Solve `3x + 8 gt 2`, when (i) x is an integer. (ii) x is a real number.

Explanation

To solve the inequality \(3x + 8 > 2\), we will break it down into two parts: (i) when \(x\) is an integer and (ii) when \(x\) is a real number.

### Step-by-Step Solution:

1. **Start with the given inequality:**
\[ 3x + 8 > 2 \]

2. **Isolate the term involving \(x\):**
Subtract 8 from both sides:
\[ 3x > 2 - 8 \]
Simplifying the right side gives:
\[ 3x > -6 \]

3. **Divide both sides by 3:**
\[ x > \frac{-6}{3} \]
This simplifies to:
\[ x > -2 \]

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

4. **Identify integer solutions:**
Since \(x\) must be greater than -2, the integer values of \(x\) that satisfy this inequality are:
\[ x = -1, 0, 1, 2, 3, \ldots \]
Therefore, the solution set for integer values is:
\[ x \in \{-1, 0, 1, 2, 3, \ldots\} \]

### (ii) When \(x\) is a real number:

5. **Identify real number solutions:**
The inequality \(x > -2\) means that \(x\) can take any real number greater than -2. Thus, the solution set for real numbers is:
\[ x \in (-2, \infty) \]

### Final Answers:
- (i) For integer values: \(x \in \{-1, 0, 1, 2, 3, \ldots\}\)
- (ii) For real values: \(x \in (-2, \infty)\)

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