WASHINGTON—The Supreme Court on Tuesday turned down a case seeking to block selective public schools from using race-neutral admissions policies that conservative activists argue are illegally designed to increase Black and Hispanic enrollment. The case, involving Thomas Jefferson High School for Science and Technology in Alexandria, Va., was seen as a follow-on to the court’s decision in June ending affirmative action in university admissions. It more directly involved a 2007 ruling that barred school boards from promoting integration by using race as a factor in pupil assignments but that suggested officials could consider the racial impact of broader policies such as where to build new campuses. Tuesday’s order, which, as is typical, was unsigned and provided no explanation, leaves school authorities free to pursue measures that may promote integration without classifying individual students by race. Two conservative justices, Samuel Alito and Clarence Thomas, dissented from the decision not to hear the case. TJ, as the school is known, has been ranked among the nation’s best public high schools. Admission to the 1,900-student school has required tests, an essay, teacher recommendations and other criteria, in addition to a $100 application fee. The Supreme Court’s order, as is typical, was unsigned and provided no explanation. After the nationwide reckoning on race relations that followed the police murder of George Floyd in 2020, the Fairfax County School Board revised TJ’s admissions criteria to diversify a student body dominated by Asian-Americans from a handful of affluent neighborhoods. The new system, which the board said was race neutral, ended the application fee and allocated most slots proportionally among the district’s middle schools, with applicants given a holistic evaluation based on grades, a problem-solving essay and “experience factors,” among other measures. The school board chairman, Karl Frisch, said the new policy was in the interests of students across the district. “It guarantees that all qualified students from all neighborhoods in Fairfax County have a fair shot at attending this exceptional high school,” he said. “The Supreme Court missed an important opportunity to end race-based discrimination in K-12 admissions,” said Joshua Thompson, an attorney with the Pacific Legal Foundation, which represented a group of Asian-American parents who alleged the new admissions criteria discriminated against their children. A federal district judge in Alexandria agreed with the parents in 2022, noting statements from school officials unhappy that TJ’s enrollment didn’t reflect the racial composition of Northern Virginia’s population. In May 2023, however, a divided federal appeals court in Richmond, Va., found no violation of the Constitution’s equal-protection guarantee. Although the share of Asian-American admissions to TJ fell to 54% from 73%, the Fourth U.S. Circuit Court of Appeals noted that applications had skyrocketed and that Asian-Americans, including a larger number from low-income families, continued to outperform the pool. Alito’s dissent Tuesday, joined by Thomas, warned that “TJ’s model itself has been trumpeted to potential replicators as a blueprint for evading” the court’s decision ending racial preferences at universities. In 2007, a divided Supreme Court ruled unconstitutional pupil assignment formulas adopted by the Seattle and Jefferson County, Ky., school districts because they took a student’s race into account to create integrated campuses. Justice Anthony Kennedy cast the deciding vote, and in a concurring opinion wrote that public schools should be free to promote student diversity through means that don’t classify individuals by race. Race-conscious policies should be allowed if they “do not lead to different treatment based on a classification that tells each student he or she is to be defined by race,” Kennedy, who retired in 2018, wrote. There are fewer than 200 selective high schools nationwide, many of them on the East Coast, according to Chester E. Finn Jr., a senior fellow at the Thomas B. Fordham Institute. In recent years, communities have pushed back against attempts to overhaul admissions standards at selective schools, changes often driven by a desire to diversify the racial and socioeconomic makeup of the campuses. In general, such schools skew more white and Asian than the broader district populations.