- Solve 24x < 100, when
(i) x is a natural number
(ii) x is an integer
Solution:
We have the given inequality 24x < 100.
then, 24x/24 < 100/24
Thus, x < 25/6
(i) We know that 1, 2, 3, and 4 are the only natural numbers less than 25/6.
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 25/6 are …–3, –2, –1, 0, 1, 2, 3, 4.
Thus, when x is an integer, the solutions of the given inequality are …–3, –2, –1, 0, 1, 2, 3, 4.
Hence, in this case, the solution set is {…–3, –2, –1, 0, 1, 2, 3, 4}.
2: Solve –12x > 30, when
(i) x is a natural number
(ii) x is an integer
Solution:
The given inequality –12x > 30.
Then -12x > 30
⇒ -12x/-12 < 30/-12
⇒ x < –5/2
(i) There is no natural number less than –5/2
Thus, when x is a natural number, there is no solution of the given inequality.
(ii) The integers less than –5/2 are …, –5, –4, –3.
Thus, when x is an integer, the solutions of the given inequality are …, –5, –4, –3.
Hence, in this case, the solution set is {…, –5, –4, –3}.
3: Solve 5x– 3 < 7, when
(i) x is an integer
(ii) x is a real number
Solution:
The given inequality is 5x – 3 < 7
⇒ 2x – 3 < 7
⇒ 5x – 3 + 3 < 7 + 3
⇒ 5x < 10
⇒ x < 2
(i)We know the integers less than 2 are …, –4, –3, –2, –1, 0, 1.
Thus, when x is an integer, the solutions of the given inequality are …, –4, –3, –2, –1, 0, 1.
Hence, in this case, the solution set is {…, –4, –3, –2, –1, 0, 1}.
(ii) When x is a real number, the solutions of the inequality 5x – 3 <7 are given by x < 2 i.e., all real numbers x which are less than 2.
Thus, the solution set of the given inequality is x € (–∞, 2).
4: Solve 3x + 8 > 2, when
(i) x is an integer
(ii) x is a real number
Solution:
The given inequality is 3x + 8 > 2
⇒ 3x + 8 > 2
⇒ 3x + 8 – 8 > 2 – 8
⇒ 3x > -6
⇒ 3x/3 > -6/3
⇒ x > -2
(i)The integers greater than –2 are –1, 0, 1, 2, …
Thus, when x is an integer, the solutions of the given inequality are –1, 0, 1, 2 …
Hence, in this case, the solution set is {–1, 0, 1, 2, …}
(ii) When x is a real number, the solutions of the given inequality are all the real numbers, which are greater than –2.
Thus, in this case, the solution set is (– 2, ∞).
5: Solve the given inequality for real x: 4x + 3 < 5x + 7
Solution:
4x + 3 < 5x + 7
⇒ 4x + 3 – 7 < 5x + 7 – 7
⇒ 4x – 4 < 5x
⇒ 4x – 4 – 4x < 5x – 4x
⇒ –4 < x
Thus, all real numbers x, which are greater than –4, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–4, ∞).
6: Solve the given inequality for real x: 3x – 7 > 5x – 1
Solution:
3x – 7 > 5x – 1
⇒ 3x – 7 + 7 > 5x – 1 + 7
⇒ 3x > 5x + 6
⇒ 3x – 5x > 5x + 6 – 5x
⇒ – 2x > 6
⇒-2x/-2 < 6/-2
⇒ x < -3
Thus, all real numbers x, which are less than –3, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, –3).
7: Solve the given inequality for real x: 3(x – 1) ≤ 2 (x – 3)
Solution:
We have the given inequality 3(x – 1) ≤ 2(x – 3)
⇒ 3x – 3 ≤ 2x – 6
⇒ 3x – 3 + 3 ≤ 2x – 6 + 3
⇒ 3x ≤ 2x – 3
⇒ 3x – 2x ≤ 2x – 3 – 2x
⇒ x ≤ – 3
Thus, all real numbers x, which are less than or equal to –3, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, –3].
8: Solve the given inequality for real x: 3(2 – x) ≥ 2(1 – x)
Solution:
We have the given inequality 3(2 – x) ≥ 2(1 – x)
⇒ 6 – 3x ≥ 2 – 2x
⇒ 6 – 3x + 2x ≥ 2 – 2x + 2x
⇒ 6 – x ≥ 2
⇒ 6 – x – 6 ≥ 2 – 6
⇒ –x ≥ –4
⇒ x ≤ 4
Thus, all real numbers x, which are less than or equal to 4, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, 4].
- Solve the given inequality for real x: x + x/2 + x/3 < 11
Solution:
The given inequality is x + x/2 + x/3 < 11
x + x/2 + x/3 < 11
⇒ x(1 + 1/2 + 1/3) < 11
⇒ x(6+3+2/6) < 11
⇒11x/6 < 11
⇒x/6 < 11
⇒ x < 6
Thus, all real numbers x, which are less than 6, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, 6).
- Solve the given inequality for real x: x/3 > x/2 + 1
Solution:
We have the given inequality x/3 > x/2 + 1
x/3 > x/2 + 1
⇒ x/3 – x/2 > 1
⇒ (2x – 3x)/6 > 1
⇒ –x/6 > 1
⇒ -x > 6
⇒ x < -6
Thus, all real numbers x, which are less than –6, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, –6).
