To determine for which value of [tex]\( x \)[/tex] the given rational expression is equal to zero, we need to analyze when the numerator of the expression is zero, while ensuring that the denominator is non-zero to avoid division by zero.
The given rational expression is:
[tex]\[
\frac{x-4}{(x+5)(x-1)}
\][/tex]
Here are the steps to find the value of [tex]\( x \)[/tex]:
1. Set the numerator equal to zero:
The numerator of the rational expression is [tex]\( x - 4 \)[/tex]. For the overall expression to be zero, the numerator must be zero. Therefore, we set:
[tex]\[
x - 4 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
We solve the equation [tex]\( x - 4 = 0 \)[/tex]:
[tex]\[
x = 4
\][/tex]
Thus, [tex]\( x = 4 \)[/tex] is a candidate for making the rational expression equal to zero.
3. Check the denominator:
The denominator of the rational expression is [tex]\( (x + 5)(x - 1) \)[/tex]. We need to ensure that this denominator is not zero when [tex]\( x = 4 \)[/tex]:
[tex]\[
(4 + 5)(4 - 1) = 9 \cdot 3 \neq 0
\][/tex]
Since the denominator is non-zero when [tex]\( x = 4 \)[/tex], the rational expression is defined at this point.
Thus, the rational expression [tex]\(\frac{x-4}{(x+5)(x-1)}\)[/tex] is equal to zero for:
[tex]\[
\boxed{4}
\][/tex]
