# McCormick envelopes and nonlinear constraints

I have a problem with a nonlinear constraint. The non-linearity stems from a term of the form $$xb$$, where $$x \in \mathbb{R}^+$$, $$x < M$$ and $$b \in \{0, 1\}$$. I am able to remove this non-linearity by using McCormick envelopes.

If I solve the problem using this relaxation, am I actually solving the problem or am I just finding a very tight lower bound?. Remember that one of the variables is binary.

What if $$b \in \mathbb{N}$$ instead?

For real $$x\in[l,u]$$ and binary $$b\in\{0,1\}$$ the McCormick envelope gives you bounds on $$w=xy$$
\begin{align} lb & \leq w \leq ub,\\ ub+x-u& \leq w\leq x+lb-l. \end{align}
By case analysis you can see that this is equal to $$w=xb$$, so you will indeed solve the problem.