Let x and b are real numbers. If ` b gt 0 and |x| gtb`, then

Options

A. `x in (-b, infty)`
B. ` x in [-infty, b)`
C. `x in (-b, b)`
D. `x in (- infty, -b) cup(b, infty)`

Explanation

To solve the problem, we need to analyze the given inequality involving the absolute value of \( x \) and the real number \( b \).

### Given:
- \( b > 0 \)
- \( |x| > b \)

### Step-by-step Solution:

1. **Understanding the Absolute Value Inequality**:
The inequality \( |x| > b \) means that the distance of \( x \) from 0 is greater than \( b \). This can be interpreted in two parts:
- \( x > b \)
- \( x < -b \)

2. **Breaking Down the Inequality**:
Since \( |x| > b \), we can express this as:
\[ x > b \quad \text{or} \quad x < -b \]

3. **Combining the Results**:
The solutions to the inequality can be combined using the union of the two intervals derived from the inequalities:
- From \( x > b \), we have the interval \( (b, \infty) \).
- From \( x < -b \), we have the interval \( (-\infty, -b) \).

4. **Final Solution**:
Therefore, the solution set for \( x \) can be expressed as:
\[ x \in (-\infty, -b) \cup (b, \infty) \]

### Conclusion:
The final answer is:
\[ x \in (-\infty, -b) \cup (b, \infty) \]