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I was in a car and waved my hand to signal the car behind me to move back just a bit so that I could reverse.

The dictionary suggests us to use this structure

wave + somebody/something + adverb / Preposition

wave somebody through/on/away etc

"back off" is a verb but "back" and "off" are also adverbs

If I wanted him/his car to move completely away from me, I would say "I waved him/his car away".

However, I didn't actually ask him to move completely away from me, I just wanted him to move away just a bit so that I could revere my car.

Is it correct to say "I waved the car back off"?

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As you noted, "to back off" is a verb, distinct from its component parts which may be prepositions or adverbs. Using the structure you are looking at, the correct phrase is "I waved the car back" without including "off".

If you wish to use the exact phrase "back off", you will need to change the structure and add some words:

I waved for the car to back off.

Now I'd like to take a moment to note that the phrase "back off" is very strong, and could be seen as aggressive or angry. You would generally expect to use "back off" when someone is physically or verbally attacking you, or if you are doing so to another person.

A gentler phrase would be "back up", but it would be used with the same grammatical formulation.

I waved for the car to back up.

This structure in these two sentences is not quite the same as you are seeing in your reference book. Here, we are simply saying that "you waved" in an intransitive way, and then separately explaining the reason for the waving with "for...". Your reference book is providing you with instructions on how to use "wave/waved" in a transitive way.

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  • I think "back up" is the best because the dictionary has this example "Can you back your car up so that I can get through?"
    – Tom
    Nov 15 '21 at 6:26

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