Mark 1:14 seems to say that the ministry of Jesus started after John the Baptist was arrested (and presumably, imprisoned):
Matthew 4:12-17 seems to say the same thing:
It should therefore be impossible for Jesus and John to be out and about preaching together at the same time, but this is John 3:22-24:
I imagine that inerrantists will want to say that Mark 1:14 and Matthew 4:12-17 do not explicitly say that the ministry was just beginning, and allow the possibility that the ministry had been going on for some time. While it's arguable that they don't explicitly say that the ministry was just starting, that's surely the natural reading (especially Matthew's statement that Jesus "began to preach").
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