Step 1:
Let $z=x+iy$
$\mid 2z-3\mid=2$
$\mid 2(x+iy)-3\mid=2$
$\Rightarrow \mid (2x-3)+2yi\mid=2$
Step 2:
On squaring both sides
$\mid (2x-3)+2yi\mid^2=4$
$(2x-3)^2+(2y)^2=4$
$4x^2-12x+9+4y^2=4$
$4x^2+4y^2-12x+5=0$
$x^2+y^2-3x+\large\frac{5}{4}$$=0$
Step 3:
The locus of $P$ is a circle with centre at $(\large\frac{3}{2}$$,0)$ .
Radius=$\sqrt{\big(\large\frac{3}{2}\big)^2-\large\frac{5}{4}}$
$\quad\;\;\;\;\;=\sqrt{\large\frac{9}{4}-\frac{5}{4}}$$=1$
The locus of $P$ is a circle with centre at $(\large\frac{3}{2}$$,0)$ and radius $1$unit. .