The Zoning Board of Appeals on Tuesday approved plans by the family that has long owned two one-story commercial building at Dudley and Virginia streets in Dorchester's Uphams Corner with a six-story, 48-unit apartment building.

Under the approval, the Salzberg brothers will tear down the commercial buildings, once of which now houses Rainbow Shops, for the new $20.6-million building, which will include ground-floor retail space, 9 parking spaces and indoor space to store 52 bicycles. The building would also have a roof deck for residents.

Six of the units would be rented as affordable, rented to people making no more than 70% of the Boston area median income, which in this year's dollars would mean rents of $1,559 for a one-bedroom apartment and $1,776 for a two-bedroom unit. The units will be split between 26 studios, 16 two-bedroom apartments, 3 one-bedroom apartments and 3 three-bedroom units.

Through an aide, City Councilor John FitzGerald supported the project.

Joanne Tuller of Monadnock Street supported the project as well, although she said she would have preferred to see more than the minimum 13% of units be rented as affordable, given the importance of such housing in Uphams Corner.

The project needed zoning-board approval for several conditions, including the overall density of the building on its lot, being too tall under the lot's zoning and not having enough setback in the rear.

The BPDA board approved the project in April.

749-759 Dudley St. filings.