If x is a real number and `|x | lt 5 ` , then

Options

A. `x le 5`
B. ` - 5 lt x lt 5`
C. ` x le -5`
D. `- 5 le x le 5`

Answer

To solve the inequality \( |x| < 5 \), we need to interpret what the absolute value means. The absolute value of a number represents its distance from zero on the number line, regardless of direction. Thus, the inequality \( |x| < 5 \) indicates that the distance of \( x \) from zero is less than 5.

### Step-by-Step Solution:

1. **Understanding the Absolute Value Inequality**:
The inequality \( |x| < 5 \) can be split into two separate inequalities:
\[ -5 < x < 5 \]
This means that \( x \) must be greater than \(-5\) and less than \(5\).

2. **Writing the Compound Inequality**:
From the above step, we can write the compound inequality:
\[ -5 < x < 5 \]

3. **Interpreting the Result**:
This tells us that \( x \) can take any value between \(-5\) and \(5\), not including \(-5\) and \(5\) themselves.

4. **Identifying the Correct Option**:
Based on the options provided:
- Option A: \( x \leq 5 \)
- Option B: \( -5 < x < 5 \) (This is correct)
- Option C: \( x \leq -5 \)
- Option D: \( -5 \leq x \leq 5 \)

The correct option that matches our derived inequality is:
\[ -5 < x < 5 \]

### Final Answer:
The solution to the inequality \( |x| < 5 \) is:
\[ -5 < x < 5 \]