these are always a little hard to stomach, but if you break it up bit by bit, it's not as hard as it looks.
"one half a number increased by three is greater than zero or less than or equal to negative three. find the number."
so break it up:
"one half a number" ... (1/2)x
"increased by 3" ... addition ... +3
"is greater than zero" ... >
"or less than or equal to" ... ≤
"negative three" ... -3
then, throw all of that together.
(1/2)x + 3 > 0
(1/2)x + 3 ≤ -3
you have two inequalities. from here, you just solve them as you would any other inequality.
(1/2)x + 3 > 0 ... subtract 3
(1/2)x > -3 ... divide by (1/2)
x > -6
(1/2)x + 3 ≤ -3 ... subtract 3
(1/2)x ≤ -6 ... divide by (1/2)
x ≤ -12
so, you have two solutions. x must be greater than -6, or less than or equal to -12. the question says "find the number" as if there... is just one number that is the answer, but i'm not sure of how to find it, if there is a way.