Problem welding vertices/edges As you can see in the gif, my problem is that that one of the vertices is loose while the other is connected, the loose vertex is located on the edge. My question is, how can i weld the vertex of the loose mesh to the edge ?

The blendfile:
If i understood the question vertex is not join.

Select the edge and subdivide once to get a single vertex

Select the verts and press G twice so that it slide on the edge.

Activate snap transform (shift+tab) and snap both the verts and remove doubles.