I've read that the rule is 1/4" per foot of drain line. From what I know, the rule is based on the height of the P-trap to where you tie in. What part of the P-trap do you measure from? How low can you tie into an existing pipe?
You're limited by the size of the materials you're using.
2012 International Residential Code (IRC)
SECTION P3105 FIXTURE VENTS
P3105.1 Distance of trap from vent. Each fixture trap shall have a protecting vent located so that the slope and the developed length in the fixture drain from the trap weir to the vent fitting are within the requirements set forth in Table P3105.1.
Exception: The developed length of the fixture drain from the trap weir to the vent fitting for self-siphoning fixtures, such as water closets, shall not be limited.
If you're using a 1 1/2" trap, there must be a vent within 6' of the trap weir. The overall length of the drain is also limited based on the pipe size (P3105.2).
P3105.2 Fixture drains. The total fall in a fixture drain resulting from pipe slope shall not exceed one pipe diameter, nor shall the vent pipe connection to a fixture drain, except for water closets, be below the weir of the trap.
So if you're using 2" PVC, the drain must be less than 8' long horizontally.
1/4" * 8' = 2"