Lebesgue measure of a set $A \in (a,b)$ can be bounded by $b+a + |t|$ where $t$ is a translation constant

Question

Let $A \subset (a,b)$ a Lebesgue measurable set such that $A\cap (A+t) = \emptyset$. Show that $2m(A) \leq b-a + |t|$. I've been trying to get the inequality by writing $m(A\cup (A+t)) = m(A) + m(A+t) = 2m(A)$ but don't know if this a correct procedure.