This is the 5943rd most frequent Japanese word.
ゆる
"ゆる" (yuru) means "loose" or "relaxed."
In this sentence, ゆるやかな modifies 坂 to describe a gentle slope.
ゆるやかな坂を登った。
I climbed the gentle slope.
Here, ゆるい describes the clothing as being loose or not tight.
先生はゆるい服を着ていた。
The teacher was wearing loose clothing.
ゆるみ in this sentence conveys the notion of looseness or slack, applied figuratively to the seriousness or quality of results.