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Question: What is the case for interpreting the Genesis creation account literally?
Answer: No textual reason for interpreting Genesis figuratively

Whenever we ask why we should take something in the Bible as literal, we must start by asking ask why we should not take it as literal.  Does the text give us any reason to interpret it as being figurative?  In the book of Revelation, and other places where mortal man is given a glimpse of eternity, descriptions employ the use of comparison quite frequently.  It was like this or had the appearance of that.  The Genesis account employs no such language.  

(By the way, the assumption that the Song of Solomon is figuratively is certainly debatable.)

The Order of Genesis

In fact, the Genesis account is written as if it were an actual account with a very logical order.  There are six distinct units in which creative works take place.  Each of these units is described as a day, with an evening and a morning.  (The Jewish and Muslim traditions still follow this pattern by marking sundown as the beginning of the next day.)

Genesis 1:2 identifies the nature of the earth:


  The earth was without form and void, and darkness was over the
  face of the deep. And the Spirit of God was hovering over the face of
  the waters.  Genesis 1:2 ESV


I have heard this original state of "without form and void" paraphrased as unformed and unfilled.  The first three days of Genesis are dedicated to forming while the second three days are dedicated to filling.  What is formed on the first day is filled on the 4th days, whatever is formed on the 2nd day is filled on the 5th, and so forth.  So, there are specific divisions that represent a very orderly process, which aligns better with an actual record rather than a figurative story.

Confirmation of Scripture

As Warren noted, other places in Scripture affirm the six days of creation.


  Six days you shall labor and do all your work, but the seventh day is
  a sabbath of the LORD your God .. For in six days the LORD made the
  heavens and the earth, the sea and all that is in them, and rested on
  the seventh day.  Exodus 20:8-11 NASB


Problems with figurative interpretations

Finally, the figurative interpretation breaks down.  If a days is a billion years, then how did flowers get pollinated when there were no bees in existence for another billion years?  Also, if the sixth day is a billion years from Adam, then we have not yet reached the 7th day of God's rest, since we have a genealogy from Adam that is roughly 6,000 years.  Fictitious characters do not have genealogies.  No one can trace his lineage back to Elmer Fudd or Pecos Bill, yet the Bible details a genealogy from Adam to Jesus Himself.

What reason is there to take it as figurative?

So, we might ask, from where does anyone get the notion that the Genesis account ought to be taken figuratively.  Quite simply, it comes from teachings based on a portion of scientific inquiry that purports irrefutable evidence of an old earth.  

It is fairly easy to see that a literal interpretation of Genesis does not fit with a very old earth.  The assumption is made, for whatever reason, that it must be the irrefutable scientific evidence that is correct.  The question, then, is what to do with Genesis, and this is answered by reinterpreting it figuratively.

The main problem with this is the assumption of the truthfulness regarding the selective data that purports irrefutable evidence of an old earth.  The problems and inconsistencies in the evidence are not given to the general public, but appear to be deliberately withheld.  Neither are the lines of scientific evidence that point to a young earth given to the general public.  To be truly scientific, all the evidence and all the data should be made available to all students.  Theological assumptions made by individual scientist, whether they be Theistic or Atheistic, should be withheld.  True science has no ax to grind other than the truth it seeks to find, so it should welcome all inquiry and debate.

Conclusion

So, the Genesis account does not give any indication of requiring figurative interpretation.  It actually details a very logical and orderly creation.  Other places in Scripture reference the six days of creation.  Finally, the figurative interpretation doesn't make any sense.