# Solution of Chapter 15. Linear Inequations (RD Sharma - Mathematics Book)

## Exercise 15.1

1

Solve the following linear inequations in R

12x < 50, when

i. x R

ii. x Z

iii. x N

2

Solve the following linear inequations in R

–4x > 30, when

i. x R

ii. x Z

iii. x N

3

Solve the following linear inequations in R

4x – 2 < 8, when

i. x R

ii. x Z

iii. x N

4

Solve the following linear inequations in R

3x – 7 > x + 1

5

Solve the following linear inequations in R

x + 5 > 4x – 10

6

Solve the following linear inequations in R

3x + 9 ≥ –x + 19

7

Solve the following linear inequations in R

8

Solve the following linear inequations in R

9

Solve the following linear inequations in R

–(x – 3) + 4 < 5 – 2x

10

Solve the following linear inequations in R

11

Solve the following linear inequations in R

12

Solve the following linear inequations in R

13

Solve the following linear inequations in R

14

Solve the following linear inequations in R

15

Solve the following linear inequations in R

16

Solve the following linear inequations in R

17

Solve the following linear inequations in R

18

Solve the following linear inequations in R

19

Solve the following linear inequations in R

20

Solve the following linear inequations in R

21

Solve the following linear inequations in R

22

Solve the following linear inequations in R

23

Solve the following linear inequations in R

24

Solve the following linear inequations in R

25

Solve the following linear inequations in R

26

Solve the following linear inequations in R

27

Solve the following linear inequations in R

28

Solve the following linear inequations in R

## Exercise 15.2

1

Solve each of the following system of inequations in R

x + 3 > 0, 2x < 14

2

Solve each of the following system of inequations in R

2x – 7 > 5 – x, 11 – 5x ≤ 1

3

Solve each of the following system of inequations in R

x – 2 > 0, 3x < 18

4

Solve each of the following system of inequations in R

2x + 6 ≥ 0, 4x – 7 < 0

5

Solve each of the following system of inequations in R

3x – 6 > 0, 2x – 5 > 0

6

Solve each of the following system of inequations in R

2x – 3 < 7, 2x > –4

7

Solve each of the following system of inequations in R

2x + 5 ≤ 0, x – 3 ≤ 0

8

Solve each of the following system of inequations in R

5x – 1 < 24, 5x + 1 > –24

9

Solve each of the following system of inequations in R

3x – 1 ≥ 5, x + 2 > –1

10

Solve each of the following system of inequations in R

11 – 5x > –4, 4x + 13 ≤ –11

11

Solve each of the following system of inequations in R

4x – 1 ≤ 0, 3 – 4x < 0

12

Solve each of the following system of inequations in R

x + 5 > 2(x + 1), 2 – x < 3(x + 2)

13

Solve each of the following system of inequations in R

2(x – 6) < 3x – 7, 11 – 2x < 6 – x

14

Solve each of the following system of inequations in R

5x – 7 < 3(x + 3),

15

Solve each of the following system of inequations in R

, 2(2x + 3) < 6(x – 2) + 10

16

Solve each of the following system of inequations in R

17

Solve each of the following system of inequations in R

18

Solve each of the following system of inequations in R

19

Solve each of the following system of inequations in R

10 ≤ –5(x – 2) < 20

20

Solve each of the following system of inequations in R

–5 < 2x – 3 < 5

21

Solve each of the following system of inequations in R

## Exercise 15.3

1

Solve each of the following system of equations in R.

2

Solve each of the following system of equations in R.

3

Solve each of the following system of equations in R.

4

Solve each of the following system of equations in R.

5

Solve each of the following system of equations in R.

6

Solve each of the following system of equations in R.

7

Solve each of the following system of equations in R.

8

Solve each of the following system of equations in R.

9

Solve each of the following system of equations in R.

10

Solve each of the following system of equations in R.

11

Solve each of the following system of equations in R.

12

Solve each of the following system of equations in R.

13

Solve each of the following system of equations in R.

## Exercise 15.4

1

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

2

Find all pairs of consecutive odd natural number, both of which are larger than 10, such that their sum is less than 40.

3

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

4

The marks scored by Rohit in two tests were 65 and 70. Find the minimum marks he should score in the third test to have an average of at least 65 marks.

5

A solution is to be kept between 86o and 95oF. What is the range of temperature in degree Celsius, if the Celsius (C)/Fahrenheit (F) conversion formula is given by

6

A solution is to be kept between 30oC and 35oC. What is the range of temperature in degree Fahrenheit?

7

To receive grade ‘A’ in a course, one must obtain an average of 90 marks or more in five papers each of 100 marks. If Shikha scored 87, 95, 92 and 94 marks in first four papers, find the minimum marks that she must score in the last paper to get grade ‘A’ in the course.

8

A company manufactures cassettes and its cost and revenue functions for a week are and R = 2x respectively, where x is the number of cassettes produced and sold in a week. How many cassettes must be sold for the company to realize a profit?

9

The longest side of a triangle is three times the shortest side, and the third side is 2 cm shorter than the longest side if the perimeter of the triangle at least 61 cm, Find the minimum length of the shortest-side.

10

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

11

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 there are 640 liters of the 8% solution, how many liters of 2% solution will have to be added?

12

The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between 7.2 and 7.8. If the first two pH reading is 7.48 and 7.85, find the range of pH value for the third reading that will result in the acidity level is normal.

## Exercise 15.5

1

Represent to solution set of the following inequations graphically in two dimensional plane:

x + 2y – 4 ≤ 0

2

Represent to solution set of the following inequations graphically in two dimensional plane:

x + 2y ≥ 6

3

Represent to solution set of the following inequations graphically in two dimensional plane:

x + 2 ≥ 0

4

Represent to solution set of the following inequations graphically in two dimensional plane:

x – 2y < 0

5

Represent to solution set of the following inequations graphically in two dimensional plane:

– 3x + 2y ≤ 6

6

Represent to solution set of the following inequations graphically in two dimensional plane:

x ≤ 8 – 4y

7

Represent to solution set of the following inequations graphically in two dimensional plane:

0 ≤ 2x – 5y + 10

8

Represent to solution set of the following inequations graphically in two dimensional plane:

3y > 6 – 2x

9

Represent to solution set of the following inequations graphically in two dimensional plane:

y> 2x – 8

10

Represent to solution set of the following inequations graphically in two dimensional plane:

3x – 2y ≤ x + y – 8

## Exercise 15.6

1

Solve the following systems of linear inequaitons graphically.

2x + 3y ≤ 6, 3x + 2y ≤ 6, x ≥ 0, y ≥ 0

1

Solve the following systems of linear inequaitons graphically.

2x + 3y≤ 6, x + 4y ≤ 4, x ≥ 0, y ≥ 0

1

Solve the following systems of linear inequaitons graphically.

x – y ≤ 1, x + 2y≤ 8, 2x + y ≥ 2, x ≥ 0, y ≥ 0

1

Solve the following systems of linear inequaitons graphically.

x + y ≥ 1, 7x + 9y ≤ 63, x ≤6, y ≤ 5, x ≥ 0, y ≥ 0

1

Solve the following systems of linear inequaitons graphically.

2x + 3y≤35, y ≥ 3, x ≥ 2, x ≥ 0, y ≥ 0

2

how that the solution set of the following linear inequations is empty set :

x – 2y ≥ 0, 2x – y ≤ –2, x ≥ 0, y ≥ 0

2

how that the solution set of the following linear inequations is empty set :

x + 2y≤ 3, 3x + 4y≥ 12, y ≥ 1, x ≥ 0, y ≥ 0

3

Find the linear inequations for which the shaded area in Fig. 15.41 is the solution set. Draw the diagram of the solution set of the linear inequations.

4

Find the linear inequations for which the solution set is the shaded region given in Fig. 15.42.

5

Show that the solution set of the following linear inequations is an unbounded set :

x + y ≥ 9, 3x + y ≥ 12, x ≥ 0, y ≥ 0.

6

solve the following systems of inequations graphically :

2x + y ≥ 8, x + 2y ≥ 8, x + y ≤ 6

6

solve the following systems of inequations graphically :

12 + 12y ≤ 840, 3x + 6y ≤ 300,8x + 4y ≤ 480, x ≥ 0, y ≥ 0

6

solve the following systems of inequations graphically :

x + 2y≤ 40,3x + y ≥ 30, 4x + 3y≥ 60, x ≥ 0, y ≥ 0

6

solve the following systems of inequations graphically :

5x + y ≥ 10, 2x + 2y ≥ 12, x + 4y ≥ 12, x ≥ 0, y ≥ 0

7

Show that the following system of linear equations has no solution :

x + 2y ≥ 3, 3x + 4y ≥ 12, x ≥ 0, y ≥ 1.

8

Show that the solution set of the following system of linear inequalities is an unbounded region 2x + y≥ 8, x + 2y≥ 10, x ≥ 0, y ≥ 0.

1

Write the solution set of the inequation .

2

Write the solution set of the inequation .

3

Write the set of values of x satisfying the inequation .

4

Write the solution set of the inequation .

5

Write the set of values of x satisfying and .

6

Write the solution set of the inequation .

7

Write the number of integral solutions of .

8

Write the set of values of x satisfying the inequations 5x + 2 < 3x + 8 and .

9

Write the solution set of .

10

Write the solution set of the inequation .

## MCQ

1

Mark the Correct alternative in the following:

If x < 7, then

2

Mark the Correct alternative in the following:

If 3x + 17 < 13, then

3

Mark the Correct alternative in the following:

Given that x, y and b are real numbers and x < y, b > 0, then

4

Mark the Correct alternative in the following:

If x is a real number and , then

5

Mark the Correct alternative in the following:

If x and a are real numbers such that a > 0 and , then

6

Mark the Correct alternative in the following:

If , then

7

Mark the Correct alternative in the following:

If , then

8

Mark the Correct alternative in the following:

The inequality representing the following graph is

9

Mark the Correct alternative in the following:

The linear inequality representing the solution set given in Fig. 15.44 is

10

Mark the Correct alternative in the following:

The solution set of the inequation is

11

Mark the Correct alternative in the following:

If , then