
abkaufen
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.
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.

to believe
Colloquial and figurative for believing what someone says; to accept a story or excuse as true. Very often used in the negative.
Seine Entschuldigung war so unglaubwürdig, die kauft ihm doch niemand ab.
His excuse was so unbelievable, no one is going to believe him.
'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
Here, kaufen and abkaufen both refer to acquiring something in exchange for money. With kaufen, the previous owner is usually identified with von. Abkaufen directly connects this person to the item as a dative object: Man kauft jemandem etwas ab.

Ich kaufe meiner Nachbarin das gebrauchte Fahrrad ab.
I am buying the used bicycle from my neighbor.
Ich kaufe das gebrauchte Fahrrad von meiner Nachbarin.
I am buying the used bicycle from my neighbor.