Linear programming problem neither max nor min Heres the actual question:

television provider broadcasts two movie channels, A and B. Channel A broadcasts 1 romantic
  movie, 3 action movies and 3 comedies per month at a cost of 50 Euro. Channel B broadcasts
  3 romantic movies, 4 action movies and 1 comedy per month at a cost of 25 Euro. Suppose that
  you like to see 8 romantic movies, 19 action movies and 7 comedies at minimal cost in the coming
  months. For how many months should you request both channels?
a. Formulate this problem as an LP problem

Now my way of thinking was that I would first write it down more clearly, A and B are the television channels.
$A: 1R,3A,3C => 50 euro$
$B: 3R, 4A, 1C => 25 euro$
And I would define a new variable $x_i :=$ cost of taking a channel one month.
But then I stumble upon the problem of defining the hard limits they have set, the 8 R, 19 A, and  7 C.
In short: how do I define a constraint that is hard (e.g. not a maximum)
 A: $min \quad 50A + 25B$
Such that
$A + 3B \geq 8$
$3A + 4B \geq 19$
$3A + B \geq 7$
$A,B\geq 0$
A: Normally I'd just give you hints but I actually have a couple of problems with the problem formulation. First of all, the proper model here is not a linear program (LP), but an integer linear program (IP). I'm assuming that you cannot subscribe to fractional months, after all, since that is standard practice in the cable industry. This is the model I'm thinking of:
$$\begin{array}{ll}
\text{minimize}   & 50 A + 25 B \\
\text{subject to} & A + 3 B \geq 8 \\
                  & 3 A + 4 B \geq 19 \\
                  & 3 A + B \geq 7 \\
                  & A, B \geq 0 \\
                  & A, B ~ \text{integer}
\end{array}$$
Now of course, it's possible they're permitting you to subscribe to fractional months. But you'd better ask the person requesting the model if that's the case. If they do allow fractional months, then indeed you can go ahead and drop the integer constraint, and you have an LP. (LATE EDIT: actually, fractional months really don't make sense, because then you'd get fractional movies!)
But even this is not enough, because this only tells you the total number of months to subscribe to A and B, not the number of months you must subscribe to both channels. To answer that question, you need more information than is provided. Specifically, you need to know the maximum number of months you are allowed to wait to complete your movie watching task. 
For instance, suppose the model yields $A=3$, $B=2$. If you're allowed to take 5 months to watch the movies on your list, then you don't need to subscribe to both channels at the same time ever. Just subscribe to A for three months, then switch to B. 
On the other hand, if you're given an additional rule that you must subscribe for a minimum number of total months, then of course the total number of months required is $\max\{A,B\}$, and the number of months subscribed to both channels is $\min\{A,B\}$. But again, without additional information, the answer cannot be given---and neither answer is obtained directly from the LP or IP model, but rather from post processing the model output.
So the task "Formulate this problem as an LP problem" is, actually, ill-posed. Not only does it likely require an IP, but it requires additional modeling work after the optimization model is solved. "Solve this problem using an LP" would be OK, if they make it clear that fractional months are permitted.
