\( | z - 3 + i | < | z + 2 i | \)

To understand the inequality \( | z - 3 + i | < | z + 2 i | \), let's delve into its geometric interpretation. This inequality describes a region in the complex plane. The expression \( | z - 3 + i | \) represents the distance from the point \( z \) to the point \( (3, -1) \), while \( | z + 2 i | \) represents the distance from \( z \) to the point \( (0, -2) \). The inequality specifies that points in this region are closer to \( (3, -1) \) than to \( (0, -2) \). If we want to solve this inequality and visualize it more clearly, we can compare the distances. To do this, rewrite \( z \) as \( x + yi \) and square both sides of the inequality: \[ (x - 3)^2 + (y + 1)^2 < x^2 + (y + 2)^2 \] Simplifying this will help find specific boundaries or regions on the complex plane that satisfy this inequality, leading you to a more comprehensive understanding of how these regions interact.