True or False $A - C = B - C $ if and only if $A \cup C = B \cup C$ True or False
$A - C = B - C $ if and only if $A \cup C = B \cup C$ 
I am in an introduction to proofs class.  I think this is a true statement. I have began the proof and realize I have to do this both directions.  I am stuck on the second part of the proof.
Assume $A - C = B - C$
Let $x \in A$ then $x \in A$ but $x\notin C$
By the assumption $x \in B - C$ then $x \in B$ and $x\notin C$ 
If $x \in A$ then $x \in A \cup C$.  If $x\in B$ then $x \in B \cup C$. Therefore,  $A \cup C = B\cup C$. 
Assume $A \cup C = B \cup C$.
Let $x \in A \cup C$. Then $x \in A$ or $x\in C$.
If $x \in A$ then $x \in A - C$
If $x\notin A$ then $x \in C$. 
This is where I get stuck... I know I am trying to show $A-C = B-C$ for the second part but I don't know how to get there... Am I on the right track?

Hint:You can use $A\cup B=(A-B)\cup B$ 
