White Americans tried to justify segregation by using the "separate but equal" clause. They believed that segregation was okay, as long as the facilities that contained segregation were equal.
The "separate but equal" idea got debunked numerous times. An example of where "separate but equal" turned out to be false was the Brown v. Board of Education case. This case started when a young African American girl had to walk over a mile to her black school when there was a white school only a couple blocks away. The Brown v. Board of Education case ended up going all of the way to the Supreme Court, which ruled that "separate but equal" was inherently false and not equal whatsoever.