**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).