$6=\frac{2}{x}+\frac{2}{y}=\frac{4}{2x}+\frac{4}{2y}\geq \frac{(2+2)^2}{2(x+y)}=\frac{8}{x+y}\Rightarrow x+y\geq \frac{4}{3}$
$6=\frac{2}{x}+\frac{2}{y}=\frac{4}{2x}+\frac{4}{2y}\geq \frac{(2+2)^2}{2(x+y)}=\frac{8}{x+y}\Rightarrow x+y\leq \frac{4}{3}$
$6=\frac{2}{x}+\frac{2}{y}=\frac{4}{2x}+\frac{4}{2y}\geq \frac{(2+2)^2}{2(x+y)}=\frac{8}{x+y}\Rightarrow x+y\
geq
\frac{4}{3}$