To determine how many values of [tex]\( x \)[/tex] must be excluded in the expression [tex]\(\frac{x-2}{(x+9)(x-5)}\)[/tex], we need to identify the values of [tex]\( x \)[/tex] that make the denominator zero. This is because the expression is undefined when the denominator is equal to zero.
The denominator of the given rational expression is [tex]\((x+9)(x-5)\)[/tex]. We need to solve for [tex]\( x \)[/tex] when the denominator is zero:
[tex]\[
(x + 9)(x - 5) = 0
\][/tex]
This equation will be satisfied when either [tex]\( x + 9 = 0 \)[/tex] or [tex]\( x - 5 = 0 \)[/tex].
Solving these individual equations:
1. [tex]\( x + 9 = 0 \)[/tex]
[tex]\[
x = -9
\][/tex]
2. [tex]\( x - 5 = 0 \)[/tex]
[tex]\[
x = 5
\][/tex]
So, the values of [tex]\( x \)[/tex] that make the denominator zero are [tex]\( x = -9 \)[/tex] and [tex]\( x = 5 \)[/tex].
Since the expression is undefined for these values, they must be excluded.
Thus, there are 2 values of [tex]\( x \)[/tex] that must be excluded.
Therefore, the answer is:
[tex]\[
\boxed{2}
\][/tex]
