Separation does not necessarily create inequality.
Laws that permit separation can be constitutional.
Plessy v. Ferguson (1896) is a case in which the Supreme Court of the United States argued that the "separate but equal" doctrine was constitutional. The case was the first major inquiry into the meaning of the Fourteenth Amendment's equal protection clause.
Writing for the majority, Associate Justice Brown argued that segregation did not violate the Thirteenth Amendment because it did not reestablish slavery, nor was it a "badge" of slavery. Moreover, he argued it also did not violate the equal protection clause because it does not by itself imply the inferiority of one race as compared to the other one, as long as the facilities are equal in quality.