If `a!=0` then complete set of solution of `(x^2-2x+2^(|a|))/(x^2-a^2)>0` is

Options

A. `(-oo,-|a|)uu(|a|,oo)`
B. `(-oo,-a)uu(a,oo)`
C. `(-oo,-a)uu[a,oo)`
D. `(-|a|,|a|)

Answer

To solve the inequality \((x^2 - 2x + 2^{|a|})/(x^2 - a^2) > 0\) given that \(a \neq 0\), we can follow these steps:

### Step 1: Analyze the numerator
The numerator is \(x^2 - 2x + 2^{|a|}\). We can rewrite it as:
\[ x^2 - 2x + 2^{|a|} = (x - 1)^2 + (2^{|a|} - 1) \]
Since \((x - 1)^2 \geq 0\) for all \(x\), the expression \(x^2 - 2x + 2^{|a|}\) is always greater than or equal to \(2^{|a|} - 1\).

- If \(2^{|a|} - 1 > 0\) (which is true for \(|a| > 0\)), then the numerator is always positive.

### Step 2: Analyze the denominator
The denominator is \(x^2 - a^2\), which can be factored as:
\[ x^2 - a^2 = (x - a)(x + a) \]
This expression is zero when \(x = a\) or \(x = -a\). The sign of the denominator changes at these points.

### Step 3: Determine intervals
To find the intervals where the entire expression is positive, we need to analyze the sign of the denominator:
1. For \(x < -a\): both factors \((x - a)\) and \((x + a)\) are negative, so the denominator is positive.
2. For \(-a < x < a\): \((x - a)\) is negative and \((x + a)\) is positive, so the denominator is negative.
3. For \(x > a\): both factors are positive, so the denominator is positive.

### Step 4: Combine results
The expression \((x^2 - 2x + 2^{|a|})/(x^2 - a^2) > 0\) holds true when:
- The numerator is positive (which it is for all \(x\) since \(2^{|a|} > 1\)), and
- The denominator is positive.

Thus, the solution to the inequality is:
\[ x \in (-\infty, -a) \cup (a, \infty) \]

### Final Answer
The complete set of solutions is:
\[ (-\infty, -a) \cup (a, \infty) \]