Which of the following is equal to the rational expression when [tex][tex]$x \neq 2$[/tex][/tex] or -4?

[tex]
\frac{5(x-2)}{(x-2)(x+4)}
[/tex]

A. [tex][tex]$\frac{1}{x-4}$[/tex][/tex]

B. [tex][tex]$\frac{1}{x-2}$[/tex][/tex]

C. [tex][tex]$\frac{5}{x-2}$[/tex][/tex]

D. [tex][tex]$\frac{5}{x+4}$[/tex][/tex]



Answer :

To simplify the given rational expression
[tex]\[ \frac{5(x-2)}{(x-2)(x+4)}, \][/tex]
we need to factor and then cancel out any common terms in the numerator and the denominator.

Step-by-step process:

1. Identify common factors: Notice that the numerator [tex]\(5(x-2)\)[/tex] and the denominator [tex]\((x-2)(x+4)\)[/tex] both have the factor [tex]\((x-2)\)[/tex].
2. Cancel the common factor: Since [tex]\(x \neq 2\)[/tex] and [tex]\(x \neq -4\)[/tex], we can safely cancel [tex]\((x-2)\)[/tex] from both the numerator and the denominator.
[tex]\[ \frac{5(x-2)}{(x-2)(x+4)} = \frac{5 \cancel{(x-2)}}{\cancel{(x-2)} (x+4)} \][/tex]

This leaves us with:
[tex]\[ \frac{5}{x+4} \][/tex]

So, the simplified expression is:
[tex]\[ \frac{5}{x+4} \][/tex]

Among the given options, this corresponds to:
D. [tex]\(\frac{5}{x+4}\)[/tex]

Therefore, the correct answer is [tex]\(\boxed{D}\)[/tex].