I have two problem areas in my lawn. The grass is thin and grows poorly under a large tree. I also have a sloping area, which is difficult to mow. What could I grow in these two areas that would be better than grass?
You may want to consider mulching the area around the shade tree and perhaps planting a few shade tolerant shrubs or flowers. Most labels on shrubs and flowers indicate whether they are shade tolerant.
Removing some of the lower branches on your shade tree would allow more light to reach the ground underneath and make it possible to grow a greater variety of plants.
There are suitable ground cover plants for both of these areas that do not require mowing. Before planting ground cover or other plants under your tree you will need to remove the grass. You could rent a sod cutter or if the area is not too large, use a shovel.
