A) If the median is 3 and sum of the integers is 10 then there are 2 possibilities: x=1,y=3 and z=6 or x=2,y=3 and z=5. Since we can find the greatest value in both the cases I chose A.

B) Since range is 5, y-x = 5 and we know x+y+z=10 so adding both the equations will give me 2y+z=15 but we have 2 unknown variables so we cannot solve this.

C) If x+z=7 then that means the median is 3 which is same as A. So since I could find the greatest value in (A), then (C) also has to be correct.

What exactly went wrong in my thought process? Did I misunderstand the question? How do I prevent this from happening? Any tips or resources would be very helpful

Indeed, thatâ€™s actually the problem. As you noted, you could find two different values of z, the greatest number. Hence you canâ€™t actually be sure on what the greatest number is, and hence A is incorrect.

The question says individually, which means that you need to consider each of the statements on their own, not combining them together. Hence that reasoning is not correct either.

The range is 5 means that z - x is 5, not y - x. Also, you can assume without loss of generality that x < y < z.

Spoiler Alert (POWERPREP +) ! My theory is a lacking when it comes to questions like these? How do I answer this question mathematically or conceptually? Is there any quick reference that covers all such concepts relevant to GRE quant?

I think youâ€™ve to build an intuition for these type of questions and for that your foundation should be solid, thus, I would recommend you either go through GRE math review or Gregâ€™s concept series at first & after completing that watch the strategies series.