The first inequality can be solved by finding the values of x that make the expression greater than or equal to 4:

x^2 ≥ 4
x ≥ 2 or x ≤ -2

Similarly, for the second inequality:

(2 - 5x)^2 ≥ 16
4 - 20x + 25x^2 ≥ 16
25x^2 - 20x - 12 ≥ 0

This inequality can be solved by factoring or using the quadratic formula to find the roots:

x = (20 ± sqrt(20^2 - 425(-12))) / (2*25)
x = (20 ± sqrt(400 + 1200)) / 50
x = (20 ± sqrt(1600)) / 50
x = (20 ± 40) / 50
x = 0.4 or x = -0.32

Therefore, the solution to the second inequality is:
-0.32 ≤ x ≤ 0.4

For the third inequality:

x^2 ≤ 16
-4 ≤ x ≤ 4

And for the fourth inequality:

(x - 2)^2 ≥ 9
(x - 2)(x - 2) ≥ 9
(x - 2) ≥ 3 or (x - 2) ≤ -3
x ≥ 5 or x ≤ -1

Therefore, the solutions to the system of inequalities are:
-4 ≤ x ≤ -1 or 0.4 ≤ x ≤ 4.