why $\{a+x: x \in I \}$ is the set of congruence class of $a \in R$ with $R$ a ring and $I$ an ideal

Question

I know that if we have a ring $R$ and $I$ an ideal of R, a congruence relation is

$a \equiv b (mod I)$ if $a-b\in I$

but why this set

$\{a+x: x \in I \}$ is the set of congruence class of $a $

I can see that if we take $b \in $$\{a+x: x \in I \}$ then $b= a+i$ for some $i \in I$

but if $a\equiv b $ (mod $I$) then $a-b=a-(a+i)=-i $ but if $-i\in I$ then $I=R$ because it contains $-i$

is not better define $\{a-x: x \in I \}$