Correct set notation for "all integers which are not multiples of 7"?
Solution 1:
I'd generally go with $\{x\in\mathbb{Z}: 7 \nmid x\}$, though this does presuppose familiarity with the $\mid \, \nmid$ notation.
Solution 2:
For starters, you should be using curly braces $\{\}$ for sets, and what you have is fine, but generally the clean abbreviation would be: $$ \mathbb{Z}-7\mathbb{Z}$$
This is because $-$ denotes set difference and $7\mathbb{Z}$ denotes the set of integer multiples of $7$.