NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequations (रैखिक असामिकाएँ) Exercise 6.1 or Exercise 6.2 or Exercise 6.3 or Miscellaneous in English to view online or download in PDF format free.

Class: | 11 |

Subject: | Maths – गणित |

Chapter 6: | Linear Inequations |

Table of Contents

- 1 NCERT Solutions for Class 11 Maths Chapter 6
- 1.1 Class 11 Maths Chapter 6 Linear Inequations
- 1.1.1 Important Terms Related to Linear Inequations
- 1.1.2 Solve 3x + 8 >2, when x is an integer.
- 1.1.3 Solve the following inequality for real x: 4x+3
- 1.1.4 Solve the following inequality for real x: 3x-7>5x-1.
- 1.1.5 Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.
- 1.1.6 Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.
- 1.1.7 The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.
- 1.1.8 A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second?
- 1.1.9 Solve the following inequality: 2 ≤ 3x – 4 ≤ 5.
- 1.1.10 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?

- 1.1 Class 11 Maths Chapter 6 Linear Inequations

## NCERT Solutions for Class 11 Maths Chapter 6

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### Class 11 Maths Chapter 6 Linear Inequations

All solutions are in accordance with Latest CBSE Curriculum for CBSE, UP Board, MP Board, Gujrat Board and all other board following NCERT Books.

- View Online Exercise 6.1 Sols
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- View Online Exercise 6.2 Sols
- Download Exercise 6.2 in PDF
- View Online Exercise 6.3 Sols
- Download Exercise 6.3 in PDF
- View Online Miscellaneous Exercise
- Download Miscellaneous Exercise 6
- NCERT Books for Class 11
- Revision Books for Class 11

##### 11 Maths Exercise 6.1 Solutions

NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequations Exercise 6.1 is given below. For other questions, please visit to Exercise 6.2 or Exercise 6.3 or Miscellaneous Solutions. Visit to Class 11 Maths main page or Top of the page.

##### 11 Maths Exercise 6.2 Solutions

Class 11 Maths Chapter 6 Linear Inequations Exercise 6.2 is given below. For other questions, please visit to Exercise 6.1 or Exercise 6.3 or Miscellaneous Solutions. Visit to Class 11 Maths main page or Top of the page.

##### 11 Maths Exercise 6.3 Solutions

11 Maths Chapter 6 Exercise 6.3 is given below. For other questions, please visit to Exercise 6.1 or Exercise 6.2 or Miscellaneous Solutions. Visit to Class 11 Maths main page or Top of the page.

##### 11 Maths Miscellaneous Exercise Sols

NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequations Miscellaneous Exercise 6 is given below. For other questions, please visit to Exercise 6.1 or Exercise 6.2 or Exercise 6.3 Solutions. Visit to Class 11 Maths main page or Top of the page.

Visit to Class 11 Maths main page or Top of the page

*Solution Set*: A solution to an inequality is a number which when substituted for the variable, makes the inequality true. The set of all solutions of an inequality is called the solution set of the inequality.*Replacement Set*: The set from which values of the variable (involved in the inequality) are chosen is called replacement set.- Procedure to solve a linear inequality in one variables.
- Simplify both sides by removing graph symbols and collecting like terms.
- Remove fractions (or decimals) by multiplying both sides by appropriate factor (L.C.M of denomination or a power of 10 in case of decimals.)
- Isolate the variable on one side and all constants on the other side. Collect like terms whenever possible.
- Make the coefficient of the variable.
- Choose the solution set from the replacement set.

###### Know More

- The graph of the inequality ax + by > c is one of the half planes and is called the solution region.
- When the inequality involves the sign ≤ or ≥ then the points on the line are included in the solution region but if it has the sign < or > then the points on the line are not included in the solution region and it has to be drawn as a dotted line.
- The common values of the variable form the required solution of the given system of linear inequalities in one variable.
- The common part of coordinate plane is the required solution of the system of linear inequations in two variables when solved by graphical method.
- Ask your doubts and share it with your friends.
- Download NCERT Solutions Offline Apps for Class 11.

Table of Contents

- 1 Solve 3x + 8 >2, when x is an integer.
- 2 Solve the following inequality for real x: 4x+3
- 3 Solve the following inequality for real x: 3x-7>5x-1.
- 4 Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.
- 5 Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.
- 6 The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.
- 7 A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second?
- 8 Solve the following inequality: 2 ≤ 3x – 4 ≤ 5.
- 9 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?

#### Solve 3x + 8 >2, when x is an integer.

⇒3x>-6

⇒x>(-6)/3

⇒x>-2

x is an integer.

The integers greater than –2 are –1, 0, 1, 2, …

Thus, when x is an integer, the solutions of the given inequality are –1, 0, 1, 2 …

Hence, in this case, the solution set is {–1, 0, 1, 2, …}.

#### Solve the following inequality for real x: 4x+3<6x+7

#### Solve the following inequality for real x: 3x-7>5x-1.

⇒ 3x – 7 + 7 > 5x – 1 + 7

⇒ 3x > 5x + 6

⇒ 3x – 5x > 5x + 6 – 5x

⇒ – 2x > 6

⇒-x>6/2

⇒-x>3

⇒x<-3 Thus, all real numbers x, which are less than –3, are the solutions of the given inequality. Hence, the solution set of the given inequality is (–∞, –3).

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

Since both the integers are smaller than 10, therefore

x + 2 < 10 ⇒ x < 10 – 2 ⇒ x < 8 … (i) Also, the sum of the two integers is more than 11. ∴ x + (x + 2) > 11

⇒ 2x + 2 > 11

⇒ 2x > 11 – 2

⇒ 2x > 9

⇒x>9/2

⇒x>4.5 …(ii)

From (i) and (ii), we obtain that the value of x can be 4,5,6 or 7. .

Since x is an odd number, x can take the values, 5 and 7.

Hence, the required possible pairs of numbers are (5, 7) and (7, 9).

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

Since both the integers are larger than 5,

x > 5 … (1)

Also, the sum of the two integers is less than 23.

x + (x + 2) < 23 ⇒ 2x + 2 < 23 ⇒ 2x < 23 – 2 ⇒2x<21 ⇒x<21/2 ⇒x<10.5 … (2) From (1) and (2), we obtain 5 < x < 10.5. Since x is an even number, so x can take the values, 6, 8 and 10. Hence, the required possible pairs are (6, 8), (8, 10) and (10, 12).

#### The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.

Then, length of the longest side = 3x cm and the length of the third side = (3x – 2) cm

Since the perimeter of the triangle is at least 61 cm,

⇒x+3x+(3x-2)≥61

⇒7x-2≥61

⇒7x≥63

⇒x≥9

Hence, the minimum length of the shortest side is 9 cm.

#### A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second?

Since the three lengths are to be cut from a single piece of board of length 91 cm,

x + (x + 3) + 2x≤ 91

⇒ 4x + 3 ≤ 91

⇒ 4x ≤ 91 – 3

⇒ 4x ≤ 88

⇒x≤22 … (1)

Also, the third piece is at least 5 cm longer than the second piece.

∴ 2x ≥ (x + 3) + 5

⇒ 2x ≥ x + 8

⇒ x ≥ 8 … (2)

From (1) and (2), we obtain, 8 ≤ x ≤ 22

Thus, the possible length of the shortest board is greater than or equal to 8 cm but less than or equal to 22 cm.

#### Solve the following inequality: 2 ≤ 3x – 4 ≤ 5.

⇒ 2 + 4 ≤ 3x – 4 + 4 ≤ 5 + 4

⇒ 6 ≤ 3x ≤ 9

⇒ 2 ≤ x ≤ 3

Therefore, all the real numbers, x, which are greater than or equal to 2 but less than or equal to 3, are the solutions of the given inequality. The solution set for the given inequality is [2, 3].

#### 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?

It is evident that the amount of acid contained in the resulting mixture is 45% of 1125 litres.

This resulting mixture will contain more than 25% but less than 30% acid content.

∴ 25% of (1125 + x) < 45% of 1125 < 30% of (1125 + x) ⇒25/100 (1125+x) < 45/100×1125<30/100 (1125+x) ⇒25(1125+x) < 45×1125<30(1125+x) ⇒5(1125+x) < 9×1125<6(1125+x) ⇒5625+5x < 10125<6750+6x ⇒5x<10125-5625 < 6750-5625+6x ⇒5x<4500 < 1125+6x ⇒5x<4500 and 4500<1125+6x ⇒x<900 and 3375<6x ⇒x<900 and 562.5