Sorry to bother with this rather trivial question, but nowhere in my lectures or books can I quite find out what the topmost line means. Maybe I'm forgetting something. Anyway: Line 2 and 3 are clear. Line 4 just means
$x_{1} \leq 0$
$x_{2} \leq 0$
$x_{3} \leq 0$
is that right?
But what can I get from line 1? Thanks deeply for your help.
This notation doesn’t only describe a system of linear inequalities but a linear optimization problem. So line 1 is the objective function and the other lines are a system of linear inequalities that describe the feasible region.
By the way, this description is a use of set builder notation (with cleverly sized braces): The problem is to find the minimum of the set $\{~f(\mathbf{x}) : P(\mathbf{x})~\}$ where $f$ is the linear function given by $f(\mathbf{x}) = 2x_1 + 4x_2 + 7x_3$ and $P$ is the predicate described by the other lines.
Your interpretation of the last line is correct (provided you meant to write $\geq$ instead of $\leq$).