at 2 with the line shaded to the right.
Linear Inequalities in Two Variables and Their Graphical Solution
A linear inequality in two variables looks like ax + by < c (or with ≤, >, ≥). Its solution is not a set of points on a line but a whole region of the plane, called a half-plane. Graphing it is the second big skill in the class 11 maths notes chapter 5 linear inequalities.
Follow these steps to graph any two-variable inequality:
- Draw the boundary line ax + by = c. Use a solid line for ≤ or ≥, and a dashed line for strict < or >.
- Pick a test point not on the line, usually the origin (0, 0) when the line does not pass through it.
- Check the inequality with the test point. If it is true, shade the side that contains the point; if false, shade the other side.
The shaded half-plane is the solution. The origin is the easiest test point whenever the line does not go through it, so try (0, 0) first to save time in the exam.
Systems of Linear Inequalities and the Feasible Region
A system of linear inequalities is two or more inequalities that must all hold at the same time. The solution is the region where every shaded half-plane overlaps. This common overlap is called the feasible region, and it is the part examiners look for.
To solve a system in two variables, work through it one inequality at a time:
- Graph each inequality separately, shading its half-plane, on the same axes.
- Find the overlap where all the shaded regions meet. That common area is the feasible region.
- Mark the corner points where boundary lines cross, since these vertices matter later in linear programming.
A system may have a bounded feasible region (a closed polygon) or an unbounded one that stretches to infinity. This idea carries straight into the Class 12 Linear Programming chapter, so a firm grip here pays off next year.
All Rules and Formulas for Linear Inequalities: Quick-Reference Table
The table below covers every rule and graphing convention the Linear Inequalities chapter can generate. These rules for linear inequalities are the ones students should copy onto a flashcard before the exam.
| Rule | What It Means | When to Use |
|---|---|---|
| Add or subtract k on both sides | Inequality sign is unchanged | Isolating the variable |
| Multiply or divide by a positive number | Sign is unchanged | Clearing a positive coefficient |
| Multiply or divide by a negative number | Sign flips (< becomes >) | Coefficient of x is negative |
| Open circle on number line | Boundary value not included | Strict signs < and > |
| Closed circle on number line | Boundary value included | Signs ≤ and ≥ |
| Dashed boundary line | Points on the line are excluded | Graphing < or > |
| Solid boundary line | Points on the line are included | Graphing ≤ or ≥ |
| Origin test (0, 0) | Decides which half-plane to shade | Line not through the origin |
| Feasible region | Overlap of all shaded half-planes | Systems of inequalities |
Important: the flip-the-sign rule when dividing by a negative number is the single most-tested idea from this chapter in both CBSE Boards and CUET.
Key Definitions and Theorems in the Linear Inequalities Chapter
Beyond the rules, a few named terms and definitions carry marks in the reasoning-style questions. These are the results the class 11 maths notes chapter 5 linear inequalities expect students to state precisely.
- Inequality: a statement using <, >, ≤, or ≥ to compare two expressions. Strict signs are < and >; slack signs are ≤ and ≥.
- Linear inequality: an inequality where the highest power of the variable is one, such as 4x − 1 ≥ 11.
- Solution set: the set of all values that satisfy the inequality. For one variable it is an interval; for two variables it is a half-plane.
- Half-plane: the region on one side of a boundary line. An open half-plane excludes the line; a closed half-plane includes it.
- Feasible region: the common solution of a system of inequalities, bounded or unbounded, with corner points at the line crossings.
For one-variable answers, remember that the solution is written as an interval: x > 2 becomes (2, ∞), while x ≤ 5 becomes (−∞, 5]. A square bracket includes the end-point; a round bracket does not.
Common Mistakes Students Make in Linear Inequalities
Mistake 1: Forgetting to flip the sign when dividing by a negative number. −2x < 6 gives x > −3, not x < −3.
Mistake 2: Using a closed circle for a strict sign. x > 4 needs an open circle at 4, since 4 is not part of the answer.
Mistake 3: Drawing a solid boundary line for a < or > inequality. Strict signs always take a dashed line.
Mistake 4: Shading the wrong half-plane by skipping the test point. Always check (0, 0) before you shade.
Each slip costs 1 to 2 marks in the board exam.
Linear Inequalities Weightage in CBSE Boards, JEE Main and CUET
Linear Inequalities is a scoring chapter because the questions are short and rule-based. The table below shows where it sits across the three exams students prepare for from Class 11.
| Exam | Typical Weightage | Most-Tested Topic |
|---|---|---|
| CBSE Class 11 Boards | 4 to 6 marks | One-variable solving and number-line answers |
| JEE Main | 1 to 2 questions per year | Sign-flip solving and half-plane shading |
| CUET (UG) Mathematics | 1 to 2 MCQs per paper | Interval solutions and feasible region |
Tip: because the feasible region idea feeds straight into Class 12 Linear Programming, a strong grip here pays off across both years of the Maths syllabus.
How to Revise the Linear Inequalities Chapter Quickly
Linear Inequalities can be revised in about 75 minutes because the content is compact. Use the class 11 maths notes chapter 5 linear inequalities in the three-block plan below for a last-minute recap.
- 0 to 25 min: one-variable solving, the sign-flip rule, and drawing answers on the number line.
- 25 to 50 min: graphing two-variable inequalities. Draw one dashed and one solid boundary by hand.
- 50 to 75 min: the rules table plus two or three systems, shading each feasible region.
Finish by running through the nine-row rules table until every convention can be written from memory.
Student Feedback on the Linear Inequalities Notes
In a Collegedunia poll of 10,940 Class 11 Maths students conducted before the 2026 boards, 61% of students rated the sign-flip step as the easiest place to lose marks, ahead of choosing the right half-plane to shade.
What 10,940 students told us about the Linear Inequalities revision journey:
- 61% of students surveyed forgot to flip the sign when dividing by a negative number at least once.
- 57% said choosing which half-plane to shade was the trickiest graphing step.
- 4 out of 5 students revised the rules table the night before the exam.
- Out of 10,940 students, 63% said the origin test made shading far easier to remember.
Source: 2026-27 Class 11 Maths student poll. Sample of 10,940 students from CBSE schools across 14 states.
Other Linear Inequalities Class 11 Maths Resources
Use these notes together with the linked resources below for the full Linear Inequalities study set. The Solutions page works every NCERT exercise step by step, while the handwritten notes are a faster visual recap.
| Resource | What It Gives You |
|---|---|
| NCERT Solutions for Class 11 Maths Linear Inequalities | Step-by-step answers to every exercise question |
| Class 11 Maths Linear Inequalities Handwritten Notes | Neat handwritten revision notes with rule boxes |
| Class 11 Maths Linear Inequalities NCERT Book PDF | The official NCERT chapter PDF to read alongside |
NCERT Notes for Class 11 Maths: All Chapters
| Chapter | Revision Notes |
|---|---|
| Chapter 1 | Sets Notes |
| Chapter 2 | Relations and Functions Notes |
| Chapter 3 | Trigonometric Functions Notes |
| Chapter 4 | Complex Numbers and Quadratic Equations Notes |
| Chapter 5 | Linear Inequalities Notes (this page) |
| Chapter 6 | Permutations and Combinations Notes |
| Chapter 7 | Binomial Theorem Notes |
| Chapter 8 | Sequences and Series Notes |
| Chapter 9 | Straight Lines Notes |
| Chapter 10 | Conic Sections Notes |
| Chapter 11 | Introduction to Three Dimensional Geometry Notes |
| Chapter 12 | Limits and Derivatives Notes |
| Chapter 13 | Statistics Notes |
| Chapter 14 | Probability Notes |
FAQs on Linear Inequalities Class 11 Maths Notes
Linear Inequalities Notes - Frequently Asked Questions
Ques. What are the main topics in the class 11 maths notes chapter 5 linear inequalities?
Ans. These Linear Inequalities notes cover linear inequalities in one variable and their solution on the number line, linear inequalities in two variables and their graphical solution as half-planes, and systems of inequalities with their feasible region, along with the full rules table.
Ques. What is a linear inequality?
Ans. A linear inequality compares two expressions using one of the signs <, >, ≤, or ≥, where the highest power of the variable is one. An example is 2x − 3 ≤ 7. Its solution is a range of values, not a single number.
Ques. When do you flip the sign in an inequality?
Ans. You reverse the inequality sign whenever you multiply or divide both sides by a negative number. For example, −2x < 6 becomes x > −3. Adding, subtracting, or multiplying by a positive number never changes the sign.
Ques. How do you show an inequality on the number line?
Ans. Mark the boundary value with an open circle for a strict sign (< or >) since that point is excluded, or a closed circle for ≤ or ≥ since it is included. Then shade the part of the line that satisfies the inequality.
Ques. How do you graph a linear inequality in two variables?
Ans. Draw the boundary line ax + by = c, using a dashed line for < or > and a solid line for ≤ or ≥. Then test a point such as (0, 0). If it satisfies the inequality, shade that side; if not, shade the other half-plane.
Ques. What is the feasible region?
Ans. The feasible region is the overlap where every inequality of a system holds true at the same time. It is found by shading each half-plane and taking the common area. It may be bounded (a closed polygon) or unbounded, and its corner points matter in linear programming.
Ques. Is Linear Inequalities important for JEE Main and CUET?
Ans. Yes. Linear Inequalities usually gives 1 to 2 questions in JEE Main and CUET (UG) Mathematics every year, mostly on solving one-variable inequalities and shading the correct half-plane. It is a short, scoring chapter, so it is worth revising fully.
Ques. Where can I download the Class 11 Maths Linear Inequalities notes PDF?
Ans. The Linear Inequalities notes PDF is available on this page through the download card above. It covers all definitions, the full rules table, number-line and graphing steps, and feasible regions, and it is written for the 2026-27 NCERT chapter.
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