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 since division by zero is undefined. Here are the steps to solve this:
1. Examine the denominator of the expression [tex]\(\frac{x-2}{(x+9)(x-5)}\)[/tex].
2. Set the denominator equal to zero:
[tex]\[
(x+9)(x-5) = 0
\][/tex]
3. Solve the equation [tex]\((x+9)(x-5) = 0\)[/tex] for [tex]\( x \)[/tex]. This equation will be zero when any of the factors are zero.
4. Break down the equation into two separate equations:
[tex]\[
x+9 = 0 \quad \text{or} \quad x-5 = 0
\][/tex]
5. Solve each equation individually:
- For [tex]\( x+9 = 0 \)[/tex]:
[tex]\[
x = -9
\][/tex]
- For [tex]\( x-5 = 0 \)[/tex]:
[tex]\[
x = 5
\][/tex]
6. Therefore, the values [tex]\( x = -9 \)[/tex] and [tex]\( x = 5 \)[/tex] will make the denominator zero.
7. We must exclude these values from the domain of the expression because division by zero is undefined.
8. Count the number of distinct values that need to be excluded: [tex]\( x = -9 \)[/tex] and [tex]\( x = 5 \)[/tex].
In conclusion, there are 2 values of [tex]\( x \)[/tex] that must be excluded to avoid division by zero.
Hence, the number of values of [tex]\( x \)[/tex] that must be excluded is:
[tex]\[
\boxed{2}
\][/tex]
