If x and a area real numbers such that `agt0 and |x|gta` then:

Options

A. `x in(-a,a)`
B. `x in(-a,a)`
C. `x in(-oo,-a)cup(a,oo)`
D. `x in(-a,oo)`

Answer

To solve the inequality given in the question, we need to analyze the expression |x| > a, where a is a positive real number.

### Step 1: Understand the absolute value inequality
The inequality |x| > a means that the distance of x from 0 is greater than a. This can be interpreted in two ways:
1. x is greater than a.
2. x is less than -a.

### Step 2: Write the two cases
From the definition of absolute value, we can break down the inequality into two separate cases:
1. \( x > a \)
2. \( x < -a \)

### Step 3: Combine the results
The solution to the inequality |x| > a can be expressed in interval notation. The values of x that satisfy either of the two cases can be combined:
- From case 1, we have \( x \) in the interval \( (a, \infty) \).
- From case 2, we have \( x \) in the interval \( (-\infty, -a) \).

Thus, the complete solution can be written as:
\[ x \in (-\infty, -a) \cup (a, \infty) \]

### Conclusion
Therefore, the solution to the inequality |x| > a, where a > 0, is:
\[ x \in (-\infty, -a) \cup (a, \infty) \]