# NCERT Solutions for Class 11 Maths Maths Chapter 6

## Linear Inequalities Class 11

### Exercise 6.1 : Solutions of Questions on Page Number : 122

Q1 :

Solve 24x < 100, when (i) x is a natural number (ii) x is an integer

The given inequality is 24x < 100.

(i) It is evident that 1, 2, 3, and 4 are the only natural numbers less than.

Thus, when x is a natural number, the solutions of the given inequality are 1, 2, 3, and 4.

Hence, in this case, the solution set is {1, 2, 3, 4}.

(ii) The integers less than are …â€“3, â€“2, â€“1, 0, 1, 2, 3, 4.

Thus, when x is an integer, the solutions of the given inequality are

…â€“3, â€“2, â€“1, 0, 1, 2, 3, 4.

Hence, in this case, the solution set is {…â€“3, â€“2, â€“1, 0, 1, 2, 3, 4}.

Q2 :

Solve -12x> 30, when

(i) xis a natural number (ii) xis an integer

The given inequality is â€“12x> 30.

(i) There is no natural number less than.

Thus, when xis a natural number, there is no solution of the given inequality.

(ii) The integers less than are …, â€“5, â€“4, â€“3.

Thus, when xis an integer, the solutions of the given inequality are

…, â€“5, â€“4, â€“3.

Hence, in this case, the solution set is {…, â€“5, â€“4, â€“3}.

Q3 :

Solve 5x- 3 < 7, when

(i) xis an integer (ii) xis a real number

Q4 :

Solve 3x+ 8 > 2, when

(i) xis an integer (ii) xis a real number

Q5 :

Solve the given inequality for real x: 4x + 3 < 5x + 7

Q6 :

Solve the given inequality for real x: 3x - 7 > 5x - 1

Q7 :

Solve the given inequality for real x: 3(x - 1) Ã¢”°Â¤ 2 (x - 3)

Q8 :

Solve the given inequality for real x: 3(2 - x) Ã¢”°Â¥ 2(1 - x)

Q9 :

Solve the given inequality for real x:

Q10 :

Solve the given inequality for real x:

Q11 :

Solve the given inequality for real x:

Q12 :

Solve the given inequality for real x:

Q13 :

Solve the given inequality for real x: 2(2x + 3) - 10 < 6 (x - 2)

Q14 :

Solve the given inequality for real x: 37 ­- (3x + 5) Ã¢”°Â¥ 9x - 8(x - 3)

Q15 :

Solve the given inequality for real x:

Q16 :

Solve the given inequality for real x:

Q17 :

Solve the given inequality and show the graph of the solution on number line: 3x - 2 < 2x +1

Q18 :

Solve the given inequality and show the graph of the solution on number line: 5x - 3 Ã¢”°Â¥ 3x - 5

Q19 :

Solve the given inequality and show the graph of the solution on number line: 3(1 - x) < 2 (x + 4)

Q20 :

Solve the given inequality and show the graph of the solution on number line:

Q21 :

Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks.

Q22 :

To receive Grade 'A' in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita's marks in first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade 'A' in the course.

Q23 :

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

Q24 :

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.

Q25 :

A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm 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 5 cm longer than the second?

[Hint: If xis the length of the shortest board, then x, (x+ 3) and 2xare the lengths of the second and third piece, respectively. Thus, x= (x + 3) + 2x Ã¢”°Â¤91 and 2x Ã¢”°Â¥(x + 3) + 5]

### Exercise 6.2 : Solutions of Questions on Page Number : 127

Q1 :

Solve the given inequality graphically in two-dimensional plane: x + y < 5

Q2 :

Solve the given inequality graphically in two-dimensional plane: 2x + y Ã¢”°Â¥ 6

Q3 :

Solve the given inequality graphically in two-dimensional plane: 3x + 4y Ã¢”°Â¤ 12

Q4 :

Solve the given inequality graphically in two-dimensional plane: y + 8 Ã¢”°Â¥ 2x

Q5 :

Solve the given inequality graphically in two-dimensional plane: x - y Ã¢”°Â¤ 2

Q6 :

Solve the given inequality graphically in two-dimensional plane: 2x - 3y > 6

Q7 :

Solve the given inequality graphically in two-dimensional plane: -3x + 2y Ã¢”°Â¥ -6

Q8 :

Solve the given inequality graphically in two-dimensional plane: 3y - 5x < 30

Q9 :

Solve the given inequality graphically in two-dimensional plane: y < -2

Q10 :

Solve the given inequality graphically in two-dimensional plane: x > -3

### Exercise 6.3 : Solutions of Questions on Page Number : 129

Q1 :

Solve the following system of inequalities graphically: x Ã¢”°Â¥ 3, y Ã¢”°Â¥ 2

Q2 :

Solve the following system of inequalities graphically: 3x + 2y Ã¢”°Â¤ 12, x Ã¢”°Â¥ 1, y Ã¢”°Â¥ 2

Q3 :

Solve the following system of inequalities graphically: 2x + yÃ¢”°Â¥ 6, 3x + 4y Ã¢”°Â¤ 12

Q4 :

Solve the following system of inequalities graphically: x + yÃ¢”°Â¥ 4, 2x - y > 0

Q5 :

Solve the following system of inequalities graphically: 2x - y > 1, x - 2y < -1

Q6 :

Solve the following system of inequalities graphically: x + y Ã¢”°Â¤ 6, x + y Ã¢”°Â¥ 4

Q7 :

Solve the following system of inequalities graphically: 2x + yÃ¢”°Â¥ 8, x + 2y Ã¢”°Â¥ 10

Q8 :

Solve the following system of inequalities graphically: x + y Ã¢”°Â¤ 9, y > x, x Ã¢”°Â¥ 0

Q9 :

Solve the following system of inequalities graphically: 5x + 4y Ã¢”°Â¤ 20, x Ã¢”°Â¥ 1, y Ã¢”°Â¥ 2

Q10 :

Solve the following system of inequalities graphically: 3x + 4y Ã¢”°Â¤ 60, x + 3y Ã¢”°Â¤ 30, x Ã¢”°Â¥ 0, y Ã¢”°Â¥ 0

Q11 :

Solve the following system of inequalities graphically: 2x + yÃ¢”°Â¥ 4, x + y Ã¢”°Â¤ 3, 2x - 3y Ã¢”°Â¤ 6

Q12 :

Solve the following system of inequalities graphically:

x - 2y ≤ 3, 3x + 4y Ã¢â€°Â¥ 12, x Ã¢â€°Â¥ 0, y Ã¢â€°Â¥ 1

Q13 :

[[Q]]

Solve the following system of inequalities graphically:

4x + 3y ≤ 60, y Ã¢â€°Â¥ 2x, x Ã¢â€°Â¥ 3, x, y Ã¢â€°Â¥ 0

Q14 :

Solve the following system of inequalities graphically: 3x + 2y Ã¢”°Â¤ 150, x + 4y Ã¢”°Â¤ 80, x Ã¢”°Â¤ 15, y Ã¢”°Â¥ 0, x Ã¢”°Â¥ 0

Q15 :

Solve the following system of inequalities graphically: x + 2y Ã¢”°Â¤ 10, x + y Ã¢”°Â¥ 1, x - y Ã¢”°Â¤ 0, x Ã¢”°Â¥ 0, y Ã¢”°Â¥ 0

### Exercise Miscellaneous : Solutions of Questions on Page Number : 132

Q1 :

Solve the inequality 2 Ã¢”°Â¤ 3x - 4 Ã¢”°Â¤ 5

Q2 :

Solve the inequality 6 Ã¢”°Â¤ -3(2x - 4) < 12

Q3 :

Solve the inequality

Q4 :

Solve the inequality

Q5 :

Solve the inequality

Q6 :

Solve the inequality

Q7 :

Solve the inequalities and represent the solution graphically on number line: 5x + 1 > -24, 5x - 1 < 24

Q8 :

Solve the inequalities and represent the solution graphically on number line: 2(x - 1) < x + 5, 3(x + 2) > 2 - x

Q9 :

Solve the following inequalities and represent the solution graphically on number line:

3x - 7 > 2(x - 6), 6 - x > 11 - 2x

Q10 :

Solve the inequalities and represent the solution graphically on number line: 5(2x - 7) - 3(2x + 3) Ã¢”°Â¤ 0, 2x + 19 Ã¢”°Â¤ 6x + 47

Q11 :

A solution is to be kept between 68°F and 77°F. What is the range in temperature in degree Celsius (C) if the Celsius/Fahrenheit (F) conversion formula is given by

Q12 :

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 we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?

Q13 :

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?

Q14 :

IQ of a person is given by the formula

Where MA is mental age and CA is chronological age. If 80 IQ 140 for a group of 12 years old children, find the range of their mental age.