Exercise 6.1
1. Solve 
when:
(i) 
is a natural number
(ii) is an integer
Ans. Given:
Dividing both sides by 24, 
(i) When is a natural number then values of that make statement true are 1, 2, 3, 4. The solution set of inequality is {1, 2, 3, 4}.
(ii) When is an integer then values of that make statement true are 
The solution set of inequality is 
2. Solve 
when:
(i) is a natural number
(ii) is an integer
Ans. Given:
Dividing both sides by 
(i) When is a natural number then values of that make statement true are none.
(ii) When is an integer then values of that make statement true are 
The solution set of inequality is 
3. Solve 
when
(i) is an integer
(ii) is a real number
Ans. Given:
(i) When is an integer then values of that make statement true are 
The solution set of inequality is 
(ii) When is a real number then solution set of inequality is 
4. Solve 
when:
(i) is an integer
(ii) is a real number
Ans. Given:
(i) When is an integer then values of that make statement true are 
The solution set of inequality is 
(ii) When is a real number then solution set of inequality is 
Solve the inequalities in Exercises 5 to 16 for real 
5. 
Ans. Here
Therefore, the solution set is 
6. 
Ans. Here
Therefore, the solution set is 
7. 
Ans. Here
Therefore, the solution set is
8. 
Ans. Here
Therefore, the solution set is 
9. 
Ans. Here
Therefore, the solution set is 
10. 
Ans. Here
Therefore, the solution set is 
11. 
Ans. Here
Therefore, the solution set is 
12. 
Ans. Here
Therefore, the solution set is 
13. 
Ans. Here
Therefore, the solution set is 
14. 
Ans. Here
Therefore, the solution set is
15. 
Ans. Here
Therefore, the solution set is
16. 
Ans. Here
Therefore, the solution set is
Solve the inequalities in Exercises 17 to 20 and show the graph of the solution in each case on number line:
17. 
Ans. Here
The solution set is 
18. 
Ans. Here
The solution set is 
19. 
Ans. Here
The solution set is
20. 
Ans. Here
The solution set is 
21. Ravi obtained 70 and 75 marks in first two unit tests. Find the minimum marks he should get in the third test to have an average of at least 60 marks.
Ans. Let the marks obtained by Ravi in third test be 
According to question,
Therefore, minimum marks needed to be obtained by Ravi is 35.
22. To receive Grade ‘A’ in a course, one must obtain an average 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.
Ans. Let the marks obtained by Sunita in fifth examination be
According to question,
Therefore, minimum marks needed to be obtained by Sunita is 82.
23. Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.
Ans. Let and 
be two consecutive odd positive integers.
and 
and 
and 
and 7
Therefore, the required pairs of odd positive integers are (5, 7) and (7, 9).
24. Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.
Ans. Let and be two consecutive even positive integers.
and 
and 
and 
and 10
Therefore, the required pairs of even positive integers are (6,8), (8 10) and (10, 12).
25. The longest side of a triangle s 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.
Ans. Let the length of the shortest side be cm.
Length of longest side = 
cm and length of third side = 
cm
Perimeter of triangle = 
= 
cm
Now, 
Therefore, the minimum length of shortest side is 9 cm.
26. 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?
Ans. Let the length of the shortest side be cm.
Length of the second board = 
cm and length of third side = 
cm
Now, 
and 
and 
and 
and
Therefore, minimum length of shortest board is 8 cm and maximum length is 22 cm.
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