complex numbers

[cazza90]

Find the Cartesian inequation for the region represented by
Re ((9+2 i) z +7 ) < 0

I have no idea what this is asking!

 

[soroban]

Hello, cazza90!

Find the Cartesian inequation for the region represented by: .\(\displaystyle \text{Re} [(9+2i)z +7] \:<\:0\)

\(\displaystyle \text{Find all complex numbers }\,z \,\text{ so that the real component of }(9\,+\,2i)z\,+\,7\,\text{ is negative.}\)


\(\displaystyle \text{Let }z \:=\:z+yi\)

\(\displaystyle \text{We have: }\;(9\,+\,2i)(x\,+\,yi)\,+\,7 \;<\;0 \quad\Rightarrow\quad 9x\,+\,9yi\,+\,2xi\,-\,2y\,+\,7 \;<\;0 \quad\Rightarrow\quad (9x\,-\,2y\,+\,7)\,+\,(2x\,+\,9y)i \;<\; 0\)

\(\displaystyle \text{Hence, we have: }\:9x-2y + 7 \;<\;0 \quad\Rightarrow\quad -2y \;<\;-9x - 7 \quad\Rightarrow\quad y \;>\;\frac{9}{2}x + \frac{7}{2}\)


\(\displaystyle \text{The }line \;y \;=\;\frac{9}{2}x + \frac{7}{2}\,\text{ has }x\text{-intercept }\left(-\frac{7}{9},\,0\right)\:\text{ and }\;y\text{-intercept }\left(0,\,\frac{7}{2}\right)\)


\(\displaystyle \text{And we want all the poihts }above\text{ that line.}\)

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