To solve the given problem, the distance between two points $(-5, 4)$ and $(-5, -7)$ is calculated using the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substituting the given points:

$d = \sqrt{((-5) - (-5))^2 + ((-7) - 4)^2}$

Simplify each term:

$d = \sqrt{(0)^2 + (-7 - 4)^2}$

$d = \sqrt{0 + (-11)^2}$

$d = \sqrt{121}$

$d = 11$

So, the distance between the points is 11 units.

Let me know if you want a detailed explanation of the formula or have more questions. Here are some related questions to explore:

1. How is the distance formula derived from the Pythagorean theorem?
2. What is the significance of absolute value in this calculation for vertical or horizontal points?
3. Can the distance formula be applied in three-dimensional space? If yes, how does it change?
4. How can this calculation be visualized on a graph?
5. What is the geometric meaning when the $x$-coordinates of two points are the same?

Tip: Always ensure the order of subtraction in the formula is consistent to avoid errors with signs.