Class 11 Maths NCERT Solutions Chapter 5 Linear Inequalities | Boundless Maths

Chapter 5 Class 11 Maths NCERT Solutions

Class 11 Maths NCERT Solutions Chapter 5: Linear Inequalities

Free, step-by-step Class 11 Maths NCERT Solutions for Chapter 5, Linear Inequalities — every question from Exercise 5.1 and the Miscellaneous Exercise solved in full, with number-line graphs wherever the question asks for one, and every answer independently verified.

Start Exercise 5.1 → 📐 Formula Cards

2Exercises (incl. Misc.)

40Total Questions

2026-27CBSE Syllabus

100%Solved

Class 11 Maths NCERT Solutions Chapter 5 — Overview

Solving an inequality works almost exactly like solving an equation — with one rule that changes everything: multiplying or dividing both sides by a negative number flips the inequality sign. This chapter covers solving linear inequalities over naturals, integers, and reals; representing the solution on a number line using open or closed circles; solving double inequalities and systems of two inequalities together; and translating real situations — test averages, consecutive integers, triangle sides, mixture percentages — into inequalities.

How the Chapter Builds

One Idea Leads to the Next

What an Inequality Is

Statements involving <, >, ≤, ≥ instead of "=", and what it means for a value to satisfy one. §5.1.

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Algebraic Solutions

Isolating the variable using the addition and multiplication rules, over N, Z, and R. Exercise 5.1.

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Graphical Representation

Plotting the solution set on a number line with open or closed circles. Exercise 5.1.

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Double Inequalities & Systems

Expressions like 2 ≤ 3x−4 ≤ 5, and solving two inequalities together. Miscellaneous Exercise.

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Word Problems

Temperature conversion, acid-mixture percentages, and other real situations. Miscellaneous Exercise.

Quick Reference

Important Formulas — Chapter 5

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Linear Inequalities.

Rules for Solving Inequalities (§5.2)

Rule 1 — addition/subtraction

\text{Adding or subtracting the same number from both sides never changes the inequality sign.}

Works exactly like an equation — move terms across freely.

Rule 2 — positive multiplier

\text{Multiplying or dividing both sides by a positive number never changes the sign.}

Also works exactly like an equation.

Rule 2 — negative multiplier

\text{Multiplying or dividing both sides by a negative number reverses the sign.}

The single rule most students forget — it's the whole reason this chapter needs separate treatment from equations.

Open interval

x\lt a \text{ or } x\gt a \Rightarrow \text{open circle at } a

The endpoint itself is not part of the solution.

Closed interval

x\le a \text{ or } x\ge a \Rightarrow \text{filled circle at } a

The endpoint itself is included in the solution.

Double inequality

a\le\text{expression}\le b \Rightarrow \text{apply the same operation to all three parts at once}

Keep all three parts lined up through every step.

x ≥ −1 — a filled circle means the endpoint is included; the bold line shows every solution in that direction.

Decision Guide

Which Linear Inequalities Technique Applies?

A quick way to decide, once you know what the question is actually asking for.

What the question is asking	Use this	Why
Solve a simple linear inequality in one variable	Isolate the variable	Same steps as solving an equation, but flip the sign if you multiply or divide by a negative (§5.2).
"Graph the solution on a number line"	Open or closed circle	Closed circle + shaded line for ≤/≥; open circle + shaded line for </> (§5.2).
An expression like a ≤ 3x−4 ≤ b	Double inequality	Apply the same operation to all three parts simultaneously, keeping them aligned (§5.3, Misc.).
Two inequalities to be satisfied together	Solve separately, then intersect	Find each solution set on its own number line first, then take the overlap (Misc.).
A word problem with an unknown range (marks, sides, mixture %)	Translate to an inequality first	Identify the constraint in words, write it as ≤/≥/</>, then solve algebraically (Misc.).

Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks in this chapter.

Forgetting to flip the sign — multiplying or dividing both sides by a negative number reverses ≤/≥/</>; this single slip accounts for most lost marks in the chapter.
Mixing up open and closed circles — a filled circle means the endpoint is included (≤, ≥); a hollow circle means it is excluded (<, >).
Losing track of a part in a double inequality — when solving a ≤ 3x−4 ≤ b, every operation must be applied to all three sections at once, not just two of them.
Intersecting instead of combining incorrectly in systems — when two inequalities must both hold, the final answer is the overlap of their solution sets, not the union.
Translating a word problem the wrong direction — "at least" means ≥, "at most" means ≤, "more than" means >; misreading the direction gives a correct-looking but wrong inequality.
Dropping the domain restriction — if a question specifies the solution should hold for natural numbers or integers only, list those specific values rather than leaving the answer as a continuous range.

Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 5 — Choose an Exercise

5.1

Exercise 5.1

Linear Inequalities in One Variable. Solve inequalities over natural numbers, integers, and reals, graph solutions on a number line, and translate real situations — test averages, consecutive integers, cut lengths of a board — into inequalities · 26 questions

Solve Exercise 5.1 →

Miscellaneous Exercise

Double inequalities of the form a ≤ expression ≤ b, systems of two inequalities graphed together on a number line, and word problems on temperature conversion, acid-mixture percentages, and IQ ranges · 14 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every rule for Linear Inequalities — solving, graphing, double inequalities, systems — plus every other Class 11 Maths chapter, in one printable PDF.

Get Formula Cards →

Common Questions

Frequently Asked Questions

Quick answers from Class 11 Maths NCERT Solutions Chapter 5, Linear Inequalities.

How many exercises are there in Class 11 Maths Chapter 5, Linear Inequalities?

Chapter 5 has one main exercise, Exercise 5.1, with 26 questions, followed by a Miscellaneous Exercise with 14 further questions — 40 questions in total, covering algebraic solutions of inequalities, number-line graphs, double inequalities, and real-world word problems.

What is the key rule to remember when solving inequalities?

You can add or subtract the same number from both sides of an inequality without changing its sign, and you can multiply or divide both sides by a positive number without changing the sign. But multiplying or dividing both sides by a negative number reverses the inequality sign — this is the single most important rule in this chapter.

What is the difference between an open and a closed circle on a number-line graph?

A closed (filled) circle at a point means that value is included in the solution — used for ≤ and ≥. An open (hollow) circle means that value is excluded — used for < and >. The line or ray from that point then shows every other value that satisfies the inequality.

Where can I find the official NCERT textbook for this chapter?

Linear Inequalities is Chapter 5 of the NCERT Class 11 Mathematics textbook, published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this site follow the textbook exactly as it appears there.

Ready to Start?

Jump into Exercise 5.1 — all 26 questions solved step by step.

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