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Hoping to spur the construction of low-cost rental housing, the Cook County Board on Thursday voted to expand countywide a program that lowers the assessment rate on new and rehabbed apartment buildings where a portion of the units are set aside for low and moderate income tenants.

The unanimous vote allows for buildings with at least seven units to be assessed at 16 percent instead of the current 33 percent. To qualify, those buildings must reserve one-third or more of the units for rent at a rate that claims no more than 30 percent of a tenant’s household income.

In addition to the board’s decision to expand the Class 9 program, commissioners gave final approval to the new map setting the boundaries of the 17 County Board districts in a 15-2 vote.

“The map is a disappointment, a lost opportunity,” said Commissioner Carl Hansen, who lamented that the boundaries of some districts still meander for miles through the county.

Commissioner Roberto Maldonado, who also opposed the remap, said the new boundaries “fragmented” the political influence of the growing Latino community in Cook County.

“I think that’s wrong,” said Maldonado, who argued that mapmakers should have drawn a third county district with a strong Hispanic majority.

But while some board members complained about the new map, there was nothing but praise for expanding the Class 9 program.

“People who have lived their entire lives in a community are being forced out,” said Commissioner Mike Quigley. “This will help.”

Previously the Class 9 program had been limited to parts of the county designated in census tracts as low or moderate income.

County commissioners and housing activists praised Thursday’s vote.

Expanding Class 9, said Commissioner Bobbie Steele, who sponsored the measure, “will increase the availability of affordable housing for families who are in need of a place to live.”

Kevin Jackson, executive director of the Chicago Rehab Network, said expanding Class 9 would provide rental property owners “with an incentive to stay rental. It’s the right medicine at the right time.”

Housing activists on the North Side called for a new federal fund Thursday to subsidize the construction of affordable housing.

Low-income families are being squeezed out of Chicago neighborhoods because of soaring rents, and new federal subsidies could double the number of units built with help from the city, Rich Monocchio, deputy commissioner of the city’s Housing Department, said in a news conference.

The push to raise support for federal legislation to create the trust fund coincides with the release of a national study advocating the trust fund as a solution to a lack of rental housing.