Hello, kingkaisai2!
The problem is badly worded ... it makes no sense.
I'll take a guess at what you meant . . .
Also, in all your problems, unless the inequality is $\displaystyle \leq$
. . there is no solution.
Are you sure it's "less than"?
Find the least possible value of $\displaystyle z$ such that $\displaystyle z + 2  i\sqrt{3} \:\leq \:2$
Describe in geometrical terms.
We have: .$\displaystyle z  (2 + i\sqrt{3}) \:\leq \:2$
This is the set of points within 2 units of $\displaystyle 2 + i\sqrt{3}.$
This is the set of points in or on the circle
. . with center $\displaystyle C(2,\sqrt{3})$ and radius $\displaystyle r = 2.$ Code:

* * * 
* * 
* * 
* *
_ 
* (2,√3) *
* C* *
* \ *
\ 
* \ *
* o 
**+
* * * O

The point $\displaystyle z$ with the least magnitude
. . is the intersection of the circle and the line $\displaystyle CO.$
The line has the equation: .$\displaystyle y \:= \:\frac{\sqrt{3}}{2}x$
The circle has the equation: .$\displaystyle (x + 2)^2 + (y  \sqrt{3})^2\:=\:4$
Find their intersections and take the rightmost point.