To express the domain of the function $y = \frac{3x - 2}{(x - 5)(x - 7)}$ in interval notation, we need to identify any values of $x$ that make the denominator zero. These are the values where the function is undefined.

1. Set the denominator equal to zero: $(x - 5)(x - 7) = 0$ This gives us two solutions: $x = 5 \quad \text{and} \quad x = 7$

Thus, the function is undefined at $x = 5$ and $x = 7$.

2. The domain of the function includes all real numbers except $x = 5$ and $x = 7$. In interval notation, this is written as: $(-\infty, 5) \cup (5, 7) \cup (7, \infty)$

This represents the complete domain where the function is defined.

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Would you like more details on this or have any questions?

Here are 5 related questions for further exploration:

1. How do you find vertical asymptotes for rational functions like this one?
2. What happens to the function as $x$ approaches 5 or 7?
3. How do you determine the horizontal asymptote of this function?
4. What is the behavior of the function for very large or very small values of $x$?
5. How do you find the critical points of this function?

Tip: Always check the denominator of a rational function to find values that should be excluded from its domain!