〜なくて
Because … not… (negative giving a reason)machine translation
Formationない-form → なくて (verbs, adjectives, nouns in the negative)
〜なくて is the て-form of the ない-form, used to give a negative reason: バスが来なくて、遅刻しました (the bus didn't come, so I was late). Like the 〜て pattern for cause, the second clause is usually an emotion or an involuntary state: 困る, 大変, 心配する, 安心する, 残念.
〜なくて works with adjectives and nouns too: 難しくなくて (because it isn't hard…), 静かじゃなくて (because it isn't quiet…), 学生じゃなくて (because one isn't a student…).
The easiest mix-up is with 〜ないで. Remember: 〜なくて answers "why?" (reason), while 〜ないで answers "in what state?" (doing B without doing A). When the second clause is an emotion or an involuntary result, use 〜なくて; when it's a deliberate action by the same person, use 〜ないで.
Examples
朝バスが来なくて、遅刻しました。
The bus didn't come this morning, so I was late.
日本語が分からなくて、困りました。
I didn't understand Japanese, so I was really at a loss.
試験が難しくなくて、安心しました。
The exam wasn't hard, so I was relieved.