Proof regarding lebesgue measure (from Folland's book)

Proof regarding lebesgue measure (from Folland's book)

Q. Let $\mu$ denote the Lebesgue measure on real line. If
$E\in\mathcal{B}_{\mathbb{R}}$ and $\mu(E)<\infty$, then for every
$\epsilon>0$ there exists a set $A$ which is a finite union of open
intervals such that $$\mu((E\backslash A)\cup(A\backslash E))<\epsilon$$
Try
It is clear that $(E\backslash A)$ and $(A\backslash E)$ are disjoint.
So,$$\mu((E\backslash A)\cup(A\backslash E))=\mu(E\backslash
A)+\mu(A\backslash E)$$
But I have no idea how to proceed further.