# How do we transform a complex number inequality from the $z$-plane to the $w$-plane?

## To be clear

I know how to transform complex numbers from $z$ to $w$ planes but what I don't really grasp is the inequality sign, what decides its direction?

## Example 1:

If $z$ is any point in the region $R$ for which $$|z+2i|<2$$ Sketch: $w=z-2+5i$

## My Solution:

NOTE: I didn't shade the inequality in the $w$-plane (that's the point of my question).

And could anyone help me solve $|zw+2iw|=1$ for the same inequality above?

I'm looking for some logic explanation that would make me tackle every transformation problem. Thank you in advance.

• I've taken the liberty of adding your images to your post; once you have some number of reputation points you'll be able to do this yourself. :) – Andrew D. Hwang Jan 11 '16 at 16:13
• Thank you, its much neater now :D – Karim A. Ahmed Jan 11 '16 at 19:19

Briefly, pick a point $z_{0}$ in the shaded region of the "before" picture and calculate its image under your transformation $T$. The region of the "after" picture containing $T(z_{0})$ is the image region $T(R)$.
To handle $|zw + 2iw| = 1$, write it as $$|z + 2i| = \left|\frac{1}{w}\right|.$$