Don't undestand this particular use of ない form

Question

This is the example sentence from one of my textbooks.

時間があるから、昼ごはんを食べて行かない？

Textbooks translation - "We have some time so how about we grab a bite to eat and then go?"

I don't understand why the ない form here doesn't translate to a present negative like 'grab a bite to eat and don't go" hence my confusion over its meaning. Either the translation is somehow off (unlikely), this sentence is simply incorrect (unlikely) or theres some rule I'm forgetting or don't know about (very much likely). Please help.

Answer 1

The textbook translation is correct, but kind of roundabout. It's more like, "We have some time, so why don't we grab a bite to eat and then go?" The "ikanai" would be pronounced with a rising intonation that affords a suggestion: [行かない？]{LLLH}

It does make sense if you do some "logical shifts" of the not part of the meaning and some semantic substitution.

...昼ごはんを食べて行かない
→　Will we eat and then not go?
→　Will we not eat and then go?
→　Will not we eat and then go? (ungrammatical and/or strange in English, but added for visual reference)
→　Won't we eat and then go?
→　"Won't we eat and then go?" + rising-intonation = "Why don't we eat and then go?"