Alright. I have spent hours it seems looking this problem over and I simply just don't get how this professor wants me to set up the class intervals.

Here's the problem:

a.) Construct a frequency distribution using five class intervals

b.) Construct a frequency distribution using ten class intervals.

Ok here is my problem. I understand that the range is 10 - 85. So the class width would be 75/5 = 15. They make us use some term "under" in the class intervals. So I had this but I doubt it is correct.

Frequency part is unimportant. I just can't for the life of me figure out if I am setting up the intervals correctly.

AFAIK there is a certain degree of freedom in how you choose your intervals.
What you should make sure is that
1) all intervals have the same width
2) the union of all your intervals contains all your data points
3) if the measurement you're dealing with can only output integer numbers, then your intervals should also have the same granularity, i.e. each interval should have an integer lower and upper limit.

That said, the usual way is to choose numbers so that there are no uselessly empty classes and possibly they are bounded by "nice" numbers (5, 10 etc)

For example you chose a "minimal coverage" solution, choosing data range/nr of intervals for their size. Won't work as well for 10 classes, as each interval should measure 7.6 and you'd end with interval limits that are not integers.
I'd rather round it up to 8, meaning that you can start lower than the minimum data point and end higher than the maximum (8-15, 16-23 ... 80-87).

Also please note that in your 5 classes case you're off by one: the very first interval spans 16 numbers instead of 15, thus your frequencies are not homogeneously sampled which is a big no-no. The fact is that while 85-10=75, there are 76 integer numbers between 10 an 85 when you include both extremes. In the same way 10-25 covers 16 integer numbers, but 26-40 and all subsequent intervals only 15.
Thus, even in the "5" case, I'd say you should round the interval width up to 16.

You usually include the biggest number of one interval, as the smallest in the next:
10-25
25-40
40-55 etc.

Other than that, it seems correct to me.