Question

study the example shown.
which expression should replace the word
umerator\ 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)}\\)
\\(3x - 11\\)
\\(3x - 5\\)
\\(x - 11\\)
\\(x - 5\\)

Explanation:

Step1: Simplify the numerator

We start with the expression \(\frac{2x - 8}{x^{2}-7x + 10}-\frac{x - 3}{x^{2}-7x + 10}\). Since the denominators are the same, we can combine the numerators: \((2x - 8)-(x - 3)\).

Step2: Expand and combine like terms

Expanding the numerator: \(2x-8 - x + 3\). Combining like terms: \((2x - x)+(-8 + 3)=x-5\)? Wait, no, wait: \(2x - x= x\), \(-8+3=-5\)? Wait, no, wait the original step in the problem is \(\frac{2x - 8-(x - 3)}{x^{2}-7x + 10}\). Let's do that again: \(2x-8 - x + 3=(2x - x)+(-8 + 3)=x - 5\)? Wait, no, the options have \(3x - 11\), etc. Wait, maybe I made a mistake. Wait, the problem's middle step is \(\frac{2x - 8-(x - 3)}{x^{2}-7x + 10}\). Let's compute \(2x-8-(x - 3)=2x-8 - x + 3=(2x - x)+(-8 + 3)=x - 5\)? No, that's not matching the options. Wait, maybe the original problem is \(\frac{2x - 8}{x^{2}-7x + 10}-\frac{x - 3}{x^{2}-7x + 10}\), but maybe the first numerator is \(2x - 8\) and the second is \(x - 3\), but when subtracting, it's \(2x - 8 - (x - 3)=2x - 8 - x + 3= x - 5\)? But the options are \(3x - 11\), \(3x - 5\), \(x - 11\), \(x - 5\). Wait, maybe the original problem was \(\frac{2x - 8}{x^{2}-7x + 10}-\frac{x - 3}{x^{2}-7x + 10}\) but maybe I misread the first numerator. Wait, the user's image shows \(\frac{2x - 8}{x^{2}-7x + 10}-\frac{x - 3}{x^{2}-7x + 10}=\frac{2x - 8-(x - 3)}{x^{2}-7x + 10}\). Let's compute \(2x - 8 - x + 3\) again: \(2x - x = x\), \(-8 + 3=-5\), so \(x - 5\). Wait, but the options have \(x - 5\) as one of them (the last option). Wait, the options are:

1. \(3x - 11\)
2. \(3x - 5\)
3. \(x - 11\)
4. \(x - 5\)

So the numerator is \(2x - 8-(x - 3)=2x - 8 - x + 3=x - 5\). So the numerator is \(x - 5\).

Answer:

The expression that should replace "numerator" is \(x - 5\) (the fourth option, assuming the options are in order: first \(3x - 11\), second \(3x - 5\), third \(x - 11\), fourth \(x - 5\)). So the answer is the option with \(x - 5\).