Translation:I live with my mother and my grandparents.
If you google abitare vs. vivere you find some information information. https://italiantutornow.co.uk/abitare-vs-vivere/ In some cases they both can be used, but abitare is more the place one is living, where vivere is more the act of living which has slightly different connotations. It's interesting that the link suggests one wouldn't use abitare for living at a hotel. I think that makes more sense when one thinks of saying "I'm living out of a suitcase", which I think would be vivere. For the DL sentence, I think vivere makes more sense. Had they said my mother's house, then abitare would be appropriate. (I am not an Italian speaker, so add sale. :)
It is correct and should be accepted please report it for the sake of other learners. Also, see the post above by mukkapazza. Here is a post outlining some of the common difficulties of newcomers and giving advice and some really good Guideline sites which you should bookmark for further reference:
In the English translation, if the result of replacing the word «live» with «inhabit / abide» is awkward, awful or nonsense then «Vivere» makes good or better meaning.
È bello essere vivi ‧ It is good to be alive
Bellissimi abiti ‧ Beautiful clothes
[ Vivere ] is the verb most commonly used to indicate «To live with» someone. [ Existential, temporal, intangible - thriving or not in life is being shared ]
[Abitare] in the «To live with» context is like Coabitare, Convivere [ Locational, Physical, Tangible - co-located living arrangements are being shared ]
Abito nel peccato ‧ I inhabit / abide in sin
Vivo nel peccato ‧ I live in sin
Abitare refers to a place, residence, location, where one lives, resides, inhabits, contrastingly, Vivere has a more existential meaning.
Even when context allows [ Abitare ] or [ Vivere ] there is still a nuanced difference in meaning.
Abitare ‧ Vivere ‧ in una casa ‧ To live in a house / home
Abitare ‧ Vivere ‧ in fondo alla strada ‧ To live down the street
Abitare ‧ Vivere ‧ di fianco ‧ To live next to ‧ next door
Abitare ‧ Vivere ‧ accanto ‧ To live next to ‧ next door
Convivere; Vivere insieme ‧ To live together
Vivere; Convivere in pace ‧ To live in peace
Vivere da sola/o/i ‧ To live alone
Vivere nel peccato ‧ To live in sin