This wave does not move...
It has a definite value at each point in space which is fixed in time.
Only if you have :
Aexp(i(kx-wt)) + Bexp(-i(kx+wt))
you can have a moving wave. Note the difference of the sign in front of w.
To show in which direction it moves it suffices to study the argument being zero:
kx-wt=0 --> x=ct moving to +x
kx+wt=0--> x=-ct moving to -x
where c=w/k the velocity of the wave.