For your first example you were manually solving for y and plotting the result.
For your second problem it wasn't so obvious how to do that by hand.
Solve can do that for you.
Re is showing only the real solutions. The
y/. is extracting the solutions from the form returned by Solve.
Now to answer the question about the ratnz warning. There are often multiple different algorithms inside of functions like
Reduce. Some of those algorithms only work with exact rational or integer values, some work with decimal approximations.
Plot is using decimal values to show your graphs. So it is passing decimal values to
Solve. To find the solutions you are looking for it appears that
Solve thinks using exact values would be the best approach. So it warns you that it is turning approximate decimal values back to exact rationals, finding the solution and then passing that back to
Plot which then turns those back into decimals and displays your curves.
We can avoid that warning with this change
which first finds the exact solutions and then hands those solutions to
Does that explain enough?