Question

study the example shown. which expression should replace the word “numerator” in the work shown?
\\(\frac{2x - 8}{x^2 - 7x + 10} - \frac{x - 3}{x^2 - 7x + 10}\\)
\\(= \frac{2x - 8 - (x - 3)}{x^2 - 7x + 10}\\)
\\(\frac{\text{numerator}}{(x - 2)(x - 5)}\\)
options: (x - 5), (3x - 11), (3x - 5), (x - 11)

Explanation:

Step1: Simplify the numerator

We start with the numerator \(2x - 8-(x - 3)\). First, distribute the negative sign: \(2x - 8 - x+ 3\).

Step2: Combine like terms

Combine the \(x\) terms: \(2x - x=x\). Combine the constant terms: \(-8 + 3=-5\). So the simplified numerator is \(x - 5\)? Wait, no, wait. Wait, \(2x - 8-(x - 3)=2x - 8 - x + 3=(2x - x)+(-8 + 3)=x-5\)? Wait, that doesn't match the options. Wait, maybe I made a mistake. Wait, let's re - do it. \(2x-8-(x - 3)=2x-8 - x + 3\). \(2x-x=x\), \(-8 + 3=-5\). Wait, but the options have \(x - 5\)? Wait, the options given: \(x - 5\), \(3x-11\), \(3x - 5\), \(x - 11\). Wait, maybe I misread the original problem. Wait, the original expression is \(\frac{2x - 8}{x^{2}-7x + 10}-\frac{x - 3}{x^{2}-7x + 10}\). So the numerator of the combined fraction is \(2x-8-(x - 3)\). Let's expand it again: \(2x-8 - x+3=(2x - x)+(-8 + 3)=x - 5\). Wait, but \(x - 5\) is one of the options (the first option). Wait, maybe I made a mistake in the sign. Wait, no, \(2x-8-(x - 3)=2x-8 - x + 3=x - 5\). So the numerator is \(x - 5\).

Answer:

\(x - 5\) (the first option with text "x - 5")