Answer
The statement is true. See below.
Work Step by Step
We are given $z=a+bi$ and $w=c+di$.
We start with the left side:
$\overline{z}+\overline{w}=\overline{(a+bi)}+\overline{(c+di)}=a-bi+c-di=a+c-bi-di=(a+c)-(b+d)i$
Now the right side:
$\overline{z+w}=\overline{(a+bi)+(c+di)}=\overline{a+c+bi+di}=\overline{(a+c)+(b+d)i}=(a+c)-(b+d)i$
And the sides equal.
