L.A. County’s unemployment rate was unchanged in May at 7.6 percent amid modest job gains that left the county just short of its all-time payroll peak, state figures released Friday show. The unemployment rate has held steady since March as more county residents both entered the workforce and found work, according to data from the California Employment Development Department. The rate is down from 8.3 percent in May of last year and is now at the lowest level since June 2008. That good news is relative, however, as L.A. County unemployment levels still exceed the statewide rate of 6.4 percent and are more than two points above the national rate of 5.5 percent. Unemployment in the county’s two largest cities was slightly higher than the countywide rate, with Los Angeles at 7.8 percent and Long Beach at 8 percent. State figures show the county, which has slightly more than 4.3 million nonfarm jobs, posted a net gain of just 3,600 payroll jobs in May. The gains were spread around a number of industries, led by increases in health care/social assistance (up 6,800) and accommodation/food services (up 3,400). Government, construction, retail trade and transportation/warehousing also recorded slight gains. However, those gains were largely offset by the volatile entertainment sector, which lost nearly 12,000 jobs in May, the most of any sector. In the more closely watched year-over-year jobs figures, county payrolls swelled by 100,000 jobs over the last 12 months for a robust annual growth rate of 2.4 percent. Virtually all sectors reported strong year-over-year job gains, led by health care/social assistance (up 21,400 jobs), leisure/hospitality (up 16,000), government (up 12,000) and professional/business services (up 10,000). Only manufacturing didn’t join the party, with the sector shedding 2,100 jobs.