In this paper we describe a new method for constructing a weight structure w on a triangulated category C. For a given C and w it allows us to give a fairly comprehensive (and new) description of triangulated categories containing C as a dense subcategory (i.e., of subcategories of the idempotent completion of C that contain C; we call them idempotent extensions of C) to which w extends. In particular, any bounded above or below w extends to any idempotent extension of C; however, we illustrate by an example that w does not extend to the idempotent completion of C in general. We also describe an application of our results to certain triangulated categories of (relative) motives.