I cannot figure out how to work this problem
Generally, an inequality would be expressed in the form:
in words: something is less than or equal to X, which is less than or equal to something else
in math: -3 <= X <= 19
in other words: X is between -3 and 19 (inclusive)
Your problem is like this, except that instead of just the lone variable ("z" in your case) in the middle, it has a whole expression (1/2z+1). Your job is simply (yes, simply!) to perform standard math operations that reduce that middle term to just a lone "z".
Hint: you might first subtract 1. That would make the middle term "1/2z" rather than "1/2z + 1", which is certainly moving in the right direction. Of course, just as with an equation (or an "equality"), you must do this same operation (subtract 1) to all sides of the inequality, thus you must also subtract 1 from the left part (-12) and from the right part (4).
If you understand this, then you're well on your way!