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Page 1
15. Linear Inequations
Exercise 15.1
1. Question
Solve the following linear inequations in R
12x < 50, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given 12x < 50
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is 4.
Thus, the solution of the given inequation is {…, –2, –1, 0, 1, 2, 3, 4}.
iii. x ? N
As , when x is a natural number, the maximum possible value of x is 4 and we know the natural
numbers start from 1.
Thus, the solution of the given inequation is {1, 2, 3, 4}.
2. Question
Solve the following linear inequations in R
–4x > 30, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given –4x > 30
i. x ? R
Page 2
15. Linear Inequations
Exercise 15.1
1. Question
Solve the following linear inequations in R
12x < 50, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given 12x < 50
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is 4.
Thus, the solution of the given inequation is {…, –2, –1, 0, 1, 2, 3, 4}.
iii. x ? N
As , when x is a natural number, the maximum possible value of x is 4 and we know the natural
numbers start from 1.
Thus, the solution of the given inequation is {1, 2, 3, 4}.
2. Question
Solve the following linear inequations in R
–4x > 30, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given –4x > 30
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is –8.
Thus, the solution of the given inequation is {…, –11, –10, –9, –8}.
iii. x ? N
As natural numbers start from 1 and can never be negative, when x is a natural number, the solution of the
given inequation is Ø.
3. Question
Solve the following linear inequations in R
4x – 2 < 8, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given 4x – 2 < 8
? 4x – 2 + 2 < 8 + 2
? 4x < 10
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is 2.
Thus, the solution of the given inequation is {…, –2, –1, 0, 1, 2}.
iii. x ? N
As , when x is a natural number, the maximum possible value of x is 2 and we know the natural
numbers start from 1.
Thus, the solution of the given inequation is {1, 2}.
4. Question
Solve the following linear inequations in R
3x – 7 > x + 1
Answer
Given 3x – 7 > x + 1
? 3x – 7 + 7 > x + 1 + 7
? 3x > x + 8
Page 3
15. Linear Inequations
Exercise 15.1
1. Question
Solve the following linear inequations in R
12x < 50, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given 12x < 50
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is 4.
Thus, the solution of the given inequation is {…, –2, –1, 0, 1, 2, 3, 4}.
iii. x ? N
As , when x is a natural number, the maximum possible value of x is 4 and we know the natural
numbers start from 1.
Thus, the solution of the given inequation is {1, 2, 3, 4}.
2. Question
Solve the following linear inequations in R
–4x > 30, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given –4x > 30
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is –8.
Thus, the solution of the given inequation is {…, –11, –10, –9, –8}.
iii. x ? N
As natural numbers start from 1 and can never be negative, when x is a natural number, the solution of the
given inequation is Ø.
3. Question
Solve the following linear inequations in R
4x – 2 < 8, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given 4x – 2 < 8
? 4x – 2 + 2 < 8 + 2
? 4x < 10
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is 2.
Thus, the solution of the given inequation is {…, –2, –1, 0, 1, 2}.
iii. x ? N
As , when x is a natural number, the maximum possible value of x is 2 and we know the natural
numbers start from 1.
Thus, the solution of the given inequation is {1, 2}.
4. Question
Solve the following linear inequations in R
3x – 7 > x + 1
Answer
Given 3x – 7 > x + 1
? 3x – 7 + 7 > x + 1 + 7
? 3x > x + 8
? 3x – x > x + 8 – x
? 2x > 8
? x > 4
Thus, the solution of the given inequation is (4, 8).
5. Question
Solve the following linear inequations in R
x + 5 > 4x – 10
Answer
Given x + 5 > 4x – 10
? x + 5 – 5 > 4x – 10 – 5
? x > 4x – 15
? 4x – 15 < x
? 4x – 15 – x < x – x
? 3x – 15 < 0
? 3x – 15 + 15 < 0 + 15
? 3x < 15
? x < 5
Thus, the solution of the given inequation is (–8, 5).
6. Question
Solve the following linear inequations in R
3x + 9 = –x + 19
Answer
Given 3x + 9 = –x + 19
? 3x + 9 – 9 = –x + 19 – 9
? 3x = –x + 10
? 3x + x = –x + 10 + x
? 4x = 10
Thus, the solution of the given inequation is .
7. Question
Solve the following linear inequations in R
Page 4
15. Linear Inequations
Exercise 15.1
1. Question
Solve the following linear inequations in R
12x < 50, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given 12x < 50
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is 4.
Thus, the solution of the given inequation is {…, –2, –1, 0, 1, 2, 3, 4}.
iii. x ? N
As , when x is a natural number, the maximum possible value of x is 4 and we know the natural
numbers start from 1.
Thus, the solution of the given inequation is {1, 2, 3, 4}.
2. Question
Solve the following linear inequations in R
–4x > 30, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given –4x > 30
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is –8.
Thus, the solution of the given inequation is {…, –11, –10, –9, –8}.
iii. x ? N
As natural numbers start from 1 and can never be negative, when x is a natural number, the solution of the
given inequation is Ø.
3. Question
Solve the following linear inequations in R
4x – 2 < 8, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given 4x – 2 < 8
? 4x – 2 + 2 < 8 + 2
? 4x < 10
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is 2.
Thus, the solution of the given inequation is {…, –2, –1, 0, 1, 2}.
iii. x ? N
As , when x is a natural number, the maximum possible value of x is 2 and we know the natural
numbers start from 1.
Thus, the solution of the given inequation is {1, 2}.
4. Question
Solve the following linear inequations in R
3x – 7 > x + 1
Answer
Given 3x – 7 > x + 1
? 3x – 7 + 7 > x + 1 + 7
? 3x > x + 8
? 3x – x > x + 8 – x
? 2x > 8
? x > 4
Thus, the solution of the given inequation is (4, 8).
5. Question
Solve the following linear inequations in R
x + 5 > 4x – 10
Answer
Given x + 5 > 4x – 10
? x + 5 – 5 > 4x – 10 – 5
? x > 4x – 15
? 4x – 15 < x
? 4x – 15 – x < x – x
? 3x – 15 < 0
? 3x – 15 + 15 < 0 + 15
? 3x < 15
? x < 5
Thus, the solution of the given inequation is (–8, 5).
6. Question
Solve the following linear inequations in R
3x + 9 = –x + 19
Answer
Given 3x + 9 = –x + 19
? 3x + 9 – 9 = –x + 19 – 9
? 3x = –x + 10
? 3x + x = –x + 10 + x
? 4x = 10
Thus, the solution of the given inequation is .
7. Question
Solve the following linear inequations in R
Answer
Given
? 30 – 10x = x + 20
? x + 20 = 30 – 10x
? x + 20 – 20 = 30 – 10x – 20
? x = 10 – 10x
? x + 10x = 10 – 10x + 10x
? 11x = 10
Thus, the solution of the given inequation is .
8. Question
Solve the following linear inequations in R
Answer
Given
? 6x – 4 = 20x – 15
? 20x – 15 = 6x – 4
? 20x – 15 + 15 = 6x – 4 + 15
? 20x = 6x + 11
? 20x – 6x = 6x + 11 – 6x
Page 5
15. Linear Inequations
Exercise 15.1
1. Question
Solve the following linear inequations in R
12x < 50, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given 12x < 50
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is 4.
Thus, the solution of the given inequation is {…, –2, –1, 0, 1, 2, 3, 4}.
iii. x ? N
As , when x is a natural number, the maximum possible value of x is 4 and we know the natural
numbers start from 1.
Thus, the solution of the given inequation is {1, 2, 3, 4}.
2. Question
Solve the following linear inequations in R
–4x > 30, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given –4x > 30
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is –8.
Thus, the solution of the given inequation is {…, –11, –10, –9, –8}.
iii. x ? N
As natural numbers start from 1 and can never be negative, when x is a natural number, the solution of the
given inequation is Ø.
3. Question
Solve the following linear inequations in R
4x – 2 < 8, when
i. x ? R
ii. x ? Z
iii. x ? N
Answer
Given 4x – 2 < 8
? 4x – 2 + 2 < 8 + 2
? 4x < 10
i. x ? R
When x is a real number, the solution of the given inequation is .
ii. x ? Z
As , when x is an integer, the maximum possible value of x is 2.
Thus, the solution of the given inequation is {…, –2, –1, 0, 1, 2}.
iii. x ? N
As , when x is a natural number, the maximum possible value of x is 2 and we know the natural
numbers start from 1.
Thus, the solution of the given inequation is {1, 2}.
4. Question
Solve the following linear inequations in R
3x – 7 > x + 1
Answer
Given 3x – 7 > x + 1
? 3x – 7 + 7 > x + 1 + 7
? 3x > x + 8
? 3x – x > x + 8 – x
? 2x > 8
? x > 4
Thus, the solution of the given inequation is (4, 8).
5. Question
Solve the following linear inequations in R
x + 5 > 4x – 10
Answer
Given x + 5 > 4x – 10
? x + 5 – 5 > 4x – 10 – 5
? x > 4x – 15
? 4x – 15 < x
? 4x – 15 – x < x – x
? 3x – 15 < 0
? 3x – 15 + 15 < 0 + 15
? 3x < 15
? x < 5
Thus, the solution of the given inequation is (–8, 5).
6. Question
Solve the following linear inequations in R
3x + 9 = –x + 19
Answer
Given 3x + 9 = –x + 19
? 3x + 9 – 9 = –x + 19 – 9
? 3x = –x + 10
? 3x + x = –x + 10 + x
? 4x = 10
Thus, the solution of the given inequation is .
7. Question
Solve the following linear inequations in R
Answer
Given
? 30 – 10x = x + 20
? x + 20 = 30 – 10x
? x + 20 – 20 = 30 – 10x – 20
? x = 10 – 10x
? x + 10x = 10 – 10x + 10x
? 11x = 10
Thus, the solution of the given inequation is .
8. Question
Solve the following linear inequations in R
Answer
Given
? 6x – 4 = 20x – 15
? 20x – 15 = 6x – 4
? 20x – 15 + 15 = 6x – 4 + 15
? 20x = 6x + 11
? 20x – 6x = 6x + 11 – 6x
? 14x = 11
Thus, the solution of the given inequation is .
9. Question
Solve the following linear inequations in R
–(x – 3) + 4 < 5 – 2x
Answer
Given –(x – 3) + 4 < 5 – 2x
? –x + 3 + 4 < 5 – 2x
? –x + 7 < 5 – 2x
? –x + 7 – 7 < 5 – 2x – 7
? –x < –2x – 2
? –x + 2x < –2x – 2 + 2x
? x < –2
Thus, the solution of the given inequation is (–8, –2).
10. Question
Solve the following linear inequations in R
Answer
Given
? 4x < 2 – 5x
? 4x + 5x < 2 – 5x + 5x
? 9x < 2
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