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

## Chapter 15 : Linear Inequations

#### Page 10

Q 1.1 | Page 10

Solve: 12x < 50, when x ∈ R

Q 1.2 | Page 10

Solve: 12x < 50, when  x ∈ Z

Q 1.3 | Page 10

Solve: 12x < 50, when x ∈ N

Q 2.1 | Page 10

Solve: −4x > 30, when  x ∈ R

Q 2.2 | Page 10

Solve: −4x > 30, when x ∈ Z

Q 2.3 | Page 10

Solve: −4x > 30, when x ∈ N

Q 3.1 | Page 10

Solve: 4x − 2 < 8, when x ∈ R

Q 3.2 | Page 10

Solve: 4x − 2 < 8, when x ∈ Z

Q 3.3 | Page 10

Solve: 4x − 2 < 8, when x ∈ N

Q 4 | Page 10

3x − 7 > x + 1

Q 5 | Page 10

x + 5 > 4x − 10

Q 6 | Page 10

3x + 9 ≥ −x + 19

Q 7 | Page 10

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

Q 8 | Page 10

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

Q 9 | Page 10

−(x − 3) + 4 < 5 − 2x

Q 10 | Page 10

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

Q 11 | Page 10

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

Q 12 | Page 10

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

Q 13 | Page 10

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

Q 14 | Page 10

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

Q 15 | Page 10

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

Q 16 | Page 10

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

Q 17 | Page 10

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

Q 18 | Page 10

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

Q 19 | Page 10

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

Q 20 | Page 10

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

Q 21 | Page 10

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

Q 22 | Page 10

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

Q 23 | Page 10

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

Q 24 | Page 10

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

Q 25 | Page 10

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

Q 26 | Page 10

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

Q 27 | Page 10

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

Q 28 | Page 10

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

#### Pages 10 - 16

Q 1 | Page 15

Solve each of the following system of equations in R.

1. x + 3 > 0, 2x < 14

Q 2 | Page 10

Solve each of the following system of equations in R.

2x − 7 > 5 − x, 11 − 5x ≤ 1

Q 3 | Page 15

Solve each of the following system of equations in R.

x − 2 > 0, 3x < 18

Q 4 | Page 15

2x + 6 ≥ 0, 4x − 7 < 0

Q 5 | Page 15

Solve each of the following system of equations in R.

3x − 6 > 0, 2x − 5 > 0

Q 6 | Page 15

Solve each of the following system of equations in R.

2x − 3 < 7, 2x > −4

Q 7 | Page 15

Solve each of the following system of equations in R.

2x + 5 ≤ 0, x − 3 ≤ 0

Q 8 | Page 15

Solve each of the following system of equations in R.

5x − 1 < 24, 5x + 1 > −24

Q 9 | Page 15

Solve each of the following system of equations in R.

3x − 1 ≥ 5, x + 2 > −1

Q 10 | Page 15

Solve each of the following system of equations in R.

11 − 5x > −4, 4x + 13 ≤ −11

Q 11 | Page 15

Solve each of the following system of equations in R.

4x − 1 ≤ 0, 3 − 4x < 0

Q 12 | Page 15

Solve each of the following system of equations in R.

x + 5 > 2(x + 1), 2 − x < 3 (x + 2)

Q 14 | Page 15

Solve each of the following system of equations in R.

2 (x − 6) < 3x − 7, 11 − 2x < 6 −

Q 15 | Page 15

Solve each of the following system of equations in R.

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

Q 16 | Page 15

Solve each of the following system of equations in R.

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

Q 17 | Page 15

Solve each of the following system of equations in R.

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

Q 18 | Page 15

Solve each of the following system of equations in R.

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

Q 19 | Page 15

Solve each of the following system of equations in R.

10 ≤ −5 (x − 2) < 20

Q 20 | Page 15

Solve each of the following system of equations in R.

20. −5 < 2x − 3 < 5

Q 21 | Page 16

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

#### Page 22

Q 1 | Page 22

Solve

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

Q 2 | Page 22

Solve

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

Q 3 | Page 22

Solve

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

Q 4 | Page 22

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

Q 5 | Page 22

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

Q 6 | Page 22

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

Q 7 | Page 22

Solve

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

Q 8 | Page 22

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

Q 9 | Page 22

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

Q 10 | Page 22

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

Q 11 | Page 22

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

Q 12 | Page 22

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

Q 13 | Page 22

Solve  $\left| 3 - 4x \right| \geq 9$

#### Pages 24 - 25

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.

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.

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.

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.

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$F = \frac{9}{5}C + 32$

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?

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.

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?

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.

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?

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?

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.

#### Page 28

Q 1 | Page 28

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

x + 2y − y ≤ 0

Q 2 | Page 28

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

2. x + 2y ≥ 6

Q 3 | Page 28

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

x + 2 ≥ 0

Q 4 | Page 28

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

4. x − 2y < 0

Q 5 | Page 28

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

5. −3x + 2y ≤ 6

Q 6 | Page 28

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

6. x ≤ 8 − 4y

Q 7 | Page 28

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

0 ≤ 2x − 5y + 10

Q 8 | Page 28

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

3y ≥ 6 − 2

Q 9 | Page 28

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

y ≥ 2x − 8

Q 10 | Page 28

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

3x − 2y ≤ x + y − 8

#### Pages 0 - 31

Solve the following systems of linear inequation graphically:

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

Solve the following systems of linear inequation graphically:

2x + 3y ≤ 6, x + 4y ≤ 4, x ≥ 0, y ≥ 0

Solve the following systems of linear inequations graphically:

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

Solve the following systems of linear inequations graphically:

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

Solve the following systems of linear inequations graphically:

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

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

x − 2y ≥ 0, 2x − y ≤ −2, x ≥ 0, y ≥ 0

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

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

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:

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

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

Solve the following systems of inequations graphically:

2x + y ≥ 8, x + 2y ≥ 8, x + y ≤ 6

Solve the following systems of inequations graphically:

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

Solve the following systems of inequations graphically:

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

Solve the following systems of inequations graphically:

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

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

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

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

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

#### Pages 31 - 32

Q 1 | Page 31

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

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)

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

$\frac{x}{b < \frac{y}{b}}$

$\frac{x}{b \leq \frac{y}{b}}$

$\frac{x}{b > \frac{y}{b}}$

$\frac{x}{b \geq \frac{y}{b}}$

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

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, $\infty$)

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$\infty$.

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,$\infty$

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$

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

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)

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]

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, $\infty$)

#### Page 31

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

Q 2 | Page 31

Write the solution set of the inequation

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

Q 3 | Page 31

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

Q 4 | Page 31

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

Q 5 | Page 31

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

Q 6 | Page 31

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

Q 7 | Page 31

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

Q 8 | Page 31

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

Q 9 | Page 31

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

Q 10 | Page 31

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

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

RD Sharma solutions for Class 11 Maths 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 Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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

Using RD Sharma Class 11 solutions Linear Inequations exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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