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Page 1
15. Linear Inequations
(Including Number Lines)
Exercise 15 (A)
Question 1.
If the replacement set is the set of natural numbers, solve.
(i) x – 5 < 0
(ii) x + 1 < 7
(iii) 3x – 4 > 6
(iv) 4x + 1 > 17
Solution:
(i) x – 5 < 0
x – 5 + 5 <0 + 5 ………(Adding 5)
=> x < 5
Required answer = {1, 2, 3, 4}
(ii) x + 1 = 7 => x + 1 – 1 = 7 – 1 (Subtracting 1)
=> x = 6
Required answer = {1, 2, 3, 4, 5, 6}
(iii) 3x – 4 > 6
3x – 4 + 4 > 6 + 4 (Adding 4)
=> 3x > 10
> …(Dividing by 3)
=> x >
=> x >
Required answer = { 4, 5, 6, …}
(iv) 4x + 1 = 17
=> 4x + 1 – 1 = 17 – 1 (Subtracting)
=> 4x = 16
=> = (Dividing by 4)
=> x = 4
Required answer = {4, 5, 6, …}
Question 2.
If the replacement set = {-6, -3, 0, 3, 6, 9}; find the truth set of the following:
(i) 2x – 1 > 9
(ii) 3x + 7 < 1
Solution:
(i) 2x – 1 > 9
? 2x – 1 + 1 > 9 + 1 (Adding 1)
? 2x > 10
? x > 5 (Dividing by 2)
? x > 5
Page 2
15. Linear Inequations
(Including Number Lines)
Exercise 15 (A)
Question 1.
If the replacement set is the set of natural numbers, solve.
(i) x – 5 < 0
(ii) x + 1 < 7
(iii) 3x – 4 > 6
(iv) 4x + 1 > 17
Solution:
(i) x – 5 < 0
x – 5 + 5 <0 + 5 ………(Adding 5)
=> x < 5
Required answer = {1, 2, 3, 4}
(ii) x + 1 = 7 => x + 1 – 1 = 7 – 1 (Subtracting 1)
=> x = 6
Required answer = {1, 2, 3, 4, 5, 6}
(iii) 3x – 4 > 6
3x – 4 + 4 > 6 + 4 (Adding 4)
=> 3x > 10
> …(Dividing by 3)
=> x >
=> x >
Required answer = { 4, 5, 6, …}
(iv) 4x + 1 = 17
=> 4x + 1 – 1 = 17 – 1 (Subtracting)
=> 4x = 16
=> = (Dividing by 4)
=> x = 4
Required answer = {4, 5, 6, …}
Question 2.
If the replacement set = {-6, -3, 0, 3, 6, 9}; find the truth set of the following:
(i) 2x – 1 > 9
(ii) 3x + 7 < 1
Solution:
(i) 2x – 1 > 9
? 2x – 1 + 1 > 9 + 1 (Adding 1)
? 2x > 10
? x > 5 (Dividing by 2)
? x > 5
Required answer = {6, 9}
(ii) 3x + 7 = 1
? 3x + 7 – 7 = 1 – 7 (Subtracting 7)
? 3x = – 6
? x = – 2
Required Answer = {-6, -3}
Question 3.
Solve 7 > 3x – 8; x ? N
Solution:
7 > 3x – 8
=> 7 – 3x > 3x – 3x – 8 (Subtracting 3x)
=> 7 – 7 – 3x > 3x – 3x – 8 – 7 (Subtracting 7)
=> -3x > -15
=> x < 5 (Dividing by -3)
Required Answer = {1, 2, 3, 4}
Note : Division by negative number reverses the inequality.
Question 4.
-17 < 9y – 8 ; y ? Z
Solution:
-17 < 9y – 8
=> -17 + 8 < 9y – 8 + 8 (Adding 8)
=> -9 < 9y
=> -1 < y (Dividing by 9)
Required number = {0, 1, 2, 3, 4, …}
Question 5.
Solve 9x – 7 = 28 + 4x; x ? W
Solution:
9x – 1 = 28 + 4x
=> 9x – 4x – 7 = 28 + 4x – 4x (Subtracting 4x)
=> 5x – 7 = 28
=> 5x – 7 + 7 = 28 + 7 (Adding 7)
=> 5x = 35
=> x = 7 (Dividing by 5)
Required answer = {0, 1, 2, 3, 4, 5, 6, 7}
Question 6.
Solve : x + 8 < 12 ; x ? W
Page 3
15. Linear Inequations
(Including Number Lines)
Exercise 15 (A)
Question 1.
If the replacement set is the set of natural numbers, solve.
(i) x – 5 < 0
(ii) x + 1 < 7
(iii) 3x – 4 > 6
(iv) 4x + 1 > 17
Solution:
(i) x – 5 < 0
x – 5 + 5 <0 + 5 ………(Adding 5)
=> x < 5
Required answer = {1, 2, 3, 4}
(ii) x + 1 = 7 => x + 1 – 1 = 7 – 1 (Subtracting 1)
=> x = 6
Required answer = {1, 2, 3, 4, 5, 6}
(iii) 3x – 4 > 6
3x – 4 + 4 > 6 + 4 (Adding 4)
=> 3x > 10
> …(Dividing by 3)
=> x >
=> x >
Required answer = { 4, 5, 6, …}
(iv) 4x + 1 = 17
=> 4x + 1 – 1 = 17 – 1 (Subtracting)
=> 4x = 16
=> = (Dividing by 4)
=> x = 4
Required answer = {4, 5, 6, …}
Question 2.
If the replacement set = {-6, -3, 0, 3, 6, 9}; find the truth set of the following:
(i) 2x – 1 > 9
(ii) 3x + 7 < 1
Solution:
(i) 2x – 1 > 9
? 2x – 1 + 1 > 9 + 1 (Adding 1)
? 2x > 10
? x > 5 (Dividing by 2)
? x > 5
Required answer = {6, 9}
(ii) 3x + 7 = 1
? 3x + 7 – 7 = 1 – 7 (Subtracting 7)
? 3x = – 6
? x = – 2
Required Answer = {-6, -3}
Question 3.
Solve 7 > 3x – 8; x ? N
Solution:
7 > 3x – 8
=> 7 – 3x > 3x – 3x – 8 (Subtracting 3x)
=> 7 – 7 – 3x > 3x – 3x – 8 – 7 (Subtracting 7)
=> -3x > -15
=> x < 5 (Dividing by -3)
Required Answer = {1, 2, 3, 4}
Note : Division by negative number reverses the inequality.
Question 4.
-17 < 9y – 8 ; y ? Z
Solution:
-17 < 9y – 8
=> -17 + 8 < 9y – 8 + 8 (Adding 8)
=> -9 < 9y
=> -1 < y (Dividing by 9)
Required number = {0, 1, 2, 3, 4, …}
Question 5.
Solve 9x – 7 = 28 + 4x; x ? W
Solution:
9x – 1 = 28 + 4x
=> 9x – 4x – 7 = 28 + 4x – 4x (Subtracting 4x)
=> 5x – 7 = 28
=> 5x – 7 + 7 = 28 + 7 (Adding 7)
=> 5x = 35
=> x = 7 (Dividing by 5)
Required answer = {0, 1, 2, 3, 4, 5, 6, 7}
Question 6.
Solve : x + 8 < 12 ; x ? W
Solution:
Question 7.
Solve -5 (x + 4) > 30 ; x ? Z
Solution:
Question 8.
Solve the inquation 8 – 2x > x – 5 ; x ? N.
Solution:
x = 1, 2, 3, 4 (x ? N)
Solution set = {1, 2, 3, 4}
Page 4
15. Linear Inequations
(Including Number Lines)
Exercise 15 (A)
Question 1.
If the replacement set is the set of natural numbers, solve.
(i) x – 5 < 0
(ii) x + 1 < 7
(iii) 3x – 4 > 6
(iv) 4x + 1 > 17
Solution:
(i) x – 5 < 0
x – 5 + 5 <0 + 5 ………(Adding 5)
=> x < 5
Required answer = {1, 2, 3, 4}
(ii) x + 1 = 7 => x + 1 – 1 = 7 – 1 (Subtracting 1)
=> x = 6
Required answer = {1, 2, 3, 4, 5, 6}
(iii) 3x – 4 > 6
3x – 4 + 4 > 6 + 4 (Adding 4)
=> 3x > 10
> …(Dividing by 3)
=> x >
=> x >
Required answer = { 4, 5, 6, …}
(iv) 4x + 1 = 17
=> 4x + 1 – 1 = 17 – 1 (Subtracting)
=> 4x = 16
=> = (Dividing by 4)
=> x = 4
Required answer = {4, 5, 6, …}
Question 2.
If the replacement set = {-6, -3, 0, 3, 6, 9}; find the truth set of the following:
(i) 2x – 1 > 9
(ii) 3x + 7 < 1
Solution:
(i) 2x – 1 > 9
? 2x – 1 + 1 > 9 + 1 (Adding 1)
? 2x > 10
? x > 5 (Dividing by 2)
? x > 5
Required answer = {6, 9}
(ii) 3x + 7 = 1
? 3x + 7 – 7 = 1 – 7 (Subtracting 7)
? 3x = – 6
? x = – 2
Required Answer = {-6, -3}
Question 3.
Solve 7 > 3x – 8; x ? N
Solution:
7 > 3x – 8
=> 7 – 3x > 3x – 3x – 8 (Subtracting 3x)
=> 7 – 7 – 3x > 3x – 3x – 8 – 7 (Subtracting 7)
=> -3x > -15
=> x < 5 (Dividing by -3)
Required Answer = {1, 2, 3, 4}
Note : Division by negative number reverses the inequality.
Question 4.
-17 < 9y – 8 ; y ? Z
Solution:
-17 < 9y – 8
=> -17 + 8 < 9y – 8 + 8 (Adding 8)
=> -9 < 9y
=> -1 < y (Dividing by 9)
Required number = {0, 1, 2, 3, 4, …}
Question 5.
Solve 9x – 7 = 28 + 4x; x ? W
Solution:
9x – 1 = 28 + 4x
=> 9x – 4x – 7 = 28 + 4x – 4x (Subtracting 4x)
=> 5x – 7 = 28
=> 5x – 7 + 7 = 28 + 7 (Adding 7)
=> 5x = 35
=> x = 7 (Dividing by 5)
Required answer = {0, 1, 2, 3, 4, 5, 6, 7}
Question 6.
Solve : x + 8 < 12 ; x ? W
Solution:
Question 7.
Solve -5 (x + 4) > 30 ; x ? Z
Solution:
Question 8.
Solve the inquation 8 – 2x > x – 5 ; x ? N.
Solution:
x = 1, 2, 3, 4 (x ? N)
Solution set = {1, 2, 3, 4}
Question 9.
Solve the inequality 18 – 3 (2x – 5) > 12; x ? W.
Solution:
Question 10.
Solve : + 15 < 17; x ? W.
Solution:
Question 11.
Solve : -3 + x < 2, x ? N
Solution:
Page 5
15. Linear Inequations
(Including Number Lines)
Exercise 15 (A)
Question 1.
If the replacement set is the set of natural numbers, solve.
(i) x – 5 < 0
(ii) x + 1 < 7
(iii) 3x – 4 > 6
(iv) 4x + 1 > 17
Solution:
(i) x – 5 < 0
x – 5 + 5 <0 + 5 ………(Adding 5)
=> x < 5
Required answer = {1, 2, 3, 4}
(ii) x + 1 = 7 => x + 1 – 1 = 7 – 1 (Subtracting 1)
=> x = 6
Required answer = {1, 2, 3, 4, 5, 6}
(iii) 3x – 4 > 6
3x – 4 + 4 > 6 + 4 (Adding 4)
=> 3x > 10
> …(Dividing by 3)
=> x >
=> x >
Required answer = { 4, 5, 6, …}
(iv) 4x + 1 = 17
=> 4x + 1 – 1 = 17 – 1 (Subtracting)
=> 4x = 16
=> = (Dividing by 4)
=> x = 4
Required answer = {4, 5, 6, …}
Question 2.
If the replacement set = {-6, -3, 0, 3, 6, 9}; find the truth set of the following:
(i) 2x – 1 > 9
(ii) 3x + 7 < 1
Solution:
(i) 2x – 1 > 9
? 2x – 1 + 1 > 9 + 1 (Adding 1)
? 2x > 10
? x > 5 (Dividing by 2)
? x > 5
Required answer = {6, 9}
(ii) 3x + 7 = 1
? 3x + 7 – 7 = 1 – 7 (Subtracting 7)
? 3x = – 6
? x = – 2
Required Answer = {-6, -3}
Question 3.
Solve 7 > 3x – 8; x ? N
Solution:
7 > 3x – 8
=> 7 – 3x > 3x – 3x – 8 (Subtracting 3x)
=> 7 – 7 – 3x > 3x – 3x – 8 – 7 (Subtracting 7)
=> -3x > -15
=> x < 5 (Dividing by -3)
Required Answer = {1, 2, 3, 4}
Note : Division by negative number reverses the inequality.
Question 4.
-17 < 9y – 8 ; y ? Z
Solution:
-17 < 9y – 8
=> -17 + 8 < 9y – 8 + 8 (Adding 8)
=> -9 < 9y
=> -1 < y (Dividing by 9)
Required number = {0, 1, 2, 3, 4, …}
Question 5.
Solve 9x – 7 = 28 + 4x; x ? W
Solution:
9x – 1 = 28 + 4x
=> 9x – 4x – 7 = 28 + 4x – 4x (Subtracting 4x)
=> 5x – 7 = 28
=> 5x – 7 + 7 = 28 + 7 (Adding 7)
=> 5x = 35
=> x = 7 (Dividing by 5)
Required answer = {0, 1, 2, 3, 4, 5, 6, 7}
Question 6.
Solve : x + 8 < 12 ; x ? W
Solution:
Question 7.
Solve -5 (x + 4) > 30 ; x ? Z
Solution:
Question 8.
Solve the inquation 8 – 2x > x – 5 ; x ? N.
Solution:
x = 1, 2, 3, 4 (x ? N)
Solution set = {1, 2, 3, 4}
Question 9.
Solve the inequality 18 – 3 (2x – 5) > 12; x ? W.
Solution:
Question 10.
Solve : + 15 < 17; x ? W.
Solution:
Question 11.
Solve : -3 + x < 2, x ? N
Solution:
Question 12.
Solve : 4x – 5 > 10 – x, x ? {0, 1, 2, 3, 4, 5, 6, 7}
Solution:
Solution set = {4, 5, 6, 7}
Question 13.
Solve : 15 – 2(2x – 1) < 15, x ? Z.
Solution:
Question 14.
Solve : > , x ? W.
Solution:
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Oct 06, 2025
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