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Class 11 Maths NCERT Solutions

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Linear Inequalities Class 11 Maths NCERT Solutions

The complete NCERT exercise solutions for Chapter 5, Linear Inequalities — 26 questions from Ex 5.1, each worked through step by step in the CBSE marking pattern. Solving linear inequalities in one and two variables, and the graphical method for a system.

Class:11Subject:MathsChapter:5
3 Key Formulas
DWritten byDeep Narayan
Updated
Key Concept Summary

How many questions are in NCERT Class 11 Maths Chapter 5?

Chapter 5 carries 1 exercise question, numbered Ex 5.1. All of them are solved step by step on this page, along with the chapter's key formulas and exam pointers.

01

Chapter Overview

A linear inequality keeps the usual algebra of equations but tracks the sign of the inequality. Multiplying or dividing both sides by a NEGATIVE number reverses the sign — the one rule every student forgets. This chapter practises one-variable inequalities, number-line graphs, and compound inequalities, then applies them to real-world range problems.

Board pattern

Always state the solution set in interval or set-builder form at the end. When you divide by a negative number, flip the inequality sign — lose that and you lose the whole question. For 'natural number' / 'integer' parts, list the finite solution set explicitly; for real numbers give the interval.
02

Exercise 5.1 — Solving Linear Inequalities

26Exercise questions

Step-by-step solution

  1. 1Divide both sides by 24: x < 100/24 = 25/6 ≈ 4.17.
  2. 2(i) Natural numbers {1, 2, 3, ...} less than 4.17: x ∈ {1, 2, 3, 4}.
  3. 3(ii) Integers less than 4.17: {..., −2, −1, 0, 1, 2, 3, 4}.

Final answer

(i) {1, 2, 3, 4}, (ii) {…, −1, 0, 1, 2, 3, 4}.

Step-by-step solution

  1. 1Divide by −12 and reverse: x < 30/(−12) = −5/2.
  2. 2(i) No natural number is < −2.5 → empty set.
  3. 3(ii) Integers less than −2.5: {..., −5, −4, −3}.

Final answer

(i) No solution, (ii) {…, −5, −4, −3}.

Step-by-step solution

  1. 15x < 10 → x < 2.
  2. 2(i) Integers less than 2: {..., −1, 0, 1}.
  3. 3(ii) Real numbers: x ∈ (−∞, 2).

Final answer

(i) {…, −1, 0, 1}, (ii) (−∞, 2).

Step-by-step solution

  1. 13x > 2 − 8 = −6 → x > −2.
  2. 2(i) Integers greater than −2: {−1, 0, 1, 2, ...}.
  3. 3(ii) Real numbers: x ∈ (−2, ∞).

Final answer

(i) {−1, 0, 1, 2, …}, (ii) (−2, ∞).

Step-by-step solution

  1. 1Bring x terms together: 4x − 5x < 7 − 3.
  2. 2−x < 4 → multiply by −1 and reverse sign: x > −4.

Final answer

x ∈ (−4, ∞).

Step-by-step solution

  1. 13x − 5x > −1 + 7 → −2x > 6.
  2. 2Divide by −2 and reverse: x < −3.

Final answer

x ∈ (−∞, −3).

Step-by-step solution

  1. 1Expand: 3x − 3 ≤ 2x − 6.
  2. 23x − 2x ≤ −6 + 3 → x ≤ −3.

Final answer

x ∈ (−∞, −3].

Step-by-step solution

  1. 1Expand: 6 − 3x ≥ 2 − 2x.
  2. 26 − 2 ≥ 3x − 2x → 4 ≥ x → x ≤ 4.

Final answer

x ∈ (−∞, 4].

Step-by-step solution

  1. 1LHS = (3x + 2x)/6 = 5x/6.
  2. 25x/6 < 11 → 5x < 66 → x < 66/5 = 13.2.

Final answer

x ∈ (−∞, 66/5).

Step-by-step solution

  1. 1x/3 − x/2 > 1 → (2x − 3x)/6 > 1 → −x/6 > 1.
  2. 2−x > 6 → x < −6.

Final answer

x ∈ (−∞, −6).

Step-by-step solution

  1. 1Cross-multiply (LCM 15): 9(x − 2) ≤ 25(2 − x).
  2. 29x − 18 ≤ 50 − 25x → 34x ≤ 68 → x ≤ 2.

Final answer

x ∈ (−∞, 2].

Step-by-step solution

  1. 1Multiply by 6: 3(3x/5 + 4) ≥ 2(x − 6).
  2. 29x/5 + 12 ≥ 2x − 12 → 24 ≥ 2x − 9x/5 = (10x − 9x)/5 = x/5.
  3. 324 ≥ x/5 → x ≤ 120.

Final answer

x ∈ (−∞, 120].

Step-by-step solution

  1. 1Expand: 4x + 6 − 10 < 6x − 12 → 4x − 4 < 6x − 12.
  2. 2−4 + 12 < 6x − 4x → 8 < 2x → x > 4.

Final answer

x ∈ (4, ∞).

Step-by-step solution

  1. 1LHS = 37 − 3x − 5 = 32 − 3x. RHS = 9x − 8x + 24 = x + 24.
  2. 232 − 3x ≥ x + 24 → 8 ≥ 4x → x ≤ 2.

Final answer

x ∈ (−∞, 2].

Step-by-step solution

  1. 1RHS = [5(5x−2) − 3(7x−3)]/15 = (25x − 10 − 21x + 9)/15 = (4x − 1)/15.
  2. 2x/4 < (4x − 1)/15 → 15x < 16x − 4.
  3. 3−x < −4 → x > 4.

Final answer

x ∈ (4, ∞).

Step-by-step solution

  1. 1RHS = [5(3x−2) − 4(2−x)]/20 = (15x − 10 − 8 + 4x)/20 = (19x − 18)/20.
  2. 2(2x − 1)/3 ≥ (19x − 18)/20 → 20(2x − 1) ≥ 3(19x − 18).
  3. 340x − 20 ≥ 57x − 54 → 34 ≥ 17x → x ≤ 2.

Final answer

x ∈ (−∞, 2].

Step-by-step solution

  1. 13x − 2x < 1 + 2 → x < 3.
  2. 2Graph: open circle at 3, ray extending left to −∞.

Final answer

x ∈ (−∞, 3). Graph: open circle at 3, arrow to the left.

Step-by-step solution

  1. 15x − 3x ≥ −5 + 3 → 2x ≥ −2 → x ≥ −1.
  2. 2Graph: closed circle at −1, ray extending right.

Final answer

x ∈ [−1, ∞). Graph: filled circle at −1, arrow to the right.

Step-by-step solution

  1. 13 − 3x < 2x + 8 → −5 < 5x → x > −1.
  2. 2Graph: open circle at −1, ray extending right.

Final answer

x ∈ (−1, ∞). Graph: open circle at −1, arrow right.

Step-by-step solution

  1. 1RHS = (4x − 1)/15 (as computed in Q15).
  2. 2x/2 ≥ (4x − 1)/15 → 15x ≥ 8x − 2 → 7x ≥ −2 → x ≥ −2/7.
  3. 3Graph: closed circle at −2/7, ray extending right.

Final answer

x ∈ [−2/7, ∞). Graph: filled circle at −2/7, arrow right.

Step-by-step solution

  1. 1Let the third test marks be x.
  2. 2(70 + 75 + x)/3 ≥ 60 → 145 + x ≥ 180.
  3. 3x ≥ 35.

Final answer

At least 35 marks.

Step-by-step solution

  1. 1Let the fifth examination mark be x.
  2. 2(87 + 92 + 94 + 95 + x)/5 ≥ 90 → 368 + x ≥ 450.
  3. 3x ≥ 82.

Final answer

At least 82 marks.

Step-by-step solution

  1. 1Consecutive odd positive pairs under 10: (1,3), (3,5), (5,7), (7,9).
  2. 2Sums: 4, 8, 12, 16.
  3. 3Sum > 11 for (5,7) and (7,9).

Final answer

(5, 7) and (7, 9).

Step-by-step solution

  1. 1Let the smaller be 2n; pairs (6,8), (8,10), (10,12), ...
  2. 2Sums: 14, 18, 22, 26, ...
  3. 3Sum < 23 for (6,8), (8,10), (10,12).

Final answer

(6, 8), (8, 10) and (10, 12).

Step-by-step solution

  1. 1Let shortest = x, longest = 3x, third = 3x − 2.
  2. 2Perimeter ≥ 61: x + 3x + (3x − 2) ≥ 61 → 7x − 2 ≥ 61.
  3. 37x ≥ 63 → x ≥ 9.

Final answer

Minimum shortest side = 9 cm.

Step-by-step solution

  1. 1Let shortest = x, second = x + 3, third = 2x.
  2. 2Board limit: x + (x + 3) + 2x ≤ 91 → 4x + 3 ≤ 91 → x ≤ 22.
  3. 3Third at least 5 cm longer than second: 2x ≥ (x + 3) + 5 → x ≥ 8.
  4. 4Combine: 8 ≤ x ≤ 22.

Final answer

Shortest board between 8 cm and 22 cm.

Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Absolute value inequality

Critical points

Solution region (two variables)

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

  • The inequality flips when you divide by a negative number — that single step is the most common error in the whole chapter.
  • Every linear inequality is a half-plane, so a system of them is feasible only if the half-planes share a non-empty region; sketching them is faster than testing points.

FAQ

Frequently asked questions

How many questions are in NCERT Class 11 Maths Chapter 5 (Linear Inequalities)?

There are 1 exercise question in this chapter, numbered Ex 5.1. Every one is solved step by step on this page in the official NCERT numbering.

Which formulas come up in Linear Inequalities Class 11 Maths?

The formulas this chapter's questions actually turn on are: Absolute value inequality, Critical points, Solution region (two variables). They are listed with their expressions in the key formulas section below, and the solved questions show where each one is used.

Is Linear Inequalities important for JEE Main?

Moderate — a short chapter worth easy marks in boards, and the graphical method is also the opening step of linear programming in Class 12.

Next Chapters

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Previous: Complex Numbers and Quadratic EquationsNext: Permutations and CombinationsLinear Inequalities revision notesAll Class 11 Maths NCERT Solutions

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