We start with the given rational function:
[tex]\[
\frac{x-7}{x+9}
\][/tex]
For any rational expression, the values of [tex]\( x \)[/tex] that make the denominator zero are considered excluded values because division by zero is undefined.
Here, the denominator of the rational function is [tex]\( x + 9 \)[/tex]. We set the denominator equal to zero to find the excluded values:
[tex]\[
x + 9 = 0
\][/tex]
Solving this equation for [tex]\( x \)[/tex]:
[tex]\[
x = -9
\][/tex]
Thus, the value [tex]\( x = -9 \)[/tex] makes the denominator zero, and hence it is the excluded value.
From the list provided:
- [tex]\( x = 7 \)[/tex]: This does not make the denominator zero, so it is not an excluded value.
- [tex]\( x = 9 \)[/tex]: This does not make the denominator zero, so it is not an excluded value.
- [tex]\( x = -9 \)[/tex]: This makes the denominator zero, so it is an excluded value.
- None of the above: Since [tex]\( x = -9 \)[/tex] is indeed an excluded value, this option is incorrect.
Therefore, the correct answer is:
[tex]\[
x = -9
\][/tex]
