To determine the domain of the variable [tex]\( x \)[/tex] in the equation
[tex]\[
\frac{1}{x^2-16} - \frac{4}{x-4} = \frac{1}{x+4},
\][/tex]
we need to identify any values of [tex]\( x \)[/tex] that would make any denominators zero, since division by zero is undefined.
1. Identify the denominators:
- The first term's denominator is [tex]\( x^2 - 16 \)[/tex].
- The second term's denominator is [tex]\( x - 4 \)[/tex].
- The third term's denominator is [tex]\( x + 4 \)[/tex].
2. Find values that make the denominators zero:
- For the first term [tex]\( \frac{1}{x^2 - 16} \)[/tex]:
[tex]\[
x^2 - 16 = 0 \implies x^2 = 16 \implies x = \pm 4.
\][/tex]
- For the second term [tex]\( \frac{4}{x - 4} \)[/tex]:
[tex]\[
x - 4 = 0 \implies x = 4.
\][/tex]
- For the third term [tex]\( \frac{1}{x+4} \)[/tex]:
[tex]\[
x + 4 = 0 \implies x = -4.
\][/tex]
3. Combine the excluded values:
The values that [tex]\( x \)[/tex] cannot take are [tex]\( x = 4 \)[/tex] and [tex]\( x = -4 \)[/tex].
So, compiling these results, we conclude that the values [tex]\( x = 4 \)[/tex] and [tex]\( x = -4 \)[/tex] make at least one of the denominators zero and hence must be excluded from the domain.
Therefore, the domain of the given equation is all real numbers except [tex]\( x = 4 \)[/tex] and [tex]\( x = -4 \)[/tex].
The correct choice is:
A. The domain is [tex]\( \{ x \mid x \text{ is a real number, } x \neq -4, 4 \} \)[/tex].
So, the filled answer box should be:
[tex]\[
\boxed{-4, 4}
\][/tex]
