lines, circles and regions in z and z-bar
Because each complex number is a point in the plane, an equation or inequality in z carves out a curve or region you can draw — and learning to translate between the algebra and the picture is a core skill. The simplest examples: |z - a| = r is the circle of radius r centered at the point a (all points exactly distance r from a), and |z - a| < r is the open disk inside it. The condition |z - a| = |z - b| is the set of points equidistant from a and b — the perpendicular bisector of the segment from a to b, a straight line.
Many of these conditions read most cleanly using the conjugate. Because Re z = (z + z-bar)/2 and Im z = (z - z-bar)/(2i), a real-linear equation like 'Re z = 1' (a vertical line) becomes z + z-bar = 2, and the general straight line can be written as the real part of (b-bar z) equals c for a fixed complex direction b. A remarkable single equation, z z-bar + (a-bar z + a z-bar) + c = 0 with c real, captures every line AND every circle at once: it is a circle when a coefficient is nonzero and a line when it degenerates — which is exactly why Mobius maps treat lines and circles as one family later.
Inequalities cut out regions instead of curves: |z - a| < r is a disk, Im z > 0 is the upper half-plane, 1 < |z| < 2 is an annulus (a ring between two circles), and 0 < arg z < pi/4 is a wedge-shaped sector. These regions are the stages on which complex functions act, and being fluent at reading 'what region does this describe?' is the difference between seeing the geometry of a problem and being lost in symbols.
What region is |z - 1| <= |z + 1|? Squaring both sides (|z-1|^2 = |z|^2 - 2 Re z + 1, similarly on the right) the |z|^2 and 1 cancel, leaving -2 Re z <= 2 Re z, i.e. Re z >= 0 — the closed right half-plane. Points closer to 1 than to -1 are exactly those with non-negative real part.
Translate the algebra to geometry: a distance comparison becomes a half-plane.
Beware z-bar: any equation that genuinely involves z-bar (and not just |z|^2 = z z-bar) is generally NOT the graph of a holomorphic function — it describes geometry, not an analytic curve. The conjugate is a great descriptive tool here but a red flag for differentiability later.