#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 15: Linear Inequations

#### Chapter 15: Linear Inequations Exercise 15.10 solutions [Page 10]

Solve: 12*x* < 50, when *x* ∈ R

Solve: 12*x* < 50, when *x* ∈ Z

Solve: 12*x* < 50, when *x* ∈ N

Solve: −4*x* > 30, when *x* ∈ R

Solve: −4*x* > 30, when *x* ∈ Z

Solve: −4*x* > 30, when *x* ∈ N

Solve: 4*x* − 2 < 8, when *x* ∈ R

Solve: 4*x* − 2 < 8, when *x* ∈ Z

Solve: 4*x* − 2 < 8, when *x* ∈ N

3*x* − 7 >* x* + 1

*x* + 5 > 4*x* − 10

3*x* + 9 ≥ −*x* + 19

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

\[\frac{3x - 2}{5} \leq \frac{4x - 3}{2}\]

−(*x* − 3) + 4 < 5 − 2*x*

\[\frac{x}{5} < \frac{3x - 2}{4} - \frac{5x - 3}{5}\]

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

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

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

\[\frac{2x + 3}{4} - 3 < \frac{x - 4}{3} - 2\]

\[\frac{5 - 2x}{3} < \frac{x}{6} - 5\]

\[\frac{4 + 2x}{3} \geq \frac{x}{2} - 3\]

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

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

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

\[\frac{2x - 3}{3x - 7} > 0\]

\[\frac{3}{x - 2} < 1\]

\[\frac{1}{x - 1} \leq 2\]

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

\[\frac{5x - 6}{x + 6} < 1\]

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

\[\frac{x - 1}{x + 3} > 2\]

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

\[\frac{x}{x - 5} > \frac{1}{2}\]

#### Chapter 15: Linear Inequations Exercise 15.20 solutions [Pages 10 - 16]

Solve each of the following system of equations in R.

1. *x* + 3 > 0, 2*x* < 14

Solve each of the following system of equations in R.

2*x* − 7 > 5 − *x*, 11 − 5*x* ≤ 1

Solve each of the following system of equations in R.

*x* − 2 > 0, 3*x* < 18

2*x* + 6 ≥ 0, 4*x* − 7 < 0

Solve each of the following system of equations in R.

3*x* − 6 > 0, 2*x* − 5 > 0

Solve each of the following system of equations in R.

2*x* − 3 < 7, 2*x* > −4

Solve each of the following system of equations in R.

2*x* + 5 ≤ 0, *x* − 3 ≤ 0

Solve each of the following system of equations in R.

5*x* − 1 < 24, 5*x* + 1 > −24

Solve each of the following system of equations in R.

3*x* − 1 ≥ 5, *x* + 2 > −1

Solve each of the following system of equations in R.

11 − 5*x* > −4, 4*x* + 13 ≤ −11

Solve each of the following system of equations in R.

4*x* − 1 ≤ 0, 3 − 4*x* < 0

Solve each of the following system of equations in R.

*x* + 5 > 2(*x* + 1), 2 − *x* < 3 (*x* + 2)

Solve each of the following system of equations in R.

2 (*x* − 6) < 3*x* − 7, 11 − 2*x* < 6 − *x *

Solve each of the following system of equations in R.

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

Solve each of the following system of equations in R.

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

Solve each of the following system of equations in R.

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

Solve each of the following system of equations in R.

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

Solve each of the following system of equations in R.

\[0 < \frac{- x}{2} < 3\]

Solve each of the following system of equations in R.

10 ≤ −5 (*x* − 2) < 20

Solve each of the following system of equations in R.

20. −5 < 2*x* − 3 < 5

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

#### Chapter 15: Linear Inequations Exercise 15.30 solutions [Page 22]

Solve

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

Solve

\[\left| 4 - x \right| + 1 < 3\]

Solve

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

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

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

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

Solve

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

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

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

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

Solve \[\left| x + 1 \right| + \left| x \right| > 3\]

Solve \[1 \leq \left| x - 2 \right| \leq 3\]

Solve \[\left| 3 - 4x \right| \geq 9\]

#### Chapter 15: Linear Inequations Exercise 15.40 solutions [Pages 24 - 25]

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

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

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

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.

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\[F = \frac{9}{5}C + 32\]

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

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.

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?

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.

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?

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?

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.

#### Chapter 15: Linear Inequations Exercise 15.50 solutions [Page 28]

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

*x* + 2*y* − *y* ≤ 0

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

2. *x* + 2*y* ≥ 6

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

*x* + 2 ≥ 0

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

4. *x* − 2*y* < 0

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

5. −3*x* + 2*y* ≤ 6

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

6. *x* ≤ 8 − 4*y*

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

0 ≤ 2*x* − 5*y* + 10

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

3*y* ≥ 6 − 2*x *

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

*y* ≥ 2*x* − 8

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

3*x* − 2*y* ≤ *x* + *y* − 8

#### Chapter 15: Linear Inequations Exercise 15.60 solutions [Pages 0 - 31]

Solve the following systems of linear inequation graphically:

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

Solve the following systems of linear inequation graphically:

2*x* + 3*y* ≤ 6, *x* + 4*y* ≤ 4, *x* ≥ 0, *y* ≥ 0

Solve the following systems of linear inequations graphically:

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

Solve the following systems of linear inequations graphically:

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

Solve the following systems of linear inequations graphically:

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

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

*x* − 2*y* ≥ 0, 2*x* − *y* ≤ −2, *x* ≥ 0, *y* ≥ 0

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

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

3*x* + *y* ≥ 12*x* ≥ 0, *y* ≥ 0

Solve the following systems of inequations graphically:

2*x* + *y* ≥ 8, *x* + 2*y* ≥ 8, *x* + *y* ≤ 6

Solve the following systems of inequations graphically:

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

Solve the following systems of inequations graphically:

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

Solve the following systems of inequations graphically:

5*x* + *y* ≥ 10, 2*x* + 2*y* ≥ 12, *x* + 4*y* ≥ 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:

\[2x + y \geq 8, x + 2y \geq 10, x \geq 0, y \geq 0\]

#### Chapter 15: Linear Inequations solutions [Pages 31 - 32]

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

Write the solution set of the inequation

\[x + \frac{1}{x} \geq 2\]

Mark the correct alternative in each of the following:

If − 3*x*\[+\]17\[< -\]13, then

(a) x\[\in\](10, \[\infty\]

(b) x\[\in\][10, \[\infty\]

(c) x\[- \infty\]10]

(d)

*x*\[\in\]\[-\]10, 10)

Write the set of values of *x* satisfying the inequation (*x*^{2} − 2*x* + 1) (*x* − 4) < 0.

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

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

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

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

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

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

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

#### Chapter 15: Linear Inequations solutions [Page 32]

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

Mark the correct alternative in each of the following:

Given that *x*, *y* and *b *are real numbers and *x*\[<\]*y*, *b*\[>\]0, then

\[\frac{x}{b < \frac{y}{b}}\]

\[\frac{x}{b \leq \frac{y}{b}}\]

\[\frac{x}{b > \frac{y}{b}}\]

\[\frac{x}{b \geq \frac{y}{b}}\]

Mark the correct alternative in each of the following:

If *x *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

Mark the correct alternative in each of the following:

If *x *and *a *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*\[\in\](\[-\]*a*,*a*)(d)

*x\[\in\](\[-\]\[\infty\]\[-\]a) \[\cup\](a, \[\infty\])*

Mark the correct alternative in each of the following:

\[\left| x - 1 \right|\]\[>\]5, then

(a)

*x*\[\in\](\[-\]4, 6)(b)

*x**\[\in\][\[-\]4, 6]*(c)

*x\[\in\](\[-\]\[\infty\]\[-\]4) \[\cup\](6, \[\infty\]*(d)

*x*\[\in\](\[-\]\[\infty\]\[-\]4) \[\cup\][6\[\infty\].

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,\[\infty\]

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

\[\left| x \right|\]\[\geq\]

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

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)

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]

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

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## 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.

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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.

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