I think people short-circuit the logic. Bible is Word of God -> God is Infallible -> Bible is infallible. Problem is, you are not perfect. You think the Earth is 6000 years old because the bible says so? Ok. Show me where it gives a date for creation. It doesn't. Humans came up with that number based on interpretation of the words in the book. Words that were first written down thousands of years ago. Copied, recopied, over and over, translated and retranslated through several languages. Each time humans involved.
Worse, the idea that the Earth is 6000 years old implies a deity who is a jackass. There is overwhelming scientific evidence proving the Earth is older than that. If God really created the Earth 6000 years ago, he created it in a manner so as to deliberately deceive us into thinking it was older. One example out of very, very many: We observe stars that are more than 6000 light-years away. If the universe were younger, that light would not be observable yet unless God either altered the speed of light dramatically mid-universe or he created light already in-transit knowing full well what we'd think when it hit the telescopes. What's more likely? That God created a universe designed into deceiving you away from his own holy book, or that maybe somewhere along the lines a human misunderstood something?