Naide vs nakute

かれはいつも朝ごはんお_______しごとにいくんだ。(たべる)

Why is the answer tabenai de instead of tabenakute?

食べないで means without eating.
食べなくて just means not eat, it’s the negative て form.

So in this case if you said 食べなくてit would be like saying I did not eat and then I went to work. Rather than I went to work without eating.

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In many cases I like to think of で as indicating “how”.

So how does he go to work? Without eating breakfast.

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