Describe and sketch subsets of the complex plane defined by equations and inequalities in modulus and argument

What this dot point is asking

SCSA wants you to interpret modulus and argument conditions geometrically, sketch the resulting curves and regions, and where needed convert to Cartesian equations.

Modulus conditions give circles and distances

Since $|z - a|$ is the distance from $z$ to the fixed point $a$, the equation $|z - a| = r$ describes all points at distance $r$ from $a$, a circle of radius $r$ centred at $a$. The inequality $|z - a| \le r$ shades the closed disc; $|z - a| > r$ shades the outside.

The condition $|z - a| = |z - b|$ says $z$ is equidistant from $a$ and $b$, which is the perpendicular bisector of the segment joining them. The inequality $|z - a| < |z - b|$ shades the half-plane nearer to $a$.

Argument conditions give rays

The condition $\arg(z - a) = \alpha$ describes all points $z$ for which the displacement from $a$ points in the fixed direction $\alpha$. This is a ray (half-line) starting at $a$ (open, since $z = a$ has no argument) at angle $\alpha$ to the horizontal. A condition like $\tfrac{\pi}{6} \le \arg(z - a) \le \tfrac{\pi}{3}$ shades a wedge between two rays.

Combining conditions

Many SCSA questions intersect two conditions, for example a disc and a wedge. Sketch each boundary, decide which side each inequality shades, and the answer is the overlap. Always state whether boundaries are included (solid) or excluded (dashed).