When someone writes for instance "Solutions are x = 30, 150, 270." is this an old fashioned set notation? It obviously does not have braces.
No, this is a shorthand for "Solutions are x = 30, x = 150, and x = 270." Writing "x = {30, 150, 270}" would be a type mismatch because x is a number (presumably, a solution to an equation), while the right-hand side is a set of numbers. One could write, "The set of solutions is {30, 150, 270}," though this sounds too pedantic. Another way is, "x is a solution iff $\displaystyle x\in\{30, 150, 270\}$."
This notation is not correct because what is right of : must be a predicate, or property, i.e., something that is either true or false for each value of x. What you have is numbers, which are neither true nor false. You could write {x : x = 30 or x = 50 or x = 270}, but this is the same set as {30, 50, 270}.