In Japanese, なくて is formed by joining the negative ない form (in its adverbial conjugation, なく) with the conjunctive particle て. This essentially functions as the negative equivalent of the standard て form, connecting clauses with the meaning of '...not, and...'.

なくて is commonly used when the first clause presents a reason or cause for the state or action in the second clause. In these situations, it's typically translated as 'because (A) didn't happen, (B)' or 'since (A) is not the case, (B)'.

• ピアノが弾(ひ)けなくて悔(くや)しいです。
  It's frustrating that I'm not able to play the piano. (Being unable to play is the reason for the frustration)
• このカバンは高(たか)くはなくて安(やす)いから、買(か)いやすいね。
  Because this bag is not expensive, but cheap, it's easy to buy, isn't it? (Not being expensive is the reason that it is easy to buy)
• 彼(かれ)の話(はなし)は正直(しょうじき)ではなくてがっかりした。
  Since his story was not honest, I was disappointed.
• テストではなくて、ほっとした。
  I was relieved because it was not a test.

When なくて is used to express a reason, it is frequently followed by the speaker's emotions (often as an adjective), a verb in the potential form, or a situation that is generally outside of the speaker's direct influence. This nuance of an 'uncontrollable result' is what gives the construction its 'because' meaning in English.

• 宿題(しゅくだい)が分(わ)からなくて、困(こま)っています。
  I'm having trouble because I don't understand the homework.
• お金(かね)がなくて何(なに)も買(か)えません。
  I can't buy anything because I don't have any money.

Some examples of words that express such involuntary states are 残念(ざんねん) (to be disappointed), 悲(かな)しい (to be sad), ほっとする (to feel relieved), がっかりする (to feel let down), and 困(こま)る (to be troubled).