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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:

Z-plane (Before Transformation)

W-plane (After Transformation)

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.

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  • $\begingroup$ 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. :) $\endgroup$ – Andrew D. Hwang Jan 11 '16 at 16:13
  • $\begingroup$ Thank you, its much neater now :D $\endgroup$ – Karim A. Ahmed Jan 11 '16 at 19:19
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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 $$ |z + 2i| = \left|\frac{1}{w}\right|. $$

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  • $\begingroup$ Ok, that is helpful :) even though sometimes its hard to determine in which region the image is located. Thank you. $\endgroup$ – Karim A. Ahmed Jan 11 '16 at 19:19

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