# RD Sharma solutions for Class 11 Mathematics Textbook chapter 15 - Linear Inequations [Latest edition]

## Chapter 15: Linear Inequations

Exercise 15.1Exercise 15.2Exercise 15.3Exercise 15.4Exercise 15.5Exercise 15.6Exercise 15.7Exercise 15.8
Exercise 15.1 [Page 10]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 15 Linear Inequations Exercise 15.1 [Page 10]

Exercise 15.1 | Q 1.1 | Page 10

Solve: 12x < 50, when x ∈ R

Exercise 15.1 | Q 1.2 | Page 10

Solve: 12x < 50, when  x ∈ Z

Exercise 15.1 | Q 1.3 | Page 10

Solve: 12x < 50, when x ∈ N

Exercise 15.1 | Q 2.1 | Page 10

Solve: −4x > 30, when  x ∈ R

Exercise 15.1 | Q 2.2 | Page 10

Solve: −4x > 30, when x ∈ Z

Exercise 15.1 | Q 2.3 | Page 10

Solve: −4x > 30, when x ∈ N

Exercise 15.1 | Q 3.1 | Page 10

Solve: 4x − 2 < 8, when x ∈ R

Exercise 15.1 | Q 3.2 | Page 10

Solve: 4x − 2 < 8, when x ∈ Z

Exercise 15.1 | Q 3.3 | Page 10

Solve: 4x − 2 < 8, when x ∈ N

Exercise 15.1 | Q 4 | Page 10

3x − 7 > x + 1

Exercise 15.1 | Q 5 | Page 10

x + 5 > 4x − 10

Exercise 15.1 | Q 6 | Page 10

3x + 9 ≥ −x + 19

Exercise 15.1 | Q 7 | Page 10

$2\left( 3 - x \right) \geq \frac{x}{5} + 4$

Exercise 15.1 | Q 8 | Page 10

$\frac{3x - 2}{5} \leq \frac{4x - 3}{2}$

Exercise 15.1 | Q 9 | Page 10

−(x − 3) + 4 < 5 − 2x

Exercise 15.1 | Q 10 | Page 10

$\frac{x}{5} < \frac{3x - 2}{4} - \frac{5x - 3}{5}$

Exercise 15.1 | Q 11 | Page 10

$\frac{2\left( x - 1 \right)}{5} \leq \frac{3\left( 2 + x \right)}{7}$

Exercise 15.1 | Q 12 | Page 10

$\frac{5x}{2} + \frac{3x}{4} \geq \frac{39}{4}$

Exercise 15.1 | Q 13 | Page 10

$\frac{x - 1}{3} + 4 < \frac{x - 5}{5} - 2$

Exercise 15.1 | Q 14 | Page 10

$\frac{2x + 3}{4} - 3 < \frac{x - 4}{3} - 2$

Exercise 15.1 | Q 15 | Page 10

$\frac{5 - 2x}{3} < \frac{x}{6} - 5$

Exercise 15.1 | Q 16 | Page 10

$\frac{4 + 2x}{3} \geq \frac{x}{2} - 3$

Exercise 15.1 | Q 17 | Page 10

$\frac{2x + 3}{5} - 2 < \frac{3\left( x - 2 \right)}{5}$

Exercise 15.1 | Q 18 | Page 10

$x - 2 \leq \frac{5x + 8}{3}$

Exercise 15.1 | Q 19 | Page 10

$\frac{6x - 5}{4x + 1} < 0$

Exercise 15.1 | Q 20 | Page 10

$\frac{2x - 3}{3x - 7} > 0$

Exercise 15.1 | Q 21 | Page 10

$\frac{3}{x - 2} < 1$

Exercise 15.1 | Q 22 | Page 10

$\frac{1}{x - 1} \leq 2$

Exercise 15.1 | Q 23 | Page 10

$\frac{4x + 3}{2x - 5} < 6$

Exercise 15.1 | Q 24 | Page 10

$\frac{5x - 6}{x + 6} < 1$

Exercise 15.1 | Q 25 | Page 10

$\frac{5x + 8}{4 - x} < 2$

Exercise 15.1 | Q 26 | Page 10

$\frac{x - 1}{x + 3} > 2$

Exercise 15.1 | Q 27 | Page 10

$\frac{7x - 5}{8x + 3} > 4$

Exercise 15.1 | Q 28 | Page 10

$\frac{x}{x - 5} > \frac{1}{2}$

Exercise 15.2 [Pages 15 - 16]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 15 Linear Inequations Exercise 15.2 [Pages 15 - 16]

Exercise 15.2 | Q 1 | Page 15

Solve each of the following system of equations in R.

1. x + 3 > 0, 2x < 14

Exercise 15.2 | Q 2 | Page 10

Solve each of the following system of equations in R.

2x − 7 > 5 − x, 11 − 5x ≤ 1

Exercise 15.2 | Q 3 | Page 15

Solve each of the following system of equations in R.

x − 2 > 0, 3x < 18

Exercise 15.2 | Q 4 | Page 15

2x + 6 ≥ 0, 4x − 7 < 0

Exercise 15.2 | Q 5 | Page 15

Solve each of the following system of equations in R.

3x − 6 > 0, 2x − 5 > 0

Exercise 15.2 | Q 6 | Page 15

Solve each of the following system of equations in R.

2x − 3 < 7, 2x > −4

Exercise 15.2 | Q 7 | Page 15

Solve each of the following system of equations in R.

2x + 5 ≤ 0, x − 3 ≤ 0

Exercise 15.2 | Q 8 | Page 15

Solve each of the following system of equations in R.

5x − 1 < 24, 5x + 1 > −24

Exercise 15.2 | Q 9 | Page 15

Solve each of the following system of equations in R.

3x − 1 ≥ 5, x + 2 > −1

Exercise 15.2 | Q 10 | Page 15

Solve each of the following system of equations in R.

11 − 5x > −4, 4x + 13 ≤ −11

Exercise 15.2 | Q 11 | Page 15

Solve each of the following system of equations in R.

4x − 1 ≤ 0, 3 − 4x < 0

Exercise 15.2 | Q 12 | Page 15

Solve each of the following system of equations in R.

x + 5 > 2(x + 1), 2 − x < 3 (x + 2)

Exercise 15.2 | Q 13 | Page 15

Solve each of the following system of equations in R.

2 (x − 6) < 3x − 7, 11 − 2x < 6 −

Exercise 15.2 | Q 14 | Page 15

Solve each of the following system of equations in R.

$5x - 7 < 3\left( x + 3 \right), 1 - \frac{3x}{2} \geq x - 4$

Exercise 15.2 | Q 15 | Page 15

Solve each of the following system of equations in R.

$\frac{2x - 3}{4} - 2 \geq \frac{4x}{3} - 6, 2\left( 2x + 3 \right) < 6\left( x - 2 \right) + 10$

Exercise 15.2 | Q 16 | Page 15

Solve each of the following system of equations in R.

$\frac{7x - 1}{2} < - 3, \frac{3x + 8}{5} + 11 < 0$

Exercise 15.2 | Q 17 | Page 15

Solve each of the following system of equations in R.

$\frac{2x + 1}{7x - 1} > 5, \frac{x + 7}{x - 8} > 2$

Exercise 15.2 | Q 18 | Page 15

Solve each of the following system of equations in R.

$0 < \frac{- x}{2} < 3$

Exercise 15.2 | Q 19 | Page 15

Solve each of the following system of equations in R.

10 ≤ −5 (x − 2) < 20

Exercise 15.2 | Q 20 | Page 15

Solve each of the following system of equations in R.

20. −5 < 2x − 3 < 5

Exercise 15.2 | Q 21 | Page 16

Solve each of the following system of equations in R. $\frac{4}{x + 1} \leq 3 \leq \frac{6}{x + 1}, x > 0$

Exercise 15.3 [Page 22]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 15 Linear Inequations Exercise 15.3 [Page 22]

Exercise 15.3 | Q 1 | Page 22

Solve

$\left| x + \frac{1}{3} \right| > \frac{8}{3}$

Exercise 15.3 | Q 2 | Page 22

Solve

$\left| 4 - x \right| + 1 < 3$

Exercise 15.3 | Q 3 | Page 22

Solve

$\left| \frac{3x - 4}{2} \right| \leq \frac{5}{12}$

Exercise 15.3 | Q 4 | Page 22

Solve  $\frac{\left| x - 2 \right|}{x - 2} > 0$

Exercise 15.3 | Q 5 | Page 22

Solve  $\frac{1}{\left| x \right| - 3} < \frac{1}{2}$

Exercise 15.3 | Q 6 | Page 22

Solve  $\frac{\left| x + 2 \right| - x}{x} < 2$

Exercise 15.3 | Q 7 | Page 22

Solve

$\left| \frac{2x - 1}{x - 1} \right| > 2$

Exercise 15.3 | Q 8 | Page 22

Solve  $\left| x - 1 \right| + \left| x - 2 \right| + \left| x - 3 \right| \geq 6$

Exercise 15.3 | Q 9 | Page 22

Solve  $\frac{\left| x - 2 \right| - 1}{\left| x - 2 \right| - 2} \leq 0$

Exercise 15.3 | Q 10 | Page 22

Solve  $\frac{1}{\left| x \right| - 3} \leq \frac{1}{2}$

Exercise 15.3 | Q 11 | Page 22

Solve $\left| x + 1 \right| + \left| x \right| > 3$

Exercise 15.3 | Q 12 | Page 22

Solve $1 \leq \left| x - 2 \right| \leq 3$

Exercise 15.3 | Q 13 | Page 22

Solve  $\left| 3 - 4x \right| \geq 9$

Exercise 15.4 [Pages 24 - 25]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 15 Linear Inequations Exercise 15.4 [Pages 24 - 25]

Exercise 15.4 | Q 1 | Page 24

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

Exercise 15.4 | Q 2 | Page 24

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

Exercise 15.4 | Q 3 | Page 24

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

Exercise 15.4 | Q 4 | Page 24

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.

Exercise 15.4 | Q 5 | Page 24

A solution is to be kept between 86° and 95°F. What is the range of temperature in degree Celsius, if the Celsius (C)/ Fahrenheit (F) conversion formula is given by$F = \frac{9}{5}C + 32$

Exercise 15.4 | Q 6 | Page 24

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

Exercise 15.4 | Q 7 | Page 24

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 paper, find the minimum marks that she must score in the last paper to get grade 'A' in the course.

Exercise 15.4 | Q 8 | Page 24

A company manufactures cassettes and its cost and revenue functions for a week are $C = 300 + \frac{3}{2}x \text{ and } R = 2x$ 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?

Exercise 15.4 | Q 9 | Page 24

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

Exercise 15.4 | Q 10 | Page 24

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?

Exercise 15.4 | Q 11 | Page 25

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 litres of the 8% solution, how many litres of 2% solution will have to be added?

Exercise 15.4 | Q 12 | Page 25

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 are 7.48 and 7.85, find the range of pH value for the third reading that will result in the acidity level being normal.

Exercise 15.5 [Page 28]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 15 Linear Inequations Exercise 15.5 [Page 28]

Exercise 15.5 | Q 1 | Page 28

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

x + 2y − y ≤ 0

Exercise 15.5 | Q 2 | Page 28

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

2. x + 2y ≥ 6

Exercise 15.5 | Q 3 | Page 28

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

x + 2 ≥ 0

Exercise 15.5 | Q 4 | Page 28

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

4. x − 2y < 0

Exercise 15.5 | Q 5 | Page 28

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

5. −3x + 2y ≤ 6

Exercise 15.5 | Q 6 | Page 28

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

6. x ≤ 8 − 4y

Exercise 15.5 | Q 7 | Page 28

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

0 ≤ 2x − 5y + 10

Exercise 15.5 | Q 8 | Page 28

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

3y ≥ 6 − 2

Exercise 15.5 | Q 9 | Page 28

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

y ≥ 2x − 8

Exercise 15.5 | Q 10 | Page 28

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

3x − 2y ≤ x + y − 8

Exercise 15.6 [Pages 30 - 31]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 15 Linear Inequations Exercise 15.6 [Pages 30 - 31]

Exercise 15.6 | Q 1.1 | Page 30

Solve the following systems of linear inequation graphically:

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

Exercise 15.6 | Q 1.2 | Page 30

Solve the following systems of linear inequation graphically:

2x + 3y ≤ 6, x + 4y ≤ 4, x ≥ 0, y ≥ 0

Exercise 15.6 | Q 1.3

Solve the following systems of linear inequations graphically:

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

Exercise 15.6 | Q 1.4 | Page 30

Solve the following systems of linear inequations graphically:

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

Exercise 15.6 | Q 1.5 | Page 30

Solve the following systems of linear inequations graphically:

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

Exercise 15.6 | Q 2.1 | Page 30

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

x − 2y ≥ 0, 2x − y ≤ −2, x ≥ 0, y ≥ 0

Exercise 15.6 | Q 2.2 | Page 30

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

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

Exercise 15.6 | Q 3 | Page 30

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:

Exercise 15.6 | Q 4 | Page 31

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

Exercise 15.6 | Q 5 | Page 31

Show that the solution set of the following linear in equations is an unbounded set:
x + y ≥ 9
3x + y ≥ 12
x ≥ 0, y ≥ 0

Exercise 15.6 | Q 6.1 | Page 31

Solve the following systems of inequations graphically:

2x + y ≥ 8, x + 2y ≥ 8, x + y ≤ 6

Exercise 15.6 | Q 6.2 | Page 31

Solve the following systems of inequations graphically:

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

Exercise 15.6 | Q 6.3 | Page 31

Solve the following systems of inequations graphically:

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

Exercise 15.6 | Q 6.4 | Page 31

Solve the following systems of inequations graphically:

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

Exercise 15.6 | Q 7 | Page 31

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

$x + 2y \leq 3, 3x + 4y \geq 12, x \geq 0, y \geq 1$

Exercise 15.6 | Q 8 | Page 31

Show that the solution set of the following system of linear inequalities is an unbounded region:

$2x + y \geq 8, x + 2y \geq 10, x \geq 0, y \geq 0$

Exercise 15.7 [Page 31]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 15 Linear Inequations Exercise 15.7 [Page 31]

Exercise 15.7 | Q 1 | Page 31

Mark the correct alternative in each of the following:

If x$<$7, then

• (a) $-$x$<$$-$7

• (b) $-$x$\leq -$7

• (c) $-$x$> -$7

• (d) $-$x$\geq -$7

Exercise 15.7 | Q 2 | Page 32

Mark the correct alternative in each of the following:

If − 3x$+$17$< -$13, then

• (a) x$\in$(10, $\infty$

• (b) x$\in$[10, $\infty$

• (c) x$- \infty$10]

• (d) x$\in$$-$10, 10)

Exercise 15.7 | Q 2 | Page 31

Write the solution set of the inequation

$x + \frac{1}{x} \geq 2$

Exercise 15.7 | Q 3 | Page 31

Write the set of values of x satisfying the inequation (x2 − 2x + 1) (x − 4) < 0.

Exercise 15.7 | Q 4 | Page 31

Write the solution set of the equation |2 − x| = x − 2.

Exercise 15.7 | Q 5 | Page 31

Write the set of values of x satisfying |x − 1| ≤ 3 and |x − 1| ≥ 1.

Exercise 15.7 | Q 6 | Page 31

Write the solution set of the inequation $\left| \frac{1}{x} - 2 \right| > 4$

Exercise 15.7 | Q 7 | Page 31

Write the number of integral solutions of $\frac{x + 2}{x^2 + 1} > \frac{1}{2}$

Exercise 15.7 | Q 8 | Page 31

Write the set of values of x satisfying the inequations 5x + 2 < 3x + 8 and $\frac{x + 2}{x - 1} < 4$

Exercise 15.7 | Q 9 | Page 31

Write the solution of set of$\left| x + \frac{1}{x} \right| > 2$

Exercise 15.7 | Q 10 | Page 31

Write the solution set of the inequation |x − 1| ≥ |x − 3|.

Exercise 15.8 [Page 32]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 15 Linear Inequations Exercise 15.8 [Page 32]

Exercise 15.8 | Q 1 | Page 32

Write the solution of the inequation$\frac{x^2}{x - 2} > 0$

Exercise 15.8 | Q 3 | Page 32

Mark the correct alternative in each of the following:
Given that xy and are real numbers and x$<$yb$>$0, then

• $\frac{x}{b < \frac{y}{b}}$

• $\frac{x}{b \leq \frac{y}{b}}$

• $\frac{x}{b > \frac{y}{b}}$

• $\frac{x}{b \geq \frac{y}{b}}$

Exercise 15.8 | Q 4 | Page 32

Mark the correct alternative in each of the following:
If is a real number and  $\left| x \right|$$<$5, then

• (a) x$\geq$5

• (b) $-$5$<$x$<$5

• (c) x$\leq$$-$5

• (d) $-$5$\leq$x$\leq$5

Exercise 15.8 | Q 5 | Page 32

Mark the correct alternative in each of the following:
If and are real numbers such that a$>$0 and \\left| x \right|\]$>$a, then

• x$\in$$\in$($-$a, $\infty$)

• (b) x$\in$[$-$$\infty$a]

• (c) x$\in$($-$aa)

• (d) x$\in$($-$$\infty$$-$a) $\cup$(a, $\infty$)

Exercise 15.8 | Q 6 | Page 32

Mark the correct alternative in each of the following:

$\left| x - 1 \right|$$>$5, then

• (a) x$\in$($-$4, 6)

• (b) $\in$[$-$4, 6]

• (c) x$\in$($-$$\infty$$-$4) $\cup$(6, $\infty$

• (d) x$\in$($-$$\infty$$-$4) $\cup$[6$\infty$.

Exercise 15.8 | Q 7 | Page 32

Mark the correct alternative in each of the following:
If $\left| x + 2 \right|$$\leq$9, then

• (a) x$\in$($-$7, 11)

• (b) x$\in$[$-$11, 7]

• (c) x$\in$($-$$\infty$$-$7) $\cup$(11, $\infty$)

• (d) x$\in$($-$$\infty$$-$7) $\cup$[11,$\infty$

Exercise 15.8 | Q 8 | Page 32

Mark the correct alternative in each of the following:
The inequality representing the following graph is

• $\left| x \right|$$<$3

• $\left| x \right|$$\leq$3

• $\left| x \right|$$>$3

• $\left| x \right|$$\geq$

Exercise 15.8 | Q 9 | Page 32

Mark the correct alternative in each of the following:
The linear inequality representing the solution set given in

• $\left| x \right|$$<$5

• $\left| x \right|$$>$5

• $\left| x \right|$$\geq$5

• $\left| x \right|$$\leq$5

Exercise 15.8 | Q 10 | Page 32

Mark the correct alternative in each of the following:
The solution set of the inequation $\left| x + 2 \right|$$\leq$5 is

• (a) ($-$7, 5)

• (b) [$-$7, 3]

• (c) [$-$5, 5]

• (d) ($-$7, 3)

Exercise 15.8 | Q 11 | Page 32

Mark the correct alternative in each of the following:
If  $\frac{\left| x - 2 \right|}{x - 2}$$\geq$ then

•  x$\in$[2, $\infty$

• x$\in$(2, $\infty$)

• x$\in$($-$$\infty$ 2)

•  x$\in$($-$$\infty$2]

Exercise 15.8 | Q 12 | Page 32

Mark the correct alternative in each of the following:
If $\left| x + 3 \right|$$\geq$10, then

• x$\in$($-$13, 7]

• x$\in$13, 7)

• x$\in$($-$$\infty$$-$13) $\cup$ (7, $\infty$)

•  x$\in$($-$$\infty$$-$13] $\cup$ [7, $\infty$)

## Chapter 15: Linear Inequations

Exercise 15.1Exercise 15.2Exercise 15.3Exercise 15.4Exercise 15.5Exercise 15.6Exercise 15.7Exercise 15.8

## RD Sharma solutions for Class 11 Mathematics Textbook chapter 15 - Linear Inequations

RD Sharma solutions for Class 11 Mathematics Textbook chapter 15 (Linear Inequations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics Textbook chapter 15 Linear Inequations are Inequalities - Introduction, Algebraic Solutions of Linear Inequalities in One Variable and Their Graphical Representation, Graphical Solution of Linear Inequalities in Two Variables, Solution of System of Linear Inequalities in Two Variables.

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