Center Controller reported they assigned an aircraft an airway whose altitude was below the Minimum Enroute Altitude for that airway. The aircraft was also below the Minimum IFR Altitude for the airspace.

Narrative

I had just received a Position Relief Briefing and assumed control of sectors XX/XY combined. I had noticed in my sector preview that Aircraft X was at 110 and that his flight path looked as if it might take him close to an area where the MIA was 12000 ft. The Controller I was relieving was not running the sector with MIA outlines visible; however; so I wasn't sure of the exact proximity. I made a mental note to revisit this when I assumed the sector. The preview period and briefing took longer than average as there was a lot a clutter and some complexity down in the south of the airspace near ZZZ APCH. By the time I had assumed control of the sector and turned on my settings (which depicted MIA outlines) I saw that Aircraft X was indeed in conflict with the eastern edge of the 12000 ft. MIA. At this point I had 10 miles or less until the MIA conflict point. Since the aircraft was on a direct routing from ZZZ VOR to ZZZ1 VOR I told the pilot to join V26 which connected those VORs. I thought the MEA on the airway was 11000 ft. and therefore I was clean and I moved my scan back down south where I had some coordination and other work to do. When I came back to Aircraft X a minute later; I glanced up at the map to check my work and realized that the 11000 ft. MEA I remembered was for the next route segment on the airway (inside ZZZ1 APCH) and that the aircraft was about to enter the 12000 ft. MIA on VXX which had a segment MEA of 11700 ft. The pilot of Aircraft X must have realized this as well because at this time he called asked if I wanted him to climb to 13000 ft. for 'right for direction'. I issued the climb to 13000 ft. and a 20 degree right turn toward lower terrain; but the aircraft had already entered the 12000 ft. MIA/11700 ft. MEA area at 11000 ft. I had the 11700 ft. number in my head from the MEA so I cleared the aircraft back on course early (leaving 11800 ft.) to ZZZ1 [VOR] putting him on a direct route still inside the edge of 12000 ft. MIA. I should have questioned the Controller I was relieving about the aircraft's proximity to the MIA right when I saw it in my preview. At that point we had ample time to vector/climb the aircraft and it wouldn't have been an issue. Especially since the other traffic in the sector (dual survey missions and a departure push) was drawing attention down south. The Controller I was relieving is known for drawn out preview/briefing processes as well so I really should have brought it up sooner and dealt with it before relieving him.