to buy from / believe
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to buy from
To acquire something from a person by paying for it; to buy something from someone.
Ich habe meiner Nachbarin ihr altes Auto für einen guten Preis abgekauft.
I bought my neighbor's old car from her for a good price.
'Abkaufen' (to buy from / to believe) is a separable verb. The prefix 'ab-' stands at the end in a main clause. In a subordinate clause, the verb stays together, and in the infinitive with 'zu', 'zu' goes between 'ab' and 'kaufen'.
Er kauft seinem Freund die alten Bücher ab.
He buys the old books from his friend.
The verb is almost always used with two objects: the person from whom you buy something (or whom you believe) is in the dative. The thing that is bought (or the story) is in the accusative.
Ich kaufe dir die Konzertkarte gerne ab.
I'll gladly buy the concert ticket from you.
In the meaning 'glauben' (to believe), 'abkaufen' is almost always used colloquially and in the negative to express that you do not believe a story or excuse.
Dass du den Termin vergessen hast, kaufe ich dir nicht ab!
I don't believe you that you forgot the appointment!
The difference between the meanings becomes clear through the accusative object. If it is a concrete object, the literal meaning is intended. If it is something abstract like 'eine Geschichte' (a story) or 'eine Ausrede' (an excuse), the figurative meaning is intended.
Ich kaufe dir das alte Regal für 20 Euro ab.
I'll buy the old shelf from you for 20 euros.
to buy
'kaufen' and 'abkaufen' can both mean that you receive something in exchange for money. 'kaufen' only says that you acquire something. 'abkaufen' additionally names the person you get something from: 'Man kauft jemandem etwas ab' (you buy something from someone). That is why 'abkaufen' often needs a dative, for example 'meinem Nachbarn' (from my neighbor). The words can also be confused because 'abkaufen' has a second, colloquial meaning: to believe someone. In buying situations, the difference is simple: 'kaufen' is general, while 'abkaufen' means buying from a specific person.
Paul hat seinem Freund das alte Handy abgekauft.
Paul bought the old phone from his friend.
Sie kauft für das Projekt einen neuen Drucker.
She buys a new printer for the project.