How did the Court justify the ruling in Plessy v. Ferguson?the 14th Amendment guaranteed equal protection under the law. yet in plessy v. ferguson...

The Court justified this ruling by arguing that there is
nothing inherently unequal about segregated facilities.  They said that it was only the
feelings of the black people that made it seem as if segregation was meant to make
blacks seem inferior to whites.

The 14th Amendment says
only that people must be treated equally, not that they must all be allowed to use the
same sorts of facilities.  This is, for example, why there can be separate bathrooms for
men and women in public facilities.  Such bathrooms are not deemed to imply that women
are inferior to men.  They only imply that men and women are different and that it would
be inappropriate for them to be together in such
situations.

The Plessy decision
essentially said that blacks and whites are simply different and should be kept apart. 
This did not, the Court said, imply that blacks are inferior.  Therefore, there was no
problem because blacks and whites were being treated equally.