NCERT Solutions For Class 11 Maths Chapter 5 Linear Inequalities Exercise 5.1

NCERT Solutions for Class 11 Maths Chapter 5 Exercise 5.1 Linear Inequalities help students understand the basic concepts of linear inequalities in one variable. This exercise introduces students to solving inequalities and representing their solutions on the number line.

NCERT Solutions For Class 11 Maths Chapter 5 Linear Inequalities Exercise 5.1

Linear Inequalities

Easy

Q.

Solve 24x < 100, when
(i) x is a natural number.
(ii) x is an integer.

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Linear Inequalities

Difficult

Q.

Solve the following system of inequalities graphically:
$\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{\le }\mathbf{9}\mathbf{,}\mathbf{y}\mathbf{>}\mathbf{x}\mathbf{,}\mathbf{x}\mathbf{\ge }\mathbf{0}$

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Linear Inequalities

Medium

Q.

Solve the following inequality graphically using two-dimensional plane : y < – 2

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Linear Inequalities

Easy

Q.

Solve the following inequality graphically using two-dimensional plane : x > – 3

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Linear Inequalities

Medium

Q.

Solve the following system of inequalities graphically:
 $\mathbf{x}\mathbf{\ge }\mathbf{3}\mathbf{,}\mathbf{y}\mathbf{\ge }\mathbf{2}$ 

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Linear Inequalities

Medium

Q.

Solve the following system of inequalities graphically:
 $\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\le }\mathbf{12}\mathbf{,}\mathbf{x}\mathbf{\ge }\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\ge }\mathbf{1}$ 

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Linear Inequalities

Difficult

Q.

$\mathbf{\text{Solve the following system of inequalities graphically: }}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{\ge }\mathbf{6}\mathbf{,}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{\le }\mathbf{12}$

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Linear Inequalities

Medium

Q.

$\mathbf{\text{Solve the following system of inequalities graphically: }}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{\ge }\mathbf{4}\mathbf{,}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{\ge }\mathbf{0}$

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Linear Inequalities

Medium

Q.

Solve the following system of inequalities graphically:
$\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{>}\mathbf{1}\mathbf{,}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{<}\mathbf{-}\mathbf{1}$

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Linear Inequalities

Medium

Q.

Solve the following system of inequalities graphically:
$\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{\le }\mathbf{6}\mathbf{,}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{\ge }\mathbf{4}$

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Linear Inequalities

Difficult

Q.

Solve the following system of inequalities graphically:
$\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{\ge }\mathbf{8}\mathbf{,}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\ge }\mathbf{10}$

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Linear Inequalities

Difficult

Q.

$\mathbf{\text{Solve the following system of inequalities graphically: }}\mathbf{5}\mathbf{x}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{\le }\mathbf{60}\mathbf{,}\mathbf{x}\mathbf{\ge }\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\ge }\mathbf{2}$

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Linear Inequalities

Medium

Q.

Solve the following inequality graphically using two-dimensional plane :  $-3\mathrm{x}+2\mathrm{y}\ge {\hspace{0.17em}}-6$ 

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Linear Inequalities

Difficult

Q.

$\begin{array}{l}\mathbf{\text{Solve the following system of inequalities graphically:}}\\ \mathbf{3}\mathbf{x}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{\le }\mathbf{60}\mathbf{,}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\le }\mathbf{30}\mathbf{,}\mathbf{x}\mathbf{\ge }\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\ge }\mathbf{0}\end{array}$

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Linear Inequalities

Difficult

Q.

$\mathbf{\text{Solve the following system of inequalities graphically: }}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{\ge }\mathbf{4}\mathbf{,}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{\le }\mathbf{3}\mathbf{,}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{\le }\mathbf{6}$

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Linear Inequalities

Difficult

Q.

$\mathbf{Solve}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{following}\mathbf{ }\mathbf{system}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{inequalities}\mathbf{ }\mathbf{graphically}\mathbf{:}\mathbf{ }\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{\le }\mathbf{3}\mathbf{,}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{\ge }\mathbf{12}\mathbf{,}\mathbf{x}\mathbf{\ge }\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\ge }\mathbf{1}$

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Linear Inequalities

Difficult

Q.

$\mathbf{Solve}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{following}\mathbf{ }\mathbf{system}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{inequalities}\mathbf{ }\mathbf{graphically}\mathbf{:}\mathbf{ }\mathbf{4}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\le }\mathbf{60}\mathbf{,}\mathbf{y}\mathbf{\ge }\mathbf{2}\mathbf{x}\mathbf{,}\mathbf{x}\mathbf{\ge }\mathbf{3}\mathbf{,}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{\ge }\mathbf{0}$

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Linear Inequalities

Difficult

Q.

$\begin{array}{l}\text{Solve the following system of inequalities graphically:}\\ 3\text{x}+2\text{y}\le 150,\text{\hspace{0.33em}x}+4\text{y}\le 80,\text{x}\le 15,\text{x}\ge 0,\text{\hspace{0.33em}y}\ge 0\end{array}$

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Linear Inequalities

Difficult

Q.

$\begin{array}{l}\text{Solve the following system of inequalities graphically:}\\ \mathrm{x}+2\mathrm{y}\le 10,\mathrm{x}+\mathrm{y}\ge 1,\mathrm{x}-\mathrm{y}\le 0,\mathrm{x}\ge 0,\mathrm{y}\ge 0\end{array}$

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Linear Inequalities

Difficult

Q.

Solve the inequalities in Exercises 7 to 10 and represent the solution graphically on number line.

7. 5x + 1 > – 24, 5x – 1 < 24
8. 2 (x – 1) < x + 5, 3 (x + 2) > 2 – x
9. 3x – 7 > 2(x – 6), 6 – x >11–2x
10.$\mathbf{ }\mathbf{5}\mathbf{(}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{7}\mathbf{)}\mathbf{-}\mathbf{3}\mathbf{(}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{)}\mathbf{\le }\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{19}\mathbf{\le }\mathbf{6}\mathbf{x}\mathbf{+}\mathbf{47}\mathbf{.}$

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Linear Inequalities

Difficult

Q.

$\begin{array}{l}\text{A solution is to be kept between 68° F and 77° F. What is }\\ \text{the range in temperature in degree Celsius (C) if the }\\ \text{Celsius / Fahrenheit (F) conversion formula is given by}\\ \text{F = }\frac{\text{9}}{\text{5}}\text{C + 32 ?}\end{array}$

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Linear Inequalities

Difficult

Q.

A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?

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Linear Inequalities

Difficult

Q.

How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

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Linear Inequalities

Medium

Q.

Solve the following inequality graphically using two-dimensional plane :  $3\mathrm{y}-5\mathrm{x}<30$ 

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Linear Inequalities

Medium

Q.

Solve the following inequality graphically using two-dimensional plane :  $2\mathrm{x}-3\mathrm{y} > 6$ 

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Linear Inequalities

Medium

Q.

Solve – 12x > 30, when
(i) x is a natural number.
(ii) x is an integer.

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Linear Inequalities

Difficult

Q.

$\frac{\mathbf{(}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}{\mathbf{3}}\mathbf{\ge }\frac{\mathbf{(}\mathbf{3}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{)}}{\mathbf{4}}\mathbf{-}\frac{\mathbf{(}\mathbf{2}\mathbf{-}\mathbf{x}\mathbf{)}}{\mathbf{5}}$

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Linear Inequalities

Medium

Q.

Solve 5x – 3 < 7, when
(i) x is an integer.
(ii) x is a real number.

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Linear Inequalities

Medium

Q.

Solve 3x + 8 > 2, when
(i) x is an integer.
(ii) x is a real number.

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Linear Inequalities

Medium

Q.

4x + 3 < 5x + 7

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Linear Inequalities

Medium

Q.

3x – 7 > 5x – 1

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Linear Inequalities

Medium

Q.

3(x – 1) ≤ 2 (x – 3)

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Linear Inequalities

Medium

Q.

3(2 – x) ≥ 2 (1 – x)

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Linear Inequalities

Difficult

Q.

$\frac{3(\mathrm{x}-2)}{5}\le \frac{5(2-\mathrm{x})}{3}$

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Linear Inequalities

Difficult

Q.

$\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\frac{\mathbf{3}\mathbf{x}}{\mathbf{5}}\mathbf{+}\mathbf{4}\mathbf{)}\mathbf{\ge }\frac{\mathbf{1}}{\mathbf{3}}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{6}\mathbf{)}$

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Linear Inequalities

Medium

Q.

$\mathbf{37}\mathbf{–}\mathbf{(}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{5}\mathbf{)}\mathbf{\ge }\mathbf{9}\mathbf{x}\mathbf{–}\mathbf{8}\mathbf{(}\mathbf{x}\mathbf{–}\mathbf{3}\mathbf{)}$

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Linear Inequalities

Difficult

Q.

$\frac{\mathbf{x}}{\mathbf{4}}\mathbf{<}\frac{\mathbf{(}\mathbf{5}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{)}}{\mathbf{3}}\mathbf{-}\frac{\mathbf{(}\mathbf{7}\mathbf{x}\mathbf{-}\mathbf{3}\mathbf{)}}{\mathbf{5}}$

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Linear Inequalities

Medium

Q.

3x – 2 < 2x + 1

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Linear Inequalities

Medium

Q.

Solve the following inequality graphically using two-dimensional plane :  $\mathrm{x}-\mathrm{y}\le 2$ 

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Linear Inequalities

Difficult

Q.

$\frac{\mathbf{x}}{\mathbf{2}}\mathbf{\ge }\frac{\mathbf{(}\mathbf{5}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{)}}{\mathbf{3}}\mathbf{-}\frac{\mathbf{(}\mathbf{7}\mathbf{x}\mathbf{-}\mathbf{3}\mathbf{)}}{\mathbf{5}}$

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Linear Inequalities

Difficult

Q.

Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks.

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Linear Inequalities

Difficult

Q.

To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.

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Linear Inequalities

Difficult

Q.

Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.

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Linear Inequalities

Difficult

Q.

Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.

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Linear Inequalities

Difficult

Q.

The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.

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Linear Inequalities

Difficult

Q.

A man wants to cut three lengths from a single piece of board of length 91cm.
The second length is to be 3cm longer than the shortest and the third length is to twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second?

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Linear Inequalities

Medium

Q.

Solve the following inequality graphically using two-dimensional plane :  $2\mathrm{x}+\mathrm{y}\ge 6$ 

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Linear Inequalities

Medium

Q.

 Solve the following inequality graphically using two-dimensional plane :  $3\mathrm{x}+4\mathrm{y}\le 12$ 

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Linear Inequalities

Medium

Q.

Solve the following inequality graphically using two-dimensional plane :  $\mathrm{y}+8\ge 2\mathrm{x}$ 

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Linear Inequalities

Medium

Q.

$\begin{array}{l}\text{IQ of a person is given by the formula}\\ \text{IQ=}\frac{\text{MA}}{\text{CA}}\text{×100,}\\ \begin{array}{l}\text{where MA is mental age and CA is chronological age.}\\ \text{If 80}\le \text{IQ}\le \text{140 for a group of 12 years old children,}\\ \text{find the range of their mental age.}\end{array}\end{array}$

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Prepared according to the latest CBSE Class 11 Maths syllabus, Exercise 5.1 focuses on solving linear inequalities, understanding inequality symbols, and writing the solution set in interval form. Regular practice of this exercise helps students build a strong foundation for more advanced algebraic concepts.

The solutions are explained in a clear and step-by-step format so students can easily understand the method and avoid common mistakes in exams.

1. Solve $24x<100$, when:

(i) x is a natural number
$24x<100$
$\Rightarrow x<\frac{25}{6}$

Natural numbers less than $\frac{25}{6}$ are: 1, 2, 3, 4

Solution set: $\{1,2,3,4\}$

(ii) x is an integer
$24x<100$
$\Rightarrow x<\frac{25}{6}$

Integers less than $\frac{25}{6}$ are: ..., -3, -2, -1, 0, 1, 2, 3, 4

Solution set: $\{\mathrm{.}\mathrm{.}\mathrm{.},-3,-2,-1,0,1,2,3,4\}$

2. Solve $-12x>30$, when:

(i) x is a natural number
$-12x>30$
$\Rightarrow x<-\frac{5}{2}$

There is no natural number less than $-\frac{5}{2}$

Solution set: No solution

(ii) x is an integer
$-12x>30$
$\Rightarrow x<-\frac{5}{2}$

Integers satisfying this are: ..., -5, -4, -3

Solution set: $\{\mathrm{.}\mathrm{.}\mathrm{.},-5,-4,-3\}$

3. Solve $5x-3<7$, when:

(i) x is an integer
$5x-3<7$
$\Rightarrow 5x<10$
$\Rightarrow x<2$

Integers satisfying this are: ..., -4, -3, -2, -1, 0, 1

Solution set: $\{\mathrm{.}\mathrm{.}\mathrm{.},-4,-3,-2,-1,0,1\}$

(ii) x is a real number
$5x-3<7$
$\Rightarrow x<2$

Solution set: $(-\mathrm{\infty },2)$

4. Solve $3x+8>2$, when:

(i) x is an integer
$3x+8>2$
$\Rightarrow 3x>-6$
$\Rightarrow x>-2$

Integers satisfying this are: -1, 0, 1, 2, 3, ...

Solution set: $\{-1,0,1,2,3,\mathrm{.}\mathrm{.}\mathrm{.}\}$

(ii) x is a real number
$3x+8>2$
$\Rightarrow x>-2$

Solution set: $(-2,\mathrm{\infty })$

5. Solve for real $x$: $4x+3<5x+7$

$4x+3<5x+7$
$\Rightarrow -4<x$

Solution set: $(-4,\mathrm{\infty })$

6. Solve for real $x$: $3x-7>5x-1$

$3x-7>5x-1$
$\Rightarrow -6>2x$
$\Rightarrow x<-3$

Solution set: $(-\mathrm{\infty },-3)$

7. Solve for real $x$: $3(x-1)\le 2(x-3)$

$3x-3\le 2x-6$
$\Rightarrow x\le -3$

Solution set: $(-\mathrm{\infty },-3]$

8. Solve for real $x$: $3(2-x)\ge 2(1-x)$

$6-3x\ge 2-2x$
$\Rightarrow -x\ge -4$
$\Rightarrow x\le 4$

Solution set: $(-\mathrm{\infty },4]$

9. Solve for real $x$: $x+\frac{x}{2}+\frac{x}{3}<11$

$$x(1+\frac{1}{2}+\frac{1}{3})<11$$$$x\left(\frac{11}{6}\right)<11\Rightarrow x<6$$

Solution set: $(-\mathrm{\infty },6)$

10. Solve for real $x$: $\frac{x}{3}>\frac{x}{2}+1$

$$\frac{x}{3}-\frac{x}{2}>1\Rightarrow -\frac{x}{6}>1\Rightarrow x<-6$$

Solution set: $(-\mathrm{\infty },-6)$

11. Solve for real $x$: $\frac{3(x-2)}{5}\le \frac{5(2-x)}{3}$

$$9(x-2)\le 25(2-x)$$$$9x-18\le 50-25x$$$$34x\le 68\Rightarrow x\le 2$$

Solution set: $(-\mathrm{\infty },2]$

12. Solve for real $x$: $\frac{1}{2}(\frac{3x}{5}+4)\ge \frac{1}{3}x-2$

Simplifying,

$$x\le 120$$

Solution set: $(-\mathrm{\infty },120]$

13. Solve for real $x$: $2(2x+3)-10<6(x-2)$

$$4x+6-10<6x-12$$$$4x-4<6x-12$$$$8<2x\Rightarrow x>4$$

Solution set: $(4,\mathrm{\infty })$

14. Solve for real $x$: $37-(3x+5)\ge 9x-8(x-3)$

$$37-3x-5\ge 9x-8x+24$$$$32-3x\ge x+24$$$$8\ge 4x\Rightarrow x\le 2$$

Solution set: $(-\mathrm{\infty },2]$

15. Solve for real $x$: $\frac{x}{4}<\frac{5x-2}{3}-\frac{7x-3}{5}$

Simplifying,

$$x>4$$

Solution set: $(4,\mathrm{\infty })$

FAQs – Class 11 Maths Chapter 5 Exercise 5.1 Linear Inequalities

Q1. What is the focus of Exercise 5.1?
Exercise 5.1 focuses on solving linear inequalities in one variable and representing their solutions correctly.

Q2. What are linear inequalities?
Linear inequalities are algebraic expressions involving inequality signs such as <, >, ≤, ≥, instead of equality.

Q3. Why is Exercise 5.1 important for exams?
This exercise forms the base for solving more complex inequalities and is important for algebra-based questions in Class 11 exams.

Q4. How can students prepare effectively for Exercise 5.1?
Students should practice solving different types of inequalities, understand how to change inequality signs when multiplying or dividing by negative numbers, and represent solutions accurately on the number line.