Solve `24 x<100`, when (i) x is a natural number. (ii) x is an integer.

Explanation

To solve the inequality \( 24x < 100 \), we will break it down into two parts based on the conditions given: (i) when \( x \) is a natural number and (ii) when \( x \) is an integer.

### Step-by-Step Solution:

1. **Start with the inequality**:
\[ 24x < 100 \]

2. **Divide both sides by 24**:
\[ x < \frac{100}{24} \]

3. **Simplify the fraction**:
\[ \frac{100}{24} = \frac{25}{6} \approx 4.1667 \]
Thus, we have:
\[ x < 4.1667 \]

### (i) When \( x \) is a natural number:

4. **Identify natural numbers less than 4.1667**:
The natural numbers are \( 1, 2, 3, 4, \ldots \). Since \( x \) must be less than 4.1667, the possible values for \( x \) are:
\[ x = 1, 2, 3, 4 \]

5. **Conclusion for natural numbers**:
Therefore, the solution when \( x \) is a natural number is:
\[ x \in \{1, 2, 3, 4\} \]

### (ii) When \( x \) is an integer:

6. **Identify integers less than 4.1667**:
The integers include all positive and negative whole numbers. Thus, the integers satisfying \( x < 4.1667 \) are:
\[ x = \ldots, -2, -1, 0, 1, 2, 3, 4 \]

7. **Conclusion for integers**:
Therefore, the solution when \( x \) is an integer is:
\[ x \in \{ \ldots, -2, -1, 0, 1, 2, 3, 4 \} \]

### Final Answers:
- (i) When \( x \) is a natural number: \( x \in \{1, 2, 3, 4\} \)
- (ii) When \( x \) is an integer: \( x \in \{ \ldots, -2, -1, 0, 1, 2, 3, 4 \} \)