How do I prove that numbers not divisible by 3 can be represented as 3x+1 or 3x-1? I saw that some proofs used the fact that numbers not divisible by $3$ can be represented as $3x+1$ or $3x-1$. But how do I prove that it is true?

If $k=3x+1$ then $$k\equiv 1\mod 3$$ and if $k=3x-1$ then $$k\equiv 2\mod 3.$$ Moreover $$\mathbb Z/3\mathbb Z=\{[0],[1],[2]\}$$
where $$[1]=\{3x+1\mid x\in\mathbb Z\}$$ and $$[2]=\{3x+2\mid x\in\mathbb Z\}=\{3x-1\mid x\in\mathbb Z\}$$
what conclude the proof. 
