Sketch a set in the complex plane

Question

How to sketch this set in the complex plane? $$S=\{ z\in C: Re[(4-i)z] > Re[(-5+7i)(4+6i)] \}$$ Where:

S - set
Re[x+yi] - real part of the complex number
C - complex numbers
z - complex number

I solved the Real parts inside the brackets and here what I`ve got: $$Re[(4-i)z] = Re[(4-i)(x+iy)] = Re[(4x+y) + i(4y-x)] = 4x+y$$ $$Re[(-5+7i)(4+6i)] = Re[-20-30i+28i-42] = Re[-64-2i] = -64$$ After that I have an inequality: $$4x+y>-64$$

What have you tried? Where are you stuck? Can you invent and solve a similar problem with simpler complex numbers? — Ethan Bolker, Feb 18 '24 at 23:00
You've done most of the work. Finding out where those points are in the plane is just old fashioned analytic geometry - nothing to do with complex numbers. Start with where that inequality is an equality. — Ethan Bolker, Feb 18 '24 at 23:12
@EthanBolker At first I`ve got tricked a bit but then yes I tried and everything looks fine! Thank you. — Arsenii Zarudniuk, Feb 18 '24 at 23:34
Please try not to use irrelevant tags. If you’re not sure, check their meanings. — Ted Shifrin, Feb 19 '24 at 00:55
"After that I have an inequality: 4x+y>−64" Do you know how to graph that? — fleablood, Feb 19 '24 at 16:07

score 1 · Accepted Answer · answered Feb 18 '24 at 23:39

1

So this is how it looks like plotted by Wolfram Alpha (it goes through x=-16)

answered Feb 18 '24 at 23:39

Arsenii Zarudniuk

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Sketch a set in the complex plane

1 Answers1