Stiefel-Whitney classes and immersion of a manifold

Question

From Milnor and Stasheff: according to the Whitney duality theorem, $r_M$ is the tangent bundle of a manifold in Euclidean space and $v$ is the normal bundle, then the Stiefel-Whitney class satisfy: $$ w(r_M)=w^{-1}(v) $$
Then the theorem be used to investigate when the space $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+k}$. My question is that a space immersed in $\mathbb{R}^{n+k}$ doesn't mean it is embedded in it. For example, a line immersed in $\mathbb{R}^n$ can have self-intersections, in this situation, the image of the immersion in fact is not even a manifold in Euclidean space. So how can we use this theorem to make conclusions?