# ML Aggarwal Solutions for Chapter 4 Linear Inequation Class 10 Maths ICSE

Here, we are providing the solutions for Chapter 4 Linear Inequation from ML Aggarwal Textbook for Class 10 ICSE Mathematics. Solutions of the fourth chapter has been provided in detail. This will help the students in understanding the chapter more clearly. Class 10 Chapter 4 Linear Inequation ML Aggarwal Solutions for ICSE is one of the most important chapter for the board exams which is based on the equations of algebra, word problems, representation on number line and solutions of set.

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

3x – 1 < 3

⇒ 3x < 3 + 11

⇒ 3x < 14x < 14/3

But x ∈ 6 {1, 2, 3, 4}.

Solution set on number line

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

2(x – 3) < 1

⇒ x – 3 < 1/2

⇒ x < 1/2 + 3

⇒ x < 3.1/2

But x ∈ {1, 2, 3….. 10}

Solution set = {1, 2, 3}

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

5 – 4x > 2 – 3x

⇒ -4x + 3x > 2 – 5

⇒ - x > - 3

x ∈ w,

Solution set {0, 1, 2}

Solution set on number line:

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

30 – 4 (2x – 1) < 30

⇒ 30 – 8x + 4 < 30

⇒ - 8x < 30 – 30 – 4

⇒ - 8x < - 4x > -4/-8

⇒ x > 1/2

x is a positive integer

x = {1, 2, 3, 4, …. }

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

2(x – 2) < 3x – 2

⇒ 2x – 4 < 3x – 2

⇒ 2x – 3x < - 2 + 4

⇒ - x < 2

⇒ x > - 2

Solution set = {- 1, 0, 1, 2, 3}

6. If x is a negative integer, find the solution set of 2/3 + 1/3 (x + 1) > 0.

2/3 + 1/3x + 1/3 > 0

⇒ 1/3x + 1 > 0

⇒ 1/3x > - 1

⇒ x > - 1 × 3/1 ⇒ x > - 3

7. Solve: (2x – 3)/4 ≥ 1/2, x {0, 1, 2, ….., 8}

(2x – 3)/4 ≥ 1/2

⇒ 2x – 3 ≥ 2 ⇒ 2x ≥ 2 + 3

⇒ 2x ≥ 5 ⇒ x ≥ 5/2

∵ x ∈ {0, 1, 2, ……., 8}

∴ Solution set = {3, 4, 5, 6, 7, 8}

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.

x – 3 (2 + x) > 2(3x - 1)

⇒ x – 6 – 3x > 6x – 2

⇒ x – 3x – 6x > - 2 + 6

⇒ - 8x > 4

⇒ x < -4/8 ⇒ x < - 1/2

x ∈ {-3, -2, -1, 0, 1, 2}

∴ Solution set = {-3, -2, -1}

Solution set on Number line:

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

x – 3 < 2x – 1

⇒ x – 2x < - 1 + 3

⇒ - x < 2x > - 2

But x ∈ {1, 2, 3, 4, 5, 6, 7, 9}

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

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.

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:

11. List the solution set of the inequation 1/2 + 8x > 5x – 3/2, x Z

1/2 + 8x > 5x – 3/2

⇒ 8x – 5x > - 3/2 – 1/2

⇒ 3x > - 2 ⇒ x > - 2/3

∵ x ∈ Z,

∴ Solution set = {0, 1, 2, 3, 4 ,…….}

12. List the solution set of (11 – 2x)/5 ≥ (9 – 3x)/8 + 3/4, x N

(11 – 2x)/5 ≥ (9 – 3x)/8 + 3/4

⇒ 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}

13. Find the values of x, which satisfy the inequation : - 2 ≤ 1/2 – 2x/3 ≤ 1.5/6, x N. Graph the solution set on the number line.

- 2 ≤ 1/2 – 2x/3 ≤ 1.5/6, x ∈ N

⇒ - 2 – 1/2 ≤ 1/2 – 2x/3 – 1/2 ≤ 11/6 – 1/2

[By subtracting 1/2 on both sides of inequality]

⇒ - 5/2 ≤ 2x/3 ≤ 8/6

⇒ - 15 ≤ - 4x ≤ 8

⇒ 15 ≥ 4x ≥ - 8

⇒ 15/4 ≥ x ≥ - 2

3.3/4 ≥ x ≥ - 2

But x ∈ N, hence only possible solution for x = {1, 2, 3}

14. If x W, find the solution set of 3/5x – (2x – 1)/3 > 1

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

3/5x – (2x – 1)/3 > 1

⇒ 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 = Î¦

Hence it can’t be represented on number line.

15. Solve:

(i) x/2 + 5 ≤ x/3 + 6 where x is a positive odd integer.

(ii) (2x + 3)/3 ≥ (3x – 1)/4 where x is positive even integer.

(i) x/2 + 5 ≤ x/3 + 6

⇒ x/2 – x/3, ≤ 6 – 5

⇒ (3x – 2x)/6 ≤ 1

⇒ x/6 ≤ 1

⇒ x ≤ 6

∵ x is a positive odd integer

∴ x = {1, 3, 5}

(ii) (2x + 3)/3 ≥ (3x – 1)/4

⇒ 2x/3 + 3/3 ≥ 3x/4 – 1/4

⇒ 2x/3 – 3x/4 ≥ -1/4 – 1

⇒ (8x – 9x)/12 ≥ - 5/4

⇒ -x/12 ≥ -5/4

⇒ x/12 ≤ 5/4

⇒ x ≤ 5/4 × 12

⇒ x ≤ 15

∵ x is positive even integer

∴ x = {2, 4, 6, 8, 10, 12, 14}

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

3 ≥ (x – 4)/2 + x/3 and 3 ≥ (x – 4)/2 + x/3 ≥ 2

(i) 3 ≥ (3x – 12 + 2x)/6

⇒ 3 ≥ (5x – 12)/6

⇒ 18 ≥ 5x – 12

⇒ 5x – 12 ≤ 18

⇒ 5x ≤ 18 + 12

⇒ 5x ≤ 30

⇒ x ≤ 6

(ii) (x – 4)/2 + x/2 ≥ 2

(3x – 12 + 2x)/6 ≥ 2

⇒ (5x – 12)/6 ≥ 2

⇒ 5x – 12 ≥ 12

⇒ 5x ≥ 12 + 12, x ≥ 24/5

⇒ x ≥ 4.4/5

∴ x = {5, 6}

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

- 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}

18. Solve: 1 ≥ 15 – 7x > 2x – 27, x N

1 ≥ 15 – 7x > 2x – 27

1 ≥ 15 – 7x and 15 – 7x > 2x – 27

⇒ 7x ≥ 15 – 1 and – 7x – 2x > - 27 – 15

⇒ 7x ≥ 14 and – 9x > - 42

⇒ x ≥2 and – x > - 42/9

⇒ 2 ≤ x and – x > - 14/3 and x < 14/3

2 ≤ x < 14/3

But x ∈ N

∴ Solution set = {2, 3, 4}

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

2 + 4x < 2x – 5 ≤ 3x

2 + 4x < 2x – 5 and 2x – 5 ≤ 3x

⇒ 4x – 2x < -5 – 2, and 2x – 3x ≤ 5

⇒ 2x < - 7 and – x ≤ 5

⇒ x < - (7/2) and x ≥ - 5 and – 5 ≤ x

∴ - 5 ≤ x < -(7/2)

∵ x ∈ Z

∴ Solution set = {- 5, -4}

Solution set on Number line

20. Solve the inequation = 12 + 1.5/6x ≤ 5 + 3x , x R. Represent the solution on a number line.

12 + 11/6.x ≤ 5 + 3x

⇒ 72 + 11x ≤ 30 + 18x (Multiplying by 6)

⇒ 11x – 18x ≤ 30 – 72

⇒ - 7x ≤ - 42

⇒ - x ≤ -(42/7)

⇒ - x ≤ - 6

⇒ x ≥ 6

∴ x ∈ R

∴ Solution set = {x : x ∈ R, x ≥ 6}

Solution set on Number line

21. Solve: (4x – 10)/3 ≤ (5x – 7)/2 x R and represent the solution set on the number line.

(4x – 10)/3 ≤ (5x – 7)/2

⇒ 8x – 20 ≤ 15x – 21

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

⇒ 8x – 15x ≤ - 21 + 20

⇒ - 7x ≤ - 1 ⇒ - x ≤ - (1/7)

⇒ x > 1/7

∵ x ∈ R

∴ Solution set = {x : x ∈ R, x > 1/7}

Solution set on the number line

22. Solve 3x/5 – (2x – 1)/3 > 1, x R and represent the solution set on the number line.

3x/5 – (2x – 1)/3 > 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

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

- 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

24. Solve 2 ≤ 2x – 3 ≤ 5, x R and mark it on number line.

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

5/2 ≤ 2x and 2x ≤ 8

5/2 ≤ x and x ≤ 4

∴ 5/2 ≤ x and x ≤ 4

∴ Solution set = {x : x ∈ R, 5/2 ≤ x ≤ 4}

Solution set on number line

25. Given that x R, solve the following inequation and graph the solution on the number line: - 1 ≤ 3 + 4x < 23.

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}

The graph of the solution set is shown below:

26. Solve the following inequation and graph the solution on the number line.

- 2.2/3 ≤ x + 1/3 < 3 + 1/3 x R

Given – 2. 2/3 ≤ x + 1/3 + 1/3 x ∈ R

- 8/3 ≤ x + 1/3 < 10/3

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

- 8 ≤ 3x + 1 < 10

- 8 – 1 ≤ 3x + 1 – 1 < 10 – 1 [Add – 1]

- 9 ≤ 3x < 9

- 3 ≤ x < 3 [Dividing by 3]

Hence, the solution set is {x : x ∈ R, - 3 ≤ x < 3}

The graph of the solution set is shown by the thick portion of the number line. The solid circle at – 3 indicates that the number – 3 is indicates that the number – 3 is included among the solutions whereas the open circle at 3 indicates that 3 is not included among the solutions.

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

- 3 < - (1/2) – 2x/3 ≤ 5.6, x R

- 3 < - (1/2) – 2x/3 ≤ 5/6, x ∈ R

(i) – 3 < - 1/2 – 2x/3 ⇒ - 3 < (1/2 + 2x/3

⇒ - (1/2 + 2x/3) > - 3

⇒ - 2x/3 > - 3 + 1/2

⇒ - 2x/3 > -5/2

⇒ 2x/3 < 5/2

⇒ x < 5/2 × 3/2

⇒ x < 15/4 …(i)

(ii) – (1/2) – 2x/3 ≤ 5/6

⇒ - (2x/3) ≤ 5/6 + 1/2

⇒ -2x/3 ≤ (5 + 3)/6

⇒ -2/3.x ≤ 8/6

⇒ 2/3.x ≥ -8/6

⇒ x ≥ - 8/6 ×3/2

⇒ x ≥ - 2

⇒ - 2 ≤ x …(ii)

From (i) and (ii),

- 2 ≤ x ≤ 15/4

∴ Solution = {x : x ∈ R, - 2x < 15/4}

Now solution on number line

28. Solve (2x + 1)/2 + 2(3 – x) ≤ 7, x R. Also graph the solution set on the number line.

(2x + 1)/2 + 2(3 – x) ≥ 7, x ∈ R

⇒ (2x + 1)/2 + 6 – 2x ≥ 7

⇒ (2x + 1)/2 – 2x ≥ 7 – 6

⇒ (2x + 1 – 4x)/2 ≥ 1

⇒ 2x + 1 – 4x ≥ 2

⇒ - 2x ≥ 2 – 1

⇒ - 2x ≥ 1

⇒ - x ≥ 1/2

⇒ x ≤ - 1/2

∴ Solution set {x: x ∈ R, x ≤ - 1/2}

Solution on number line:

29. Solving the following inequation, write the solution set and represent it on the number line. – 3(x – 7) ≥ 15 – 7x > (x + 1)/3, n R

Solution:

- 3(x – 7) ≥ 15 – 7x > (x + 1)/3, n ∈ R

⇒ -3(x – 7) ≥ 15 – 7x ⇒ - 3x + 21 ≥ 15 – 7x

⇒ - 3x + 7x ≥ 15 – 21 ⇒ 4x ≥ - 6

⇒ x ≥ -6/4

⇒ x ≥ -3/2

⇒ -3/2 ≤ x

And 15 – 7x > (x + 1)/3

⇒ 45 – 21x > x + 1

⇒ 45 – 1 > x + 21x

⇒ 44 > 22x

2 > x ⇒ x = 2

∴ -3/2 ≤ x < 2, x ∈ R

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.

Given, - 2 + 10x ≤ 13x + 10 < 24 + 10x, x ∉ Z

⇒ - 2 – 10 ≤ 13x – 10x

⇒ - 12 ≤ 3x

⇒ - 4 ≤ x

Also 13x + 10 < 24 + 10x

⇒ 13x – 10x < 24 – 10

3x < 14

∴ x < 4.2/3

∴ - 4 ≤ x < 4.2/3

31. Solve the inequation 2x - 5 ≤ 5x + 4 < 11, where x I. Also represent the solution set on the number line.

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 < 7/5

⇒ 3x > - 9 and x < 1.4

⇒ x > - 3

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.

2(x – 1)< 3x – 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 ≤ 11/3

Solution set B = {3, 2, 1, 0, -1, ……}

A ∩ B = {0, 1, 2, 3}

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

– 3x + 4 < 2x – 3

- 3z – 2z < - 3 – 4

⇒ - 5x < - 7

- x < - 7/5

⇒ x > 7/5

∴ Solution set P = {2, 3, 4, 5, …….}

4x – 5 < 12

4x < 12 + 5 ⇒ 4x < 17

x < 17/4

∵ x ∈ W

∴ Solution set Q = {4, 3, 2, 1, 0}

(i) P ∩ Q = {2, 3, 4}

(ii) Q – P = {1, 0}

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

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

B = 18x – 9 ≥ 15 + 12x

⇒ 18x – 12x ≥ 15 + 9

⇒ 6x ≥ 24

⇒ x ≥ 4

∴ A ∩ B = x ≥ 4, x ∈ R

Hence Range of A ∩ B = {x : x ≥ 4, x ∈ R} and its graph will be.

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.

P = {x : 5 < 2x – 1 ≤ 11}

5 < 2x – 1 ≤ 11

⇒ 5 < 2x – 1 and 2x – 1 ≤ 11

⇒ - 2x < - 5 – 1 and 2x ≤ 11 + 1

⇒ - 2x < - 6 and 2x ≤12

⇒ - x < - 3 and x ≤ 6

⇒ x > 3 or 3 < x

∴ Solution set = 3 < x ≤ 6 = {4, 5, 6}

Solution set on number line.

Q = {-1 ≤ 3 + 4x < 23}

- 1 ≤ 3 + 4x < 23

⇒ - 1 < 3 + 4x and 3 + 4x < 23

⇒ - 4x < 3 + 1 4x < 23 - 3

⇒ - 4x < 4 4x < 20

⇒ - x < 1 x < 5

⇒ x > - 1

- 1 < x

∴ - 1 < x < 5

∴ Solution set = {1, 1, 2, 3, 4}

Solution set on number line

36. If x I, find the smallest value of x which satisfies the inequation 2x + 5/2 > 5x/3 + 2

2x + 5/2 > 5x/3 + 2

⇒ 2x – 5x/3 > 2 – 5/2

⇒ 12x – 10 x > 12 – 15

⇒ 2x > - 3

⇒ x > - 3/2

Smallest value of x = - 1

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

(i) x I

(ii) x W

(iii) x N

20 – 5x < 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

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

4x - 19 < 3x/5 – 2 - 2/5 + x , x ∈ R

We have

4x – 19 < 3x/5 – 2 ≤ - 2/5 + x , x ∈ R

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

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

⇒ 4x – 19 < 3x/5 – 2 and 3x/5 – 2 ≤ -2/5 + x, x ∈ R

⇒ 4x – 3x/5 < 17 and – 2 + 2/5 ≤ x – 3x/5, x ∈ R

⇒ 17x/5 < 17 and -8/5 ≤ 2x/5, x ∈ R

⇒ x < 5 and – 4 ≤ x, x ∈ R

⇒ - 4 ≤ x < 5, x ∈ R

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

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

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

2y – 3 < y + 1 4y + 7; y ∈ R

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

⇒ y – 4y ≤ 7 – 1

⇒ 3y ≤ 6

⇒ y ≤ 6/-3

⇒ y ≥ - 2 ….(ii)

From (i) and (ii),

4 > y ≥ - 2 or – 2 ≤ y < 4

Now representing it on a number given below

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

- 3 + x 8x/3 + 2 14/3 + 2x, where x ∈ I

Given : - 3 + x ≤ 8x/3 + 2 ≤ 14/3 + 2x, Where x ∈ I

(i) – 3 + x ≤ 8x/3 + 2

⇒ - 3 – 2 ≤ 8x/3 – x

⇒ - 5 ≤ 5x/3

⇒ - 1 ≤ x/3

⇒ - 3 ≤ x ….(i)

And 8x/3 = 2 ≤ 14/3 + 2x

8x/3 – 2x ≤ 14/3 – 2

⇒ 2x/3 ≤ 8/3

⇒ x ≤ 4 ….(ii)

From (i) and (ii)

⇒ - 5 ≤ 5x/3 and 2x/3 ≤ 8/3

⇒ x ≥ - 3 and x ≤ 4

∴ - 3 ≤ x ≤ 4

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

Solution set on number line

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.

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

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.

Let the length of the shortest pole = x metre

Length of pole which is buried in mud = x/3

Length of pole which is in the water = x/6

According to this problem,

x – [x/3 + x/6] ≥ 3

⇒ x – (2x + x)/6 ≥ 3

⇒ x – x/2 ≥ 3

⇒ x/2 ≥ 3

⇒ x ≥ 6

∴ Length of pole (shortest in length) = 6 metres

Multiple Choice Questions

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, 1, 3}

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

(d) {- 3, - 2, - 1, 0, 1, 2}

(b) {- 3, - 1, 0, 1, 3}

x ∈ {- 3, - 1, 0, 1, 3, 5}

⇒ 3x – 2 ≤ 8

⇒ 3x ≤ 8 + 2

⇒ 3x ≤ 10

So ⇒ x ≤ 10/3

Therefore ⇒ x < 3.1/3

Solution set = {- 3, - 1, 0, 1, 3}

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 -(3/2)

(c) {0, 1, 2, 3,…..}

x ∈ W

3x + 11 ≥ x + 8

⇒ 3x – x ≥ 8 – 11

⇒ 2x ≥ - 3

⇒ x ≥ -3/2

⇒ x ≥ -1.1/2

Solution set = {0, 1, 2, 3,…..}

3. If x ∈ W, the 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}

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

x ∈ W

5 – 4x < 2 – 3x

⇒ 5 – 2 ≤ 3x + 4x

⇒ 3 ≤ x

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

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, - (4/3) < x 2}

(a) {- 1, 0, 1, 2}

x ∈ I

1 < 3x + 5 ≤ 11

⇒ 1 < 3x + 5

⇒ 1 – 5 < 3x

⇒ - 4 < 3x

⇒ -4/3 < x

And 3x + 5 ≤ 11 ⇒ 3x ≤ 11 – 5

⇒ 3x ≤ 6

⇒ x ≤ 6/3

⇒ x ≤ 2

∴ -4/3 < x ≤ 2

Solution set = {- 1, 0, 1, 2}

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

(a) {x : x ∈ R, 0 < x ≤ 1}

x ∈ R

⇒ 6 ≤ - 3(2x – 4) < 12

⇒ 6 ≤ - 3(2x – 4)

⇒ 6 ≤ - 6x + 12

⇒ 6x ≤ 12 – 6

⇒ 6x ≤ 6

⇒ x ≤ 6/6

⇒ x ≤ 1

And -3(2x – 4) < 12

⇒ - 6x + 12 < 12

⇒ - 6x < 0

⇒ x < 0 ………(ii)

From (i) and (iii),

∴ 0 < x ≤ 1

Solution set = {x : x ∈ R, 0 < x ≤ 1}

Chapter Test

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.

5x - 2 < 3(3 – x)

⇒ 5x – 2 ≤ 9 – 3x

⇒ 5x + 3x ⇒ 9 + 2

⇒ 8x ≤ 11

⇒ x ≤ 11/8

∵ x ∈ {-2, - 1, 0, 1, 2, 3, 4}

∴ Solution set = {- 2, - 1, 0, 1}

Solution set on number line

2. Solve the inequations:

6x – 5 < 3x + 4, x ∈ I.

6x – 5 < 3x + 4

⇒ 6x – 3x < 4 + 5

⇒ 3x < 9

⇒ x < 3

x ∈ I

Solution set = {-1, - 2, 2, 1, 0………}

3. Find the solution set of the inequation x + 5 < 2x +3 ; x ∈ R

Graph the solution set on the number line.

x + 5 ≤ 2x + 3

⇒ x – 2x ≤ 3 – 5

⇒ - x ≤ - 2

⇒ x ≥ 2

∵ x ∈ R

∴ Solution set = {2, 3, 4, 5, …….}

Solution set on number line

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

- 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

5. Solve the inequation:

(5x + 1)/7 – 4(x/7 + 2/5) 1.3/5 + (3x – 1)/7, x ∈ R

(5x + 1)/7 – 4(x/7 + 2/5) ≤ 1.3/5 + (3x – 1)/7

(5x + 1)/7 – 4(x/7 + 2/5) ≤ 8/5 + (3x – 1)/7

Multiplying by L.C.M. of 7 and 5 i.e., 35

25x + 5 – 4 (5x + 14) ≤ 56 + 15x – 5

⇒ 25 + 5 – 20x – 56 ≤ 56 + 15x – 5

⇒ 25x – 20x – 15x ≤ 56 – 5 – 5 + 56

⇒ - 10x ≤ 102

⇒ - x ≤ 102/10

⇒ - x ≤ 51/5

⇒ x ≥ - 51/5

∵ x ∈ R

∴ Solution set = {x : x ∈ R, x ≥ - 51/5}

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.

Solution:

7 < - 4x + 2 < 12

⇒ 7 < - 4x + 2 and – 4x + 2 < 12

⇒ 4x ≤ 2 – 7 and – 4x < 12 – 2

⇒ 4x ≤ - 5 and – 4x < 10

⇒ x ≤ -5/4 and – x < 10/4

⇒ x ≤ -5/4 and – x < 5/2

or x > - (5/2)

∵ x ∈ R

∴ Solution set – 5/2 < x ≤ -5/4

= {x : x ∈ R, - 5/2 < x ≤ - 5/4}

Solution set on the number line

7. If x ∈ R, solve 2x – 3 x + (1 – x)/3 > 2/5x

2x – 3 ≥ x + (1 – x)/3 > 2/5.x

⇒ 2x – 3 ≥ x + (1 – x)/3 and x + (1 – x)/3 > 2/5x

⇒ 2x – 3 ≥ (3x + 1 – x)/3 and (3x + 1 – x)/3 > 2/5x

⇒ 6x – 9 ≥ 3x + 1 – x and 15x + 5 – 5x > 6x

⇒ 6x – 3x + x ≥ 1 + 9 and 15x – 6x – 5x > - 5

⇒ 4x ≥ 10 and 4x > - 5

⇒ x ≥ 10/4 and x > - 5/4

⇒ x ≥ 5/2

∴ x ≥ 5/2

∵ x ∈ R

∴ Solution set = {x : x ∈ R, x ≥ 5/2}

Solution set on number line

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.

Let the positive integer = x

According to the problem,

5a – 6 – 4x

⇒ 5a – 4x < 6

⇒ x < 6

Solution set = {x : x < 6}

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

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.

Let first least natural number = x

Then second number = x + 1

And third number = x + 2

According to the condition 1/3(x + 2) – 1/5 (x) ≥ 3

5x + 10 – 3x ≥ 45

(Multiplying by 15 the L.C.M. of 2 and 5)

2x ≥ 45 – 10

⇒ 2x ≥ 35

x ≥ 35/2

⇒ x ≥ 17.1/2

∵ x is a natural least number

∴ x = 18

∴ find least natural number = 18

Second number = 18 + 1 = 19

And third numbers = 18 + 2 = 20

Hence, least natural numbers are 18, 19, 20

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