My question is to describe/sketch the sets of complex numbers.
a) $|z+1|-|z-1|=0$. so far for this subpart my understanding is that the distance between the point $z$ and $-1$ is equal to that between $z$ and $+1$. now do i say that the set of complex number is a straight line from $-1$ to $1$ passing through zero? i am confused as to how to describe it.
b) $iz-i \bar z =2$
for part b i am really confused, i did the algebra to find $x = z+i$ and $y=-1$. I am not sure how to proceed from that.
Hints:
a) Points $-1$ and $1$ in the complex plane are both on the real axis, and are symmetric with respect to the origin. The locus of points equidistant from two fixed points is the perpendicular bisector of the segment between the two points, which in this case is $\ldots$
b) $iz - i \bar z = 2 \iff z - \bar z = \dfrac{2}{i}=-2i \iff 2 \operatorname{Im}(z) = -2\,$. The locus of points with constant imaginary part is a line parallel to the real axis.