#### Chapters

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 6: Linear Inequalities

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 6 Linear InequalitiesSolved Examples [Pages 100 - 106]

#### Short Answer

Solve the inequality, 3x – 5 < x + 7, when x is a natural number

Solve the inequality, 3x – 5 < x + 7, when x is a whole number

Solve the inequality, 3x – 5 < x + 7, when x is an integer

Solve the inequality, 3x – 5 < x + 7, when x is a real number

Solve `(x - 2)/(x + 5) > 2`

Solve |3 – 4x| ≥ 9

Solve 1 ≤ |x –2| ≤ 3.

The cost and revenue functions of a product are given by C(x) = 20x + 4000 and R(x) = 60x + 2000, respectively, where x is the number of items produced and sold. How many items must be sold to realise some profit?

Solve for x, |x + 1| + |x| > 3.

#### Long Answer

Solve for x, `(|x + 3| + x)/(x + 2) > 1`

Solve the following system of inequalities:

`x/(2x + 1) ≥ 1/4, (6x)/(4x - 1) < 1/2`

Find the linear inequalities for which the shaded region in the given figure is the solution set

#### Objective Type Question from 10 to 13

If `|x - 2|/(x - 2) ≥ 0`, then ______.

x ∈ [2, `oo`)

x ∈ (2, `oo`)

x ∈ (– `oo`, 2)

x ∈ (– `oo`, 2]

The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is 160 cm, then ______.

Breadth > 20 cm

Length < 20 cm

Breadth x ≥ 20 cm

Length ≤ 20 cm

Solutions of the inequalities comprising a system in variable x are represented on number lines as given below, then ______.

x ∈ (– `oo`, – 4] ∪ [3, `oo)`

x ∈ [– 3, 1]

x ∈ (– `oo`, – 4) ∪ [3, `oo`)

x ∈ [– 4, 3]

If |x + 3| ≥ 10, then ______.

x ∈ (– 13, 7]

x ∈ (– 13, 7]

x ∈ (– `oo`, – 13] ∪ [7, `oo`)

x ∈ [– `oo`, – 13] ∪ [7, `oo`)

#### State whether the following statement is True or False:

If x > y and b < 0, then bx < by

True

False

If xy > 0, then x > 0, and y < 0

True

False

If xy < 0, then x > 0, and y > 0

True

False

If x > 5 and x > 2, then x ∈ (5, `oo`)

True

False

If |x| < 5, then x ∈ (– 5, 5)

True

False

Graph of x > – 2 is

True

False

Solution set of x – y ≤ 0 is

True

False

#### Fill in the blanks in the following:

If x ≥ – 3, then x + 5 ______ 2

If – x ≤ – 4, then 2x ______ 8

If `1/(x - 2) < 0`, then x ______ 2

If a < b and c < 0, then `a/c` ______ `b/c`

If |x − 1| ≤ 2, then – 1 ______ x ______ 3

If |3x – 7| > 2, then x ______ `5/3` or x ______ 3

If p > 0 and q < 0, then p + q ______ p

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 6 Linear InequalitiesExercise [Pages 107 - 113]

#### Short Answer and Solve for x, the inequalities 1 to 12

`4/(x + 1) ≤ 3 ≤ 6/(x + 1)`, (x > 0)

`(|x - 2| - 1)/(|x - 2| - 2) ≤ 0`

`1/(|x| - 3) ≤ 1/2`

|x − 1| ≤ 5, |x| ≥ 2

`5 ≤ (2 - 3x)/4 ≤ 9`

4x + 3 ≥ 2x + 17, 3x – 5 < – 2.

A company manufactures cassettes. Its cost and revenue functions are C(x) = 26,000 + 30x and R(x) = 43x, respectively, where x is the number of cassettes produced and sold in a week. How many cassettes must be sold by the company to realise some profit?

The water acidity in a pool is considerd normal when the average pH reading of three daily measurements is between 8.2 and 8.5. If the first two pH readings are 8.48 and 8.35, find the range of pH value for the third reading that will result in the acidity level being normal.

A solution of 9% acid is to be diluted by adding 3% acid solution to it. The resulting mixture is to be more than 5% but less than 7% acid. If there is 460 litres of the 9% solution, how many litres of 3% solution will have to be added?

A solution is to be kept between 40°C and 45°C. What is the range of temperature in degree fahrenheit, if the conversion formula is F = `9/5` C + 32?

The longest side of a triangle is twice the shortest side and the third side is 2cm longer than the shortest side. If the perimeter of the triangle is more than 166 cm then find the minimum length of the shortest side.

In drilling world’s deepest hole it was found that the temperature T in degree celcius, x km below the earth’s surface was given by T = 30 + 25(x – 3), 3 ≤ x ≤ 15. At what depth will the temperature be between 155°C and 205°C?

#### Long Answer

Solve the following system of inequalities `(2x + 1)/(7x - 1) > 5, (x + 7)/(x - 8) > 2`

Find the linear inequalities for which the shaded region in the given figure is the solution set.

Find the linear inequalities for which the shaded region in the given figure is the solution set.

Show that the following system of linear inequalities has no solution x + 2y ≤ 3, 3x + 4y ≥ 12, x ≥ 0, y ≥ 1

Solve the following system of linear inequalities:

3x + 2y ≥ 24, 3x + y ≤ 15, x ≥ 4

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

#### Objective Type Questions from 19 to 26

If x < 5, then ______.

– x < – 5

– x ≤ – 5

– x > – 5

– x ≥ – 5

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

`x/b < y/b`

`x/b ≤ y/b`

`x/b > y/b`

`x/b ≥ y/b`

If – 3x + 17 < – 13, then ______.

x ∈ (10, `oo`)

x ∈ [10, `oo`)

x ∈ (– `oo`, 10]

x ∈ [– 10, 10)

If x is a real number and |x| < 3, then ______.

x ≥ 3

– 3 < x < 3

x ≤ – 3

– 3 ≤ x ≤ 3

x and b are real numbers. If b > 0 and |x| > b, then ______.

x ∈ (– b, `oo`)

x ∈ [– `oo`, b)

x ∈ (– b, b)

x ∈ (– `oo`, – b) ∪ (b, `oo`)

If |x −1| > 5, then ______.

x ∈ (– 4, 6)

x ∈ [– 4, 6]

x ∈ [– `oo`, – 4) ∪ (6, `oo`)

x ∈ [– `oo`, – 4) ∪ [6, `oo`)

If |x + 2| ≤ 9, then ______.

x ∈ (– 7, 11)

x ∈ [– 11, 7]

x ∈ (– `oo`, – 7) ∪ (11, `oo`)

x ∈ (– `oo`, – 7) ∪ [11, `oo`)

The inequality representing the following graph is ______.

|x| < 5

|x| ≤ 5

|x| > 5

|x| ≥ 5

#### Objective the Questions from 27 to 30

Solution of a linear inequality in variable x is represented on number line in ______.

x ∈ (– `oo,` 5)

x ∈ (– `oo`, 5]

x ∈ [5, `oo`)

x ∈ (5, `oo`)

Solution of a linear inequality in variable x is represented on number line in ______.

x ∈ `(9/2, oo)`

x ∈ `[9/2, oo)`

x ∈ `[-oo, 9/2)`

x ∈ `(-oo, 9/2]`

Solution of a linear inequality in variable x is represented on number line in ______.

x ∈ `(- oo, 7/2)`

x ∈ `(-oo, 7/2]`

x ∈ `[7/2, -oo)`

x ∈ `(7/2, oo)`

Solution of a linear inequality in variable x is represented on number line in ______.

x ∈ `(– oo`, – 2)

x ∈ (– `oo`, – 2]

x ∈ (– 2, `oo`]

x ∈ [– 2, `oo`)

#### State whether the following statement is True or False:

If x < y and b < 0, then `x/"b" < y/"b"`

True

False

If xy > 0, then x > 0 and y < 0

True

False

If xy > 0, then x < 0 and y < 0

True

False

If xy < 0, then x < 0 and y < 0

True

False

If x < –5 and x < –2, then x ∈ (– `∞`, – 5)

True

False

If x < –5 and x > 2, then x ∈ (– 5, 2)

True

False

If x > –2 and x < 9, then x ∈ (– 2, 9)

True

False

If |x| > 5, then x ∈ (– `oo`, – 5) ∪ [5, `oo`)

True

False

If |x| ≤ 4, then x ∈ [– 4, 4]

True

False

Graph of x < 3 is

True

False

Graph of x ≥ 0 is

True

False

Graph of y ≤ 0 is

True

False

Solution set of x ≥ 0 and y ≤ 0 is

True

False

Solution set of x ≥ 0 and y ≤ 1 is

True

False

Solution set of x + y ≥ 0 is

True

False

#### Fill in the blanks of the following:

If – 4x ≥ 12, then x ______ – 3.

If `(-3)/4 x ≤ – 3`, then x ______ 4.

If `2/(x + 2) > 0`, then x ______ –2.

If x > – 5, then 4x ______ –20.

If x > y and z < 0, then – xz ______ – yz.

If p > 0 and q < 0, then p – q ______ p.

If |x + 2| > 5, then x ______ – 7 or x ______ 3.

If – 2x + 1 ≥ 9, then x ______ – 4.

## Chapter 6: Linear Inequalities

## NCERT solutions for Mathematics Exemplar Class 11 chapter 6 - Linear Inequalities

NCERT solutions for Mathematics Exemplar Class 11 chapter 6 (Linear Inequalities) 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 Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 11 chapter 6 Linear Inequalities 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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