N4
なくて
Not ~ and, Because not ~
重病じゃなくてホッとした。
I was relieved because it wasn't a serious illness.
Explanation
The negative 「~なくて」 can also give a reason: because something did not happen or is not so, a result follows.
「試験に合格できなくて、残念です」 (I couldn't pass the exam, and I'm disappointed). 「連絡が取れなくて、心配した」 (I couldn't reach them, so I was worried).
Mixed up with
What follows must be something you cannot control, such as a feeling or a result (hungry, worried, sorry). Not your own will, a request, an invitation or a command.
Do not say 「×寒くなくて、出かけましょう」. Say 「寒くないから、出かけましょう」.
「食べなくて、お腹が空いた」 (I didn't eat, so I'm hungry); 「食べないで、学校へ行った」 (I went to school without eating).
For nouns and na-adjectives the form is 「~ではなくて」 (also 「じゃなくて」, 「でなくて」).