What is the minimum slope required for horizontal toilet drain pipe?

I need to fix a bathroom remodel problem. The previous owner turned a hall closet into a small half-bath (toilet and sink). The drains from the toilet and sink meet below the room and the waste is carried horizontally in a 3" PVC pipe to the main drain pipe about 14ft away. The horizontal pipe was installed with minimal slope and has now backed up. The pipe is running in the crawl space about 24" high. The plumber who came to look at it wants to install a new horizontal pipe that will be put in a trench that must be dug in the crawl space. I'm wondering if instead of the trench, we use the space between the bottom of the joists and dirt to slope the pipe on its 14 ft run. How much slope is required for this length? A drop of 6" from beginning to end? 12"? Every inch counts here so please be as exact as you can?

Thanks for any advice or useful links.

Minimum slope for a drain is 1/4" per foot. That's 1" for 4 feet, so 14 feet would require at least 3-1/2" slope.

Thanks for the quick reply. There is room to do this above ground. I think that is better than excavating a new trench in a tight space. Do you have any thoughts on this?

As long as you don't mind the pipe hanging there, there's no reason to trench it. All that is required is the MINIMUM of the 1/4" per foot. It must run straight, or if not, at least with no dips or humps. This means over that length, it will need hangers to keep it in place and maintain the proper slope without sagging over time from its own weight.

We would have to see a photo of the installation to find out WHY he thinks he has to dig a pipe if there is enough room to do it above ground. Also, where is the vent in that 14' run? However, even with "minimal slope" it should not have "backed up", so there may be some other problem you do not know about.