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I'd like to ask you about a geometrical interpretation (2D) of this example:

$ \left| \frac{4i-3}{3i-z} \right| > 1 $

The easiest form I got:

$ \frac{5}{\left|3 i-z\right| } > 1$

But how to draw this set on the plane?

I've run the query on Wolfram Alpha

Image from Wolfram Alpha

Is it $y(1, +\infty)$?

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$$ \frac{5}{\left|3 i-z\right| } > 1$$ or, $$ 5>\left|3 i-z\right|$$ or, $$ 5>\left|z-3 i\right|$$ or, $$ \left|z-3 i\right|<5$$ which is the equation of a circular surface (not just a circle) with center at $(0,3)$ and radius $5$.
On the Argand plane,this will represent a circular region.

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    $\begingroup$ Thank you so much. Is there any way to draw it in Wolfram Alpha? How did you know about the center point? Is it something like $z - z_o = z - (3i)$? $\endgroup$ – belford Oct 16 '15 at 17:27
  • $\begingroup$ Yes, you are bang on. The equation for a circle with center $z_0$ and radius $r$ is $z-z_0 = r$ $\endgroup$ – SchrodingersCat Oct 16 '15 at 17:29
  • $\begingroup$ Aniket, by the way... Should I exclude the center point from the region? $\endgroup$ – belford Oct 17 '15 at 18:20
  • $\begingroup$ @terry_8 No. The center point should be included for the inequality holds true for all points within the circle. However you should exclude the circumference as the inequality has no equality condition i.e. $|z-3i| \not = 5$ $\endgroup$ – SchrodingersCat Oct 17 '15 at 18:45
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    $\begingroup$ Corrections are always welcome. Thank you $\endgroup$ – SchrodingersCat Oct 18 '15 at 18:28

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