One half a number increased by three is greater than zero or less than or equal to negative three

Let x be that number and half of it = x/2:

- 3 ≥ x/2 + 3 > 0 ; Reduce to same denominator

→→ - 6 ≥ x + 6 > 0 →→ - 6 ≥ x > - 6.
A number that is greater or equal to - 6 or smaller than - 6 is nothing but -6. Then x = - 6

Logical equivalent of the give question is its contrapositive