The story about Jesus forgiving an adulterous women is not in the original copy, but that's the version we most commonly have. So how does that make the Bible inerrant, given that many important parts were added or removed?
The belief in biblical inerrancy does not require that the copies or translations of the Bible be completely inerrant. It is generally accepted that:
(1) Copies are subject to human error
(2) Translations are subject to human interpretation
Biblical inerrancy is the belief that the original manuscripts are inerrant. Unfortunately, we do not have many original manuscripts (if we have any), so we study the copies that we have, compare them, and try to determine the text of the original manuscripts. We may not be able to do so perfectly, and areas of ambiguity are often noted in modern Bibles (John 8, Mark 16).
However, it stands to reason that we have a reasonably accurate representation of the original manuscripts, with the essentials preserved.