Given $$ \begin{cases} A \setminus X = B\\ X \setminus A = C \end{cases} \ \ \ B \subseteq A, A \cap C \neq \emptyset $$ I'm asked to find $X$.
My attempt:
$A \setminus X = A \setminus ((X \setminus A) \cup (X \cap A)) = B$
$A \setminus (C \ \cup (X \ \cap A)) = B$
$A = B \cup C \cup (X \cap A)$
$A \setminus (B \ \cup C) = X \setminus \overline{A}$
$(A \setminus (B \ \cup C)) \cup \overline{A} = X$
Is everything correct?
If not, what is the correct solution?
Your jump from the second to the third line is unjustified - and the third line clearly cannot be true if $C \cap A = \varnothing$
Instead, start from $X = (X\setminus A) \cup(X\cap A)$. The first term is $C$; the second is related to $B$, see if you can figure out how.
As I mentioned in the comments, I recommend following along a Venn diagram as you derive these equations.