To find the simplest form of the given expression
[tex]\[
\frac{x-1}{x^2-4 x+3} \div \frac{5 x-2}{x^2-9}
\][/tex]
we need to follow several steps, including simplification and performing the division of fractions. Here is the detailed, step-by-step process:
1. Factorize the Denominators:
Let's first factorize the denominators of the fractions.
The denominator of the first fraction is [tex]\( x^2 - 4x + 3 \)[/tex]:
[tex]\[
x^2 - 4x + 3 = (x - 1)(x - 3)
\][/tex]
The denominator of the second fraction is [tex]\( x^2 - 9 \)[/tex]:
[tex]\[
x^2 - 9 = (x + 3)(x - 3)
\][/tex]
2. Rewrite the Expression with Factorized Denominators:
Rewrite the given expression using the factorized forms:
[tex]\[
\frac{x-1}{(x - 1)(x - 3)} \div \frac{5x - 2}{(x + 3)(x - 3)}
\][/tex]
3. Simplify the Division of Fractions:
To divide by a fraction, multiply by its reciprocal:
[tex]\[
\frac{x-1}{(x - 1)(x - 3)} \cdot \frac{(x + 3)(x - 3)}{5x - 2}
\][/tex]
Now, simplify by canceling out common factors. Notice that [tex]\((x - 1)\)[/tex] and [tex]\((x - 3)\)[/tex] appear in the numerator and denominator:
[tex]\[
= \frac{(x-1) \cdot (x + 3)(x - 3)}{(x - 1)(x - 3) \cdot (5x - 2)}
\][/tex]
Cancel the common terms [tex]\((x - 1)\)[/tex] and [tex]\((x - 3)\)[/tex] from the numerator and the denominator:
[tex]\[
= \frac{x + 3}{5x - 2}
\][/tex]
So the simplest form of the given expression is:
[tex]\[
\frac{x + 3}{5x - 2}
\][/tex]
This matches the fourth option provided. Thus, the correct answer is:
[tex]\[
\boxed{\frac{x+3}{5 x-2}}
\][/tex]
