# Linear Inequations ML Aggarwal Solutions ICSE Maths Class-10

**Linear Inequations ML Aggarwal** Solutions ICSE Maths Class-10 Chapter-4 . We Provide Step by Step Answer of Exercise-4 **Linear Inequations**** **, with MCQs and Chapter-Test Questions / Problems related Exercise-4 **Linear Inequations**** ** for ICSE Class-10 APC Understanding Mathematics . Visit official Website **CISCE ** for detail information about ICSE Board Class-10.

**Linear Inequations ML Aggarwal Solutions ICSE Maths Class-10**

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**ML Solution for ICSE Maths Class 10th Linear Inequations Chapter 4**

**Exercise 4**

**Question 1**

Solve the inequation 3x -11 < 3 where x ∈ {1, 2, 3,……, 10}. Also represent its solution on a number line

**Answer 1**

3x – 11 < 3 => 3x < 3 + 11 => 3x < 14 x <

But x ∈ 6 {1, 2, 3, ……., 10}

Solution set is (1, 2, 3, 4}. Solution set on number line

**Question 2**

Solve 2(x – 3)< 1, x ∈ {1, 2, 3, …. 10}

**Answer 2**

2(x – 3) < 1 => x – 3 < => x < + 3 => x <

But x ∈ {1, 2, 3 …..10}

Solution set = {1, 2, 3} Ans.

**Question 3**

Solve : 5 – 4x > 2 – 3x, x ∈ W. Also represent its solution on the number line.

**Answer 3**

5 – 4x > 2 – 3x

– 4x + 3x > 2 – 5

=> – x > – 3

=> x < 3

x ∈ w,

solution set {0, 1, 2}

Solution set on Number Line :

**Question 4**

List the solution set of 30 – 4 (2.x – 1) < 30, given that x is a positive integer.

**Answer 4**

30 – 4 (2x – 1) < 30

30 – 8x + 4 < 30

– 8x < 30 – 30 – 4

– 8x < – 4 x >

=> x >

x is a positive integer

x = {1, 2, 3, 4…..} Ans.

**Question 5**

Solve : 2 (x – 2) < 3x – 2, x ∈ { – 3, – 2, – 1, 0, 1, 2, 3} .

**Answer 5**

2(x – 2) < 3x – 2

=> 2x – 4 < 3x – 2

=> 2x – 3x < – 2 + 4

=> – x < 2

=> x > – 2

Solution set = { – 1, 0, 1, 2, 3} Ans.

**Question 6**

If x is a negative integer, find the solution set of + (x + 1) > 0.

**Answer 6**

+ x + > 0

=> x + 1 > 0

=> x > – 1

x is a negative integer

Solution set = {- 2, – 1} Ans.

**Question 7**

Solve: ≥, x ∈ {0, 1, 2,…,8}

**Answer 7**

≥

=> 2x – 3 ≥

**Question 8**

Solve x – 3 (2 + x) > 2 (3x – 1), x ∈ { – 3, – 2, – 1, 0, 1, 2, 3}. Also represent its solution on the number line.

**Answer 8**

x – 3 (2 + x) > 2 (3x – 1)

=> x – 6 – 3x > 6x – 2

=> x – 3x – 6x > – 2 + 6

=> – 8x > 4

=> x < => x <

x ∈ { – 3, – 2, – 1, 0, 1, 2}

.’. Solution set = { – 3, – 2, – 1}

Solution set on Number Line :

**Question 9**

Given x ∈ {1, 2, 3, 4, 5, 6, 7, 9} solve x – 3 < 2x – 1.

**Answer 9**

x – 3 < 2x – 1

x – 2x < – 1 + 3 => – x < 2 x > – 2

But x ∈ {1, 2, 3, 4, 5, 6, 7, 9}

Solution set = {1, 2, 3, 4, 5, 6, 7, 9} Ans.

**Question 10**

Given A = {x : x ∈ I, – 4 ≤ x ≤ 4}, solve 2x – 3 < 3 where x has the domain A Graph the solution set on the number line.

**Answer 10**

2x – 3 < 3 => 2x < 3 + 3 => 2x < 6 => x < 3

But x has the domain A = {x : x ∈ I – 4 ≤ x ≤ 4}

Solution set = { – 4, – 3, – 2, – 1, 0, 1, 2}

Solution set on Number line :

**Question 11**

List the solution set of the inequation

+ 8x > 5x , x ∈ Z

**Answer 11**

+8x > 5x

**Question 12**

List the solution set of ≥ + ,

x ∈ N

**Answer 12**

≥ +

=> 88 – 16x ≥ 45 – 15x + 30

(L.C.M. of 8, 5, 4 = 40}

=> – 16x + 15x ≥ 45 + 30 – 88

=> – x ≥ – 13

=>x ≤ 13

x ≤ N.

Solution set = {1, 2, 3, 4, 5, .. , 13} Ans.

**Question 13**

Find the values of x, which satisfy the inequation : , x ∈ N.

Graph the solution set on the number line. (2001)

**Answer 13**

, x ∈ N

**Question 14**

If x ∈ W, find the solution set of

Also graph the solution set on the number line, if possible.

**Answer 14**

9x – (10x – 5) > 15 (L.C.M. of 5, 3 = 15)

=> 9x – 10x + 5 > 15

=> – x > 15 – 5

=> – x > 10

=> x < – 10

But x ∈ W

Solution set = Φ

Hence it can’t be represented on number line.

**Question 15**

Solve:

(i) where x is a positive odd integer.

(ii) where x is positive even integer.

**Answer 15**

(i)

**Question 16**

Given that x ∈ I, solve the inequation and graph the solution on the number line :

(2004)

**Answer 16**

and

**Question 17**

Given x ∈ {1, 2, 3, 4, 5, 6, 7, 9}, find the values of x for which -3 < 2x – 1 < x + 4.

**Answer 17**

-3 < 2x – 1 < x + 4.

=> – 3 < 2x – 1 and 2x – 1 < x + 4

=> – 2x < – 1 + 3 and 2x – x < 4 + 1

=> – 2x < 2 and x < 5

=> – x < 1

=> x > – 1

– 1 < x < 5

x ∈ {1, 2, 3, 4, 5, 6, 7, 9}

Solution set = {1, 2, 3, 4} Ans.

**Question 18**

Solve : 1 ≥ 15 – 7x > 2x – 27, x ∈ N

**Answer 18**

1 ≥ 15 – 7x > 2x – 27

1 ≥ 15 – 7x and 15 – 7x > 2x – 27

**Question 19**

If x ∈ Z, solve 2 + 4x < 2x – 5 ≤ 3x. Also represent its solution on the number line.

**Answer 19**

2 + 4x < 2x – 5 ≤ 3x

2 + 4x < 2x – 5 and 2x – 5 ≤ 3x => 4x – 2x < – 5 – 2 ,and 2x – 3x ≤ 5

**Question 20**

Solve the inequation = 12 + ≤ 5 + 3x, x ∈ R. Represent the solution on a number line. (1999)

**Answer 20**

12 + ≤ 5 + 3x

**Question 21**

Solve : ≤ x ∈ R and represent the solution set on the number line.

**Answer 21**

≤

=> 8x – 20 ≤ 15x – 21

(L.C.M. of 3, 2 = 6)

**Question 22**

Solve > 1, x ∈ R and represent the solution set on the number line.

**Answer 22**

> 1

=> 9x – (10x – 5) > 15

=> 9x – 10x + 5 > 15

=> – x > 15 – 5

=> – x > 10

=> x < – 10

x ∈ R.

.’. Solution set = {x : x ∈R, x < – 10}

Solution set on the number line

**Question 23**

Solve the inequation – 3 ≤ 3 – 2x < 9, x ∈ R. Represent your solution on a number line. (2000)

**Answer 23**

– 3 ≤ 3 – 2x < 9

– 3 ≤ 3 – 2x and 3 – 2x < 9

2x ≤ 3 + 3 and – 2x < 9 – 3

2x ≤ 6 and – 2x < 6 => x ≤ 3 and – x < 3 => x ≤ – 3 and – 3 < x

– 3 < x ≤ 3.

Solution set= {x : x ∈ R, – 3 < x ≤ 3)

Solution on number line

**Question 24**

Solve 2 ≤ 2x – 3 ≤ 5, x ∈ R and mark it on number line. (2003)

**Answer 24**

2 ≤ 2x – 3 ≤ 5 .or 2 ≤ 2x – 3 and 2x – 3 ≤ 5 or 2 + 3 ≤ 2x and 2x ≤ 5 + 3

5 ≤ 2x and 2x ≤ 8.

**Question 25**

Given that x ∈ R, solve the following inequation and graph the solution on the number line: – 1 ≤ 3 + 4x < 23. (2006)

**Answer 25**

We have

– 1 ≤ 3 + 4x < 23 => – 1 – 3 ≤ 4x < 23 – 3 => – 4 ≤ 4x < 20 => – 1 ≤ x < 5, x ∈ R

Solution Set = { – 1 ≤ x < 5; x ∈ R}

**Question 26**

Solve tlie following inequation and graph the solution on the number line. (2007)

x∈R

**Answer 26**

Given x∈R

Multiplying by 3, L.C.M. of fractions, we get

**Question 27**

Solve the following inequation and represent the solution set on the number line :

**Answer 27**

**Question 28**

Solve . Also graph the solution set on the number line

**Answer 28**

**Question 29**

Solving the following inequation, write the solution set and represent it on the number line. – 3(x – 7)≥15 – 7x > , n ∈R

**Answer 29**

– 3(x – 7)≥15 – 7x > , n ∈R

**Question 30**

Solving the following inequation , write the solution set and represent it on the real number line. -2+10x ≤13x +10<24+10x, x ∈ Z.

**Answer 30**

**Question 31**

Solve the inequation 2x – 5 ≤ 5x + 4 < 11, where x ∈ I. Also represent the solution set on the number line. (2011)

**Answer 31**

2x – 5 ≤ 5x + 4 < 11 2x – 5 ≤ 5x + 4

=> 2x – 5 – 4 ≤ 5x and 5x + 4 < 11

=> 2x – 9 ≤ 5x and 5x < 11 – 4

and 5x < 7

=> 2x – 5x ≤ 9 and x <

=> 3x > – 9 and x< 1.4

=> x > – 3

**Question 32**

If x ∈ I, A is the solution set of 2 (x – 1) < 3 x – 1 and B is the solution set of 4x – 3 ≤ 8 + x, find A ∩B.

**Answer 32**

2 (x – 1) < 3 x – 1

2x – 2 < 3x – 1

2x – 3x < – 1 + 2 => – x < 1 x > – 1

Solution set A = {0, 1, 2, 3, ..,.}

4x – 3 ≤ 8 + x

4x – x ≤ 8 + 3

=> 3x ≤ 11

=> x ≤

Solution set B = {3, 2, 1, 0, – 1…}

A ∩ B = {0, 1, 2, 3} Ans.

**Question 33**

If P is the solution set of – 3x + 4 < 2x – 3, x ∈ N and Q is the solution set of 4x – 5 < 12, x ∈ W, find

(i) P ∩ Q

(ii) Q – P.

**Answer 33**

(i) – 3 x + 4 < 2 x – 3

– 3x – 2x < – 3 – 4 => – 5x < – 7

**Question 34**

A = {x : 11x – 5 > 7x + 3, x ∈R} and B = {x : 18x – 9 ≥ 15 + 12x, x ∈R}

Find the range of set A ∩ B and represent it on a number line

**Answer 34**

A = {x : 11x – 5 > 7x + 3, x ∈R}

B = {x : 18x – 9 ≥ 15 + 12x, x ∈R}

Now, A = 11x – 5 > 7x + 3

=> 11x – 7x > 3 + 5

=> 4x > 8

=>x > 2, x ∈ R

**Question 35**

Given: P {x : 5 < 2x – 1 ≤ 11, x∈R)

Q{x : – 1 ≤ 3 + 4x < 23, x∈I) where

R = (real numbers), I = (integers)

Represent P and Q on number line. Write down the elements of P ∩ Q. (1996)

**Answer 35**

P= {x : 5 < 2x – 1 ≤ 11}

5 < 2x – 1 ≤ 11

**Question 36**

**I**f x ∈ I, find the smallest value of x which satisfies the inequation

**Answer 36**

=>

=>12x – 10x > 12 – 15

=> 2x > – 3

=>

Smallest value of x = – 1 Ans.

**Question 37**

Given 20 – 5 x < 5 (x + 8), find the smallest value of x, when

(i) x ∈ I

(ii) x ∈ W

(iii) x ∈ N.

**Answer 37**

20 – 5 x < 5 (x + 8)

⇒ 20 – 5x < 5x + 40

⇒ – 5x – 5x < 40 – 20

⇒ – 10x < 20

⇒ – x < 2

⇒ x > – 2

(i) When x ∈ I, then smallest value = – 1.

(ii) When x ∈ W, then smallest value = 0.

(iii) When x ∈ N, then smallest value = 1. Ans.

**Question 38**

Solve the following inequation and represent the solution set on the number line :

**Answer 38**

We have

Hence, solution set is {x : -4 < x < 5, x ∈ R}

The solution set is represented on the number line as below.

**Question 39**

Solve the given inequation and graph the solution on the number line :

2y – 3 < y + 1 ≤ 4y + 7; y ∈ R.

**Answer 39**

2y – 3 < y + 1 ≤ 4y + 7; y ∈ R.

(a) 2y – 3 < y + 1

⇒ 2y – y < 1 + 3

⇒ y < 4

⇒ 4 > y ….(i)

(b) y + 1 ≤ 4y + 7

**Question 40**

Solve the inequation and represent the solution set on the number line.

**Answer 40**

Given :

**Question 41**

Find the greatest integer which is such that if 7 is added to its double, the resulting number becomes greater than three times the integer.

**Answer 41**

Let the greatest integer = x

According to the condition,

2x + 7 > 3x

⇒ 2x – 3x > – 7

⇒ – x > – 7

⇒ x < 7

Value of x which is greatest = 6 Ans.

**Question 42**

One-third of a bamboo pole is buried in mud, one-sixth of it is in water and the part above the water is greater than or equal to 3 metres. Find the length of the shortest pole.

**Answer 42**

Let the length of the shortest pole = x metre

Length of pole which is burried in mud =

Length of pole which is in the water =

**Linear Inequations Chapter-4 ,ML Aggarwal Solution for ICSE Maths Class 10th**

**MULTIPLE CHOICE QUESTIONS**

**Choose the correct answer from the given four options (1 to 5) :**

**Question 1**

If x ∈ { – 3, – 1, 0, 1, 3, 5}, then the solution set of the inequation 3x – 2 ≤ 8 is

(a) { – 3, – 1, 1, 3}

(b) { – 3, – 1, 0, 1, 3}

(c) { – 3, – 2, – 1, 0, 1, 2, 3}

(d) { – 3, – 2, – 1, 0, 1, 2}

**Answer 1**

x ∈ { -3, -1, 0, 1, 3, 5}

3x – 2 ≤ 8

….⇒ 3x ≤ 8 + 2

and ⇒ 3x ≤ 10

so ⇒ x ≤

therefore ⇒ x <

Solution set = { -3, -1, 0, 1, 3} (b)

**Question 2**

If x ∈ W, then the solution set of the inequation 3x + 11 ≥ x + 8 is

(a) { – 2, – 1, 0, 1, 2, …}

(b) { – 1, 0, 1, 2, …}

(c) {0, 1, 2, 3, …}

(d) {x : x∈R,x≥}

**Answer 2**

x ∈ W

3x + 11 ≥ x + 8

⇒ 3x – x ≥ 8 – 11

**Question 3**

If x ∈ W, then the solution set of the inequation 5 – 4x ≤ 2 – 3x is

(a) {…, – 2, – 1, 0, 1, 2, 3}

(b) {1, 2, 3}

(c) {0, 1, 2, 3}

(d) {x : x ∈ R, x ≤ 3}

**Answer 3**

x ∈ W

5 – 4x < 2 – 3x

⇒ 5 – 2 ≤ 3x + 4x

⇒ 3 ≤ x

Solution set = {0, 1, 2, 3,} (c)

**Question 4**

If x ∈ I, then the solution set of the inequation 1 < 3x + 5 ≤ 11 is

(a) { – 1, 0, 1, 2}

(b) { – 2, – 1, 0, 1}

(c) { – 1, 0, 1}

(d) {x : x ∈ R, < x ≤ 2}

**Answer 4**

x ∈ I

1 < 3x + 5 ≤ 11

⇒ 1 < 3x + 5

⇒ 1 – 5 < 3x

**Question 5**

If x ∈ R, the solution set of 6 ≤ – 3 (2x – 4) < 12 is

(a) {x : x ∈ R, 0 < x ≤ 1}

(b) {x : x ∈ R, 0 ≤ x < 1}

(c) {0, 1}

(d) none of these

**Answer 5**

x ∈ R

6 ≤ – 3(2x – 4) < 12

⇒ 6 ≤ – 3(2x – 4)

⇒ 6 ≤ – 6x + 12

**Solution of ML Aggarwal Linear Inequations Chapter 4 for ICSE Maths Class 10**

**CHAPTER TEST **

**Question 1**

Solve the inequation : 5x – 2 ≤ 3(3 – x) where x ∈ { – 2, – 1, 0, 1, 2, 3, 4}. Also represent its solution on the number line.

**Answer 1**

5x – 2 < 3(3 – x)

⇒ 5x – 2 ≤ 9 – 3x

⇒ 5x + 3x ≤ 9 + 2

**Question 2**

Solve the inequations :

6x – 5 < 3x + 4, x ∈ I.

**Answer 2**

6x – 5 < 3x + 4

6x – 3x < 4 + 5

⇒ 3x <9

⇒ x < 3

x ∈ I

Solution Set = { -1, -2, 2, 1, 0….. }

**Question 3**

Find the solution set of the inequation

x + 5 < 2 x + 3 ; x ∈ R

Graph the solution set on the number line.

**Answer 3**

x + 5 ≤ 2x + 3

x – 2x ≤ 3 – 5

⇒ -x ≤ -2

⇒ x ≥ 2

**Question 4**

If x ∈ R (real numbers) and – 1 < 3 – 2x ≤ 7, find solution set and represent it on a number line.

**Answer 4**

-1 < 3 – 2x ≤ 7

-1 < 3 – 2x and 3 – 2x ≤ 7

⇒ 2x < 3 + 1 and – 2x ≤ 7 – 3

⇒ 2x < 4 and -2x ≤ 4

⇒ x < 2 and -x ≤ 2

and x ≥ -2 or -2 ≤ x

x ∈ R

Solution set -2 ≤ x < 2

Solution set on number line

**Question 5**

Solve the inequation :

**Answer 5**

**Question 6**

Find the range of values of a, which satisfy 7 ≤ – 4x + 2 < 12, x ∈ R. Graph these values of a on the real number line.

**Answer 6**

7 < – 4x + 2 < 12

7 < – 4x + 2 and – 4x + 2 < 12

**Question 7**

If x∈R, solve

**Answer 7**

and

**Question 8**

Find positive integers which are such that if 6 is subtracted from five times the integer then the resulting number cannot be greater than four times the integer.

**Answer 8**

Let the positive integer = x

According to the problem,

5a – 6 < 4x

⇒ 5a – 4x < 6

⇒ x < 6

Solution set = {x : x < 6}

= { 1, 2, 3, 4, 5, 6}

**Question 9**

Find three smallest consecutive natural numbers such that the difference between one-third of the largest and one-fifth of the smallest is at least 3.

**Answer 9**

Let first least natural number = x

then second number = x + 1

and third number = x + 2