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NCERT Solutions for Class 11 Maths Chapter 6 : Linear Inequalities

Q: Q1. How should linear inequalities in two variables be expressed?

Answer: Linear inequalities in two variables can be written as ax + by c or ax + by ≥ c.

Q: Q2. What is the difference between literal and numerical inequalities?

Answer: Literal inequalities are ones that include both variables and numbers. For example, x 8, and x 2, and 2 < 1, etc.

Q: Q3. How many exercises are there in the Linear Inequalities Class 11 NCERT Solutions?

Answer: There are 50 questions in three exercises in Class 11 Maths Chapter 6 Linear Inequalities, including 15 problems in a miscellaneous practice. These tasks largely focus on the understanding of linear inequalities, their algebraic solutions, and their graphical depiction. The solutions to the exercise-specific problems are also accessible in PDF format, with the goal of assisting students in performing well in the yearly exam.

Q: Q4. What are the key subtopics in Linear Inequalities Class 11 Maths Chapter 6 that could be tested?

Answer: NCERT Solutions Class 11 Maths Chapter 6 begins with a brief overview of linear inequalities and associated concepts. It also discusses some of the most significant algebraic ideas for solving linear inequalities in one and two variables, as well as their graphical representation and pictorial method for solving linear inequalities.

Q: Q5. Is it necessary to practice all of the questions in Linear Inequalities Class 11 NCERT Solutions?

Answer: Linear Inequalities Class 11 NCERT Solutions are well-designed to help students study Linear Inequalities easily. It thoroughly covers all essential principles, allowing students to grasp the concepts. These solutions use examples to clarify the subjects presented so that students may easily relate to the concepts being addressed.

Linear Inequalities Class 11 Maths Chapter 6 deals with the understanding of linear inequalities in one and two variables that are essential for solving problems in the field of science, mathematics, statistics, optimisation problems, economics, psychology, etc. Class 11 Maths Chapter 6 explains the key concepts related to Linear Inequalities with examples. The concepts taught in Class 11 Chapter 6 Linear Inequalities will also help students to gain a thorough understanding of this topic and its practical applications. The expert faculty of Toppr produced these solutions to assist students with their first-term exam preparations.

Linear Inequalities Class 11 Chapter 6 also covers topics such as Introduction to linear inequalities, algebraic solutions of linear inequalities in one variable, graphical representation of linear inequalities in one variable and two variables, and solution of the system of linear inequalities in two variables. Students will quickly learn about these concepts if they practice Chapter 6 Maths Class 11 NCERT Solutions of Linear Inequalities. All of these solutions have been created with the new CBSE pattern in mind, ensuring that students have a thorough understanding of their exams. It goes over all major concepts in depth, allowing students to better understand the ideas.

Chapter 6 Maths Class 11 Questions and Answers can help you get good grades in tests and properly prepare for all of the important concepts. Several examples provided in the solutions will assist students with a better understanding of linear inequalities. Linear Inequalities Class 11 NCERT Solutions Chapter 6 are available below in pdf format, and a few solutions are also included in the exercises. These solutions explain the topics covered with examples so that students can easily relate to the notion being discussed.

Table of Content

Introduction

Download PDF of NCERT Solutions for Class 11 Maths Chapter 6 : Linear Inequalities

Access NCERT Solutions for Class 11 Maths Chapter 6 : Linear Inequalities

Exercise 6.1

Question 1

Solve

24 x < 100

, when (i)

x

is a natural number. (ii)

x

is an integer

Easy

Solution

Verified by Toppr

24 x < 100

\Rightarrow x < \frac{1 0 0}{2 4}

\Rightarrow x < \frac{2 5}{6}

(i) It is evident that

1, 2, 3

and

4

are the only natural numbers less than

\frac{2 5}{6}

,
Thus when

x

is a natural number ,the solutions of the given inequality are

1, 2, 3

and

4

.
Hence, in this case, the solution set is

{1, 2, 3, 4}

.
(ii) The integers less than

\frac{2 5}{6}

are

. . . . . . . - 3, - 2, - 1, 0, 1, 2, 3, 4

.
Hence,in this case ,the solution set is

{. . . . . . . . . . . . - 3, - 2, - 1, 0, 1, 2, 3, 4} .

Question 2

Solve

- 12 x > 30

, when (i)

x

is a natural number. (ii)

x

is an integer

Easy

Solution

Verified by Toppr

The given inequality is

- 12 x > 30

\Rightarrow \frac{- 1 2 x}{- 1 2} < \frac{3 0}{- 1 2}

[ Dividing both sides by same negative number]

\Rightarrow x < - \frac{5}{2}

(i) There is no natural number less than

(- \frac{5}{2})

(ii) The integers less than

(- \frac{5}{2})

are

- \infty, . . . . . ., - 5, - 4, - 3 .

Question 3

Solve

5 x - 3 < 7,

when

(i)

x

is an integer.

(ii)

x

is a real number.

Medium

Solution

Verified by Toppr

The given inequality is,

5 x - 3 < 7

\Rightarrow 5 x - 3 + 3 < 7 + 3

\Rightarrow 5 x < 10

\Rightarrow \frac{5 x}{5} < \frac{1 0}{5}

\Rightarrow x < 2

(i) The integers less than 2 are

. . . . . . ., - 4, - 3, - 3, - 1, 0, 1 .

Thus when

x

is an integer, the solutions of the given inequality are
all the integral values of

x

which are less than

2

.
(ii) When

x

is real number the solution is given by

x < 2

,
i.e., all real numbers

x

which are less than 2.
Thus, the solution set of the given inequality is

x \in (- \infty, 2)

Question 4

Solve

3 x + 8 > 2,

when (i)

x

is an integer. (ii)

x

is a real number

Medium

Solution

Verified by Toppr

The given inequality is

3 x + 8 > 2

\Rightarrow 3 x + 8 - 8 > 2 - 8

\Rightarrow 3 x > - 6

\Rightarrow \frac{3 x}{3} > \frac{- 6}{3}

\Rightarrow x > - 2

(i) The integers greater than

- 2

are

- 1, 0, 1, 2, . . . . . . .

Thus when

x

is an integer, the solutions of the given inequality are

- 1, 0, 1, 2 . . . . . . . . . . . .

(ii) When

x

is a real number, the solutions of the given inequality are all the real numbers, which are greater than

- 2

.
Thus in this case, the solution set is

(- 2, \infty)

Question 5

Solution set of inequality

4 x + 3 < 5 x + 7

(- 4, \infty)

True

False

Easy

Solution

Verified by Toppr

4 x + 3 < 5 x + 7

4 x + 3 - 7 < 5 x + 7 - 7

4 x - 4 < 5 x

4 x - 4 - 4 x < 5 x - 4 x

- 4 < x

Thus, the solution set of the given inequality is

(- 4, \infty)

Question 6

Solve:

3 x - 7 > 5 x - 1

Medium

Solution

Verified by Toppr

G i v e n :

3 x - 7 > 5 x - 1

\Rightarrow 1 - 7 > 5 x - 3 x

\Rightarrow - 6 > 2 x

\Rightarrow ∴ x > \frac{- 6}{2} = - 3

∴

S o l u t i o n s e t

= (

- 3, \infty

)

Question 7

Solve the inequalities for real

x

3 (x - 1) \leq 2 (x - 3)

Easy

Solution

Verified by Toppr

Given,

3 (x - 1) \leq 2 (x - 3)

\Rightarrow 3 x - 3 \leq 2 x - 6

\Rightarrow 3 x - 2 x \leq - 6 + 3

\Rightarrow x \leq - 3

\Rightarrow x \in (- \infty, - 3]

Question 8

Solve:

3 (2 - x) \geq 2 (1 - x)

when

x

is an integer.

Medium

Solution

Verified by Toppr

3 (2 - x) \geq 2 (1 - x)

\Rightarrow 6 - 3 x \geq 2 - 2 x

\Rightarrow 6 - 3 x - 6 \geq 2 - 2 x - 6

\Rightarrow - 3 x \geq - 2 x - 4

\Rightarrow - 3 x + 2 x \geq - 2 x - 4 + 2 x

\Rightarrow - x \geq - 4

\Rightarrow x \leq 4

When

x \in I,

the solution set is

{, . . . ., - 2, - 1, 0, 1, 2, 3, 4}

Question 9

Solve the inequality for real

x

x + \frac{x}{2} + \frac{x}{3} < 11

Easy

Solution

Verified by Toppr

Given,

x + \frac{x}{2} + \frac{x}{3} < 11

\Rightarrow x (1 + \frac{1}{2} + \frac{1}{3}) < 11

\Rightarrow x (\frac{6 + 3 + 2}{6}) < 11

\Rightarrow \frac{1 1 x}{6} < 11

\Rightarrow x < 6

\Rightarrow x \in (- \infty, 6)

Question 10

Solve for

x

\frac{x}{2} = \frac{x}{3} + 1

4

8

6

9

Easy

Solution

Verified by Toppr

Given,

\frac{x}{2} = \frac{x}{3} + 1

Transposing

x

terms to one side, we get,

\Rightarrow \frac{x}{2} - \frac{x}{3} = 1

\Rightarrow \frac{3 x - 2 x}{6} = 1

∴ x = 6

Hence, option

C

is correct.

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NCERT Solutions for Class 11 Maths Chapter 6 : Linear Inequalities

NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequalities – Brief Overview

Linear Inequalities

Two real numbers or two algebraic expressions related by the symbol ‘<’, ‘>’, ‘≤’ or ‘≥’ form an inequality. For example:

x > 10

y ≤ 4

ax + b > 0

ax + by ≤ c

ax² + bx + c ≥ 0

Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation

The following are the rules followed for algebraic operations on linear inequalities in one variable:

An inequality can have equal numbers added or subtracted from both sides.
An inequality can be multiplied or divided by the same positive value on both sides. The inequality sign is reversed when both sides are multiplied (or divided) by a negative value.

For Graphical Representation:

- To depict x < a (or x > a) on a number line, draw a circle around the number a and a dark line to the left or right of the number a, respectively.
- To depict x ≤ a (or x ≥ a) on a number line, draw a dark circle around the number a and a dark line to the left or right of the number x, respectively.

Graphical Solution of Linear Inequalities in Two Variables

The region containing all the solutions of an inequality, which satisfies all the given inequalities is called the solution region.

Related Chapters

Chapter 1 : Sets

Chapter 2 : Relations and Functions

Chapter 3 : Trigonometric Functions

Chapter 4 : Principle of Mathematical Induction

Chapter 5 : Complex Numbers and Quadratic Equations

Chapter 7 : Permutations and Combinations

Chapter 8 : Binomial Theorem

Chapter 9 : Sequences and Series

Chapter 10 : Straight Lines

Chapter 11 : Conic Sections

Chapter 12 : Introduction to Three Dimensional Geometry

Chapter 13 : Limits and Derivatives

Chapter 14 : Mathematical Reasoning

Chapter 15 : Statistics

Chapter 16 : Probability

Frequently Asked Questions on NCERT Class 11 Maths Chapter 6 : Linear Inequalities

Q1. How should linear inequalities in two variables be expressed?

Answer: Linear inequalities in two variables can be written as ax + by < c or ax + by ≤ c or ax + by > c or ax + by ≥ c.

Q2. What is the difference between literal and numerical inequalities?

Answer: Literal inequalities are ones that include both variables and numbers. For example, x < 3, y > 8, and x < 4 etc. In contrast, numerical inequalities solely include numbers. For example, 5 < 7, 9 > 2, and 2 < 1, etc.

Q3. How many exercises are there in the Linear Inequalities Class 11 NCERT Solutions?

Answer: There are 50 questions in three exercises in Class 11 Maths Chapter 6 Linear Inequalities, including 15 problems in a miscellaneous practice. These tasks largely focus on the understanding of linear inequalities, their algebraic solutions, and their graphical depiction. The solutions to the exercise-specific problems are also accessible in PDF format, with the goal of assisting students in performing well in the yearly exam.

Q4. What are the key subtopics in Linear Inequalities Class 11 Maths Chapter 6 that could be tested?

Answer: NCERT Solutions Class 11 Maths Chapter 6 begins with a brief overview of linear inequalities and associated concepts. It also discusses some of the most significant algebraic ideas for solving linear inequalities in one and two variables, as well as their graphical representation and pictorial method for solving linear inequalities.

Q5. Is it necessary to practice all of the questions in Linear Inequalities Class 11 NCERT Solutions?

Answer: Linear Inequalities Class 11 NCERT Solutions are well-designed to help students study Linear Inequalities easily. It thoroughly covers all essential principles, allowing students to grasp the concepts. These solutions use examples to clarify the subjects presented so that students may easily relate to the concepts being addressed.

Related Chapters

Chapter 1 : Sets

Chapter 2 : Relations and Functions

Chapter 3 : Trigonometric Functions

Chapter 4 : Principle of Mathematical Induction

Chapter 5 : Complex Numbers and Quadratic Equations

Chapter 7 : Permutations and Combinations

Chapter 8 : Binomial Theorem

Chapter 9 : Sequences and Series

Chapter 10 : Straight Lines

Chapter 11 : Conic Sections

Chapter 12 : Introduction to Three Dimensional Geometry