For the rational function:

$r(x) = \frac{(x + 1)(x - 2)}{(x + 2)(x - 7)}$

Vertical asymptotes occur where the denominator equals zero, as long as the numerator is not zero at the same point. In this case, we need to find the values of $x$ that make the denominator zero.

The denominator is $(x + 2)(x - 7)$, so set each factor equal to zero and solve for $x$:

1. $x + 2 = 0 \implies x = -2$
2. $x - 7 = 0 \implies x = 7$

Thus, the vertical asymptotes occur at:

• $x = -2$ (smaller value)
• $x = 7$ (larger value)

Would you like further details or explanations?

Related Questions:

1. How do horizontal asymptotes differ from vertical asymptotes in rational functions?
2. What happens at the points where the numerator equals zero in a rational function?
3. How do you find horizontal asymptotes for rational functions like this?
4. What are the possible cases where vertical asymptotes don't exist?
5. Can you explain how to sketch a graph of a rational function with asymptotes?

Tip: Vertical asymptotes represent values of $x$ where the function grows without bound, typically resulting in very large positive or negative values.