When sektching the region $\left|\frac{2z-1}{z+i}\right|$$\geq$1 on the argrand diagram,
how should we go about identifying the region, should we take
$\left|2z-1\right|\geq\left|z+i\right|$ or try multiplying by the conjugate in the left hand side modulus?
ie something like $\left|\frac{2z-1}{z+i}\frac{z-1}{z-i}\right|$$\geq1.$ nothing is jumping out at me either way.
HINTS:
Yes, write it as $|2z-1| \ge |z+\mathrm{i}|$. This is true, if and only if $|2z-1|^2 \ge |z+\mathrm{i}|^2$.
Replace $z$ by $x+\mathrm{i}y$.
Collect the real and imaginary parts inside each modulus.
Apply the definition for the modulus: $|u+\mathrm{i}v|^2 = u^2 + v^2$
Expand your brackets, and take all of the $x$s and $y$s over to one side.
Complete the square on the $x$ terms and $y$ terms separately.
You'll find that you have either the inside or the outside of a circle, including the circle itself.
I get the radius of the circle to be a multiple of $\sqrt{5}$ and the centre is in the upper-right quadrant.