If a, b are positive real numbers, then `|x|le a hArr`

Options

A. A. `-a le x le a`
B. B. `x le -a or x ge a`
C. C. `-a lt x lt a`
D. D. `x lt -a or x gt +a`

Answer

To solve the problem, we need to understand the implications of the given condition: \( |x| \leq a \), where \( a \) is a positive real number.

### Step-by-Step Solution:

1. **Understanding the Absolute Value Inequality**:
The expression \( |x| \leq a \) means that the distance of \( x \) from 0 on the number line is at most \( a \). This can be interpreted as:
\[ -a \leq x \leq a \]

2. **Breaking Down the Inequality**:
The inequality \( |x| \leq a \) can be split into two separate inequalities:
- \( x \geq -a \) (when \( x \) is non-negative)
- \( x \leq a \) (when \( x \) is non-negative)

3. **Combining the Results**:
From the two inequalities derived from the absolute value, we can combine them:
\[ -a \leq x \leq a \]
This means \( x \) lies in the interval from \( -a \) to \( a \).

4. **Conclusion**:
Therefore, the solution to the inequality \( |x| \leq a \) is that \( x \) must be in the range:
\[ x \in [-a, a] \]

### Final Answer:
The correct interpretation of the condition \( |x| \leq a \) is that \( x \) is between \( -a \) and \( a \), which can be expressed as:
\[ x \geq -a \quad \text{and} \quad x \leq a \]