Thank you for your good question! Answering questions of this general type has been a big concern of the field of philosophy of language for the last few decades. One way of starting to understand where this tradition is coming from, is to distinguish between the literal meaning of a sentence, and the meaning that a speaker using that sentence is normally thought to convey. So suppose that you often has unkempt hair. One day you show up with nicely combed hair and I remark, "Your hair is combed!" Now it will be natural to take me to be *suggesting* that your hair is normally not tidy. But this is no part of the literal meaning of the sentence. If it were, then I'd contradict myself by saying, "Your hair is combed, though of course it is normally tidy." This might be an odd things to say, but it isn't a self-contradiction like, "Bob is a bachelor, but he is married."
So we can distinguish between what a sentence usually means and what a person uttering that sentence might be conveying, suggesting or insinuating in saying it. This distinction applies to your examples: The sentence, "Men work to support their families." does not imply that women don't (because it fails the self-contradiction test), but it is true that often speakers who utter such a sentence are suggesting or insinuating more than that they literally say, namely that women don't, or that they usually don't.
In everyday discourse we usually don't distinguish between the literal meaning of a sentence and that speakers convey in uttering it. But it helps to keep that distinction in mind when we need to be careful about what an argument actually establishes, and what a speaker is actually committed to. When in doubt, I'd say commitment is a matter of the literal meaning of the sentences uttered; anything beyond that is just suggestion, and we would need to ask the speaker whether she meant to be committing herself to something stronger than that literal meaning.