Question

warm-up
consider this rational expression:
\\(\frac{x^2 - 5x + 6}{2x^2 - 7x + 6}\\)
what is the simplified form of the expression, and what are the excluded values?
\\(\frac{x - 3}{2x - 3}\\); excluded values: \\(x = \frac{3}{2}\\) and \\(x = 2\\)
\\(\frac{x - 6}{2x - 7}\\); excluded values: \\(x = 0\\) and \\(x = \frac{7}{2}\\)
\\(\frac{x - 1}{2x - 1}\\); excluded values: \\(x = \frac{1}{2}\\) and \\(x = 5\\)
\\(\frac{x - 2}{2x - 3}\\); excluded values: \\(x = 1\\) and \\(x = 3\\)

Explanation:

Step1: Factor numerator

$x^2 - 5x + 6 = (x-2)(x-3)$

Step2: Factor denominator

$2x^2 - 7x + 6 = (2x-3)(x-2)$

Step3: Cancel common factor

$\frac{(x-2)(x-3)}{(2x-3)(x-2)} = \frac{x-3}{2x-3}$

Step4: Find excluded values

Set denominator $2x^2 -7x +6=0$, solve $(2x-3)(x-2)=0$
$2x-3=0 \implies x=\frac{3}{2}$; $x-2=0 \implies x=2$

Answer:

$\frac{x-3}{2x-3}$; excluded values: $x = \frac{3}{2}$ and $x = 2$