Question

simplify. \\(\dfrac{x - \dfrac{10}{x - 3}}{1 - \dfrac{2}{x - 3}}\\)

Explanation:

Step1: Simplify numerator

First, simplify the numerator \( x - \frac{10}{x - 3} \). To do this, get a common denominator of \( x - 3 \):
\( x = \frac{x(x - 3)}{x - 3} \), so \( x - \frac{10}{x - 3} = \frac{x(x - 3) - 10}{x - 3} \).
Expand the numerator: \( x(x - 3) - 10 = x^2 - 3x - 10 \).
Factor the quadratic: \( x^2 - 3x - 10 = (x - 5)(x + 2) \). So the numerator simplifies to \( \frac{(x - 5)(x + 2)}{x - 3} \).

Step2: Simplify denominator

Next, simplify the denominator \( 1 - \frac{2}{x - 3} \). Get a common denominator of \( x - 3 \):
\( 1 = \frac{x - 3}{x - 3} \), so \( 1 - \frac{2}{x - 3} = \frac{x - 3 - 2}{x - 3} = \frac{x - 5}{x - 3} \).

Step3: Divide numerator by denominator

Now, we have the complex fraction \( \frac{\frac{(x - 5)(x + 2)}{x - 3}}{\frac{x - 5}{x - 3}} \). Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes \( \frac{(x - 5)(x + 2)}{x - 3} \times \frac{x - 3}{x - 5} \).

Step4: Cancel common factors

Cancel out the common factors \( (x - 5) \) and \( (x - 3) \) from the numerator and denominator. This leaves us with \( x + 2 \) (assuming \( x
eq 3 \) and \( x
eq 5 \) to avoid division by zero).

Answer:

\( x + 2 \)