Question 1. Solve: 12x < 50, when
(i) x ∈ R
Solution:
Given: 12x < 50
Dividing both sides by 12, we get
12x/ 12 < 50/12
⇒ x < 25/6
When x is a real number, the solution of the given inequation is (-∞, 25/6).
(ii) x ∈ Z
Solution:
Since, 4 < 25/6 < 5
So, when x is an integer, the maximum possible value of x is 4.
The solution of the given inequation is {…, –2, –1, 0, 1, 2, 3, 4}.
(iii) x ∈ N
Solution:
Since 4 < 25/6 < 5
So, when x is a natural number, the maximum possible value of x is 4.
We know that the natural numbers start from 1.
Hence the solution of the given inequation is {1, 2, 3, 4}.
Question 2. Solve: − 4x > 30, when
(i) x ∈ R
Solution:
Given: − 4x > 30
So when we divide by 4, we get
⇒ − 4x/4 > 30/4
⇒ − x > 15/2
⇒ x < – 15/2
When x is a real number, the solution of the given inequation is (-∞, −15/2).
(ii) x ∈ Z
Solution:
Since, − 8 < − 15/2 < − 7
So, when x is an integer, the maximum possible value of x is − 8.
The solution of the given inequation is {…, –11, –10, − 9, −8}.
(iii) x ∈ N
Solution:
As natural numbers start from 1 and can never be negative.
Hence when x is a natural number, the solution of the given inequation is ∅.
Question 3. Solve: 4x-2 < 8, when
(i) x ∈ R
Solution:
Given: 4x – 2 < 8
4x – 2 + 2 < 8 + 2
⇒ 4x < 10
So dividing by 4 on both sides we get,
4x/4 < 10/4
⇒ x < 5/2
When x is a real number, the solution of the given inequation is (-∞, 5/2).
(ii) x ∈ Z
Solution:
Since, 2 < 5/2 < 3
So, when x is an integer, the maximum possible value of x is 2.
The solution of the given inequation is {…, –2, –1, 0, 1, 2}.
(iii) x ∈ N
Solution:
Since, 2 < 5/2 < 3
So, when x is a natural number, the maximum possible value of x is 2.
We know that the natural numbers start from 1.
The solution of the given inequation is {1, 2}.
Question 4. Solve: 3x – 7 > x + 1
Solution:
Given:
3x – 7 > x + 1
⇒ 3x – 7 + 7 > x + 1 + 7
⇒ 3x > x + 8
⇒ 3x – x > x + 8 – x
⇒ 2x > 8
Dividing both sides by 2, we get
2x/2 > 8/2
⇒ x > 4
∴ The solution of the given inequation is (4, ∞).
Question 5. Solve: x + 5 > 4x – 10
Solution:
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
Dividing both sides by 3, we get
3x/3 < 15/3
⇒ x < 5
∴ The solution of the given inequation is (-∞, 5).
Question 6. Solve: 3x + 9 ≥ –x + 19
Solution:
Given: 3x + 9 ≥ –x + 19
⇒ 3x + 9 – 9 ≥ –x + 19 – 9
⇒ 3x ≥ –x + 10
⇒ 3x + x ≥ –x + 10 + x
⇒ 4x ≥ 10
Dividing both sides by 4, we get
4x/4 ≥ 10/4
⇒ x ≥ 5/2
∴ The solution of the given inequation is [5/2, ∞).
Question 7. Solve: 2 (3 – x) ≥ x/5 + 4
Solution:
Given: 2 (3 – x) ≥ x/5 + 4
⇒ 6 – 2x ≥ x/5 + 4
⇒ 6 – 2x ≥ (x+20)/5
⇒ 5(6 – 2x) ≥ (x + 20)
⇒ 30 – 10x ≥ x + 20
⇒ 30 – 20 ≥ x + 10x
⇒ 10 ≥11x
⇒ 11x ≤ 10
Dividing both sides by 11, we get
11x/11 ≤ 10/11
⇒ x ≤ 10/11
∴ The solution of the given inequation is (-∞, 10/11].
Question 8. Solve:
Solution:
Given:
≤ Multiplying both the sides by 5 we get,
× 5 ≤ × 5 ⇒ (3x – 2) ≤ 5(4x – 3)/2
⇒ 3x – 2 ≤ (20x – 15)/2
Multiplying both the sides by 2 we get,
(3x – 2) × 2 ≤ (20x – 15)/2 × 2
⇒ 6x – 4 ≤ 20x – 15
⇒ 20x – 15 ≥ 6x – 4
⇒ 20x – 15 + 15 ≥ 6x – 4 + 15
⇒ 20x ≥ 6x + 11
⇒ 20x – 6x ≥ 6x + 11 – 6x
⇒ 14x ≥ 11
Dividing both sides by 14, we get
14x/14 ≥ 11/14
⇒ x ≥ 11/14
∴ The solution of the given inequation is [11/14, ∞).
Question 9. Solve: –(x – 3) + 4 < 5 – 2x
Solution:
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
∴ The solution of the given inequation is (–∞, –2).
Question 10. Solve:
Solution:
Given:
< – ⇒
< ⇒
< ⇒
< Multiplying both the sides by 20 we get,
× 20 < × 20 ⇒ 4x < 2 – 5x
⇒ 4x + 5x < 2 – 5x + 5x
⇒ 9x < 2
Divide both sides by 9, we get
9x/9 < 2/9
⇒ x < 2/9
∴ The solution of the given inequation is (-∞, 2/9).
Question 11. Solve:
Solution:
Given:
≤ ⇒
≤ Multiply both the sides by 5 we get,
× 5 ≤ × 5 ⇒ 2x – 2 ≤
⇒ 7 (2x – 2) ≤ 5 (6 + 3x)
⇒ 14x – 14 ≤ 30 + 15x
⇒ 14x – 14 + 14 ≤ 30 + 15x + 14
⇒ 14x ≤ 44 + 15x
⇒ 14x – 44 ≤ 44 + 15x – 44
⇒ 14x – 44 ≤ 15x
⇒ 15x ≥ 14x – 44
⇒ 15x – 14x ≥ 14x – 44 – 14x
⇒ x ≥ –44
∴ The solution of the given inequation is [–44, ∞).
Question 12. Solve: 5x/2 + 3x/4 ≥ 39/4
Solution:
Given: 5x/2 + 3x/4 ≥ 39/4
By taking LCM, we get:
≥ 39/4 ⇒ 13x/4 ≥ 39/4
Multiplying both the sides by 4 we get,
13x/4 × 4 ≥ 39/4 × 4
⇒ 13x ≥ 39
Divide both sides by 13, we get
13x/13 ≥ 39/13
⇒ x ≥ 39/13
⇒ x ≥ 3
∴ The solution of the given inequation is [3, ∞).
Question 13. Solve:
Solution:
Given:
+ 4 < – 2 Subtracting both sides by 4 we get,
⇒
+ 4 – 4 < – 2 – 4 ⇒
< – 6 ⇒
< ⇒
< After Cross multiplying, we get,
5 (x – 1) < 3 (x – 35)
⇒ 5x – 5 < 3x – 105
⇒ 5x – 5 + 5 < 3x – 105 + 5
⇒ 5x < 3x – 100
⇒ 5x – 3x < 3x – 100 – 3x
⇒ 2x < –100
Divide both sides by 2, we get
2x/2 < -100/2
⇒ x < -50
∴ The solution of the given inequation is (-∞, -50).
Question 14. Solve:
Solution:
Given:
– 3 < – 2 Adding 3 on both sides we get,
– 3 + 3 < – 2 + 3 ⇒
< + 1 ⇒
< ⇒
< After Cross multiplying, we get,
3(2x + 3) < 4(x – 1)
⇒ 6x + 9 < 4x – 4
⇒ 6x + 9 – 9 < 4x – 4 – 9
⇒ 6x < 4x – 13
⇒ 6x – 4x < 4x – 13 – 4x
⇒ 2x < –13
Dividing both sides by 2, we get
2x/2 < -13/2
⇒ x < -13/2
∴ The solution of the given inequation is (-∞, -13/2).