Deriving lower and upper bounds for T(n) = T(n-1) + T(n-2) + 10
The goal of a recurrence relation is to make a complicated problem easier to think about, which is very general. It allows us to express a function in terms of itself, but still reason about what the value of the function for an input is.

Sometimes you care about the exact value of the function, in which case you'd prefer to rewrite it in an exactly equivalent way (or just plug in a number and follow the recursion chain until you have your answer).

Other times, you're just worried about the extreme "if my worst enemy plugged a value into this function, what would it do?" case.

Rewriting the equation as an inequality let's us remove an entire recursive call of the function, which makes doing substitutions to figure out an easier to reason about form of the function a more achievable goal. 

So yes, $T(N) = T(N-1) + T(N-2) + 10$, may exactly describe the run time of your program, or the amount of money in my bank account after $N$ days (I wish), but we don't care about that precision, because $T(N) &lt; 2T(N-1) + 10$ is an upperbound, and we just want to know that our program will finish before the heat death of the universe, or that I have enough money in my bank account for dinner.

Note the benefit here. Perhaps because your algorithm had a recursive nature to it, it might have been really easy to come up with the $T(N) = T(N-1) + T(N-2) + 10$ description of the run time. Easy to come up with, but hard to think about what the function actually means. So now that we have this building block, we come up with an upper bound (that we know is true), but has less recursion (which our brains don't like). This is nice on its own, but it also let's us continue to expand the recursive definition until we understand the underlying pattern, and can remove the recursion completely.