I think it's long enough that I can add a full answer. By the first hint, you have that,
$$\begin{align}[x+2]\overset{1}{=}[x]+2&=6[x]-23\\5[x]&=25\\\ [x]&=5\\(2) \implies 5\le &x \lt6\end{align}$$
So, the solution is $$\boxed{5 \le x \lt6}$$
Hint:
- $[x+I]=[x]+I$ for $I$ an integer.
- $[x]=I \implies I\le x \lt I+1$ for $I$ an integer.
Functions over Relations
why differentiate between a function and a relation?
Please note that mathematical objects are usually defined because their existence is a need to facilitate dealing with some math subjects. When you see that a math concept is often appeared in mathematical literature, you can conclude that it has a significant role and many applications inside and outside of (pure) mathematics.
The concept "relation" has been defined to show mathematical relationships between mathematical objects. A "function" is a special kind of a relation that for each input there is only one output; in fact, functions are well-behaved relations because we can control the outputs of a function by controlling its inputs, which is a very important fact in developing calculus subjects such as limit, differentiation, integration, and so on. Looking at various subjects inside and outside of (pure) mathematics, we can find that almost all relations are functions or can be written as a union of some functions.
For example, consider the circle $C:y=\pm \sqrt{1-x^2}$. This relation is not a function because for each input there are two outputs. However, we can write it as the union of the following functions:$$y=\begin{cases}f_1(x)=\sqrt{1-x^2} \\ f_2(x)=-\sqrt{1-x^2} \end{cases} \quad \Rightarrow \quad C= f_1 \cup f_2.$$Now, we can apply any facts about functions to the function pieces of a relation. Please note that almost all (applied) facts about functions are local properties, so we can use them to treat a relation as a function.
Please note that there is a general principle:
Generality of a concept is inversely related to the information we know about the concept.
This principle not only holds in mathematics but also in other branches of knowledge.
Functions are less general than relations, but we have much more information about functions than relations. Almost all facts in many various (applied) mathematics subjects are expressed in terms of functions, and most of them cannot be expressed in terms of relations, and even if they can, they become awkward; also as mentioned above, many relations can be written as a union of functions.
Complexity of Inverting
Why does combining them seems to make inversion that much harder?
I think the question is a special case of the following question:
Why are there many mathematics problems which most people can understand easily but have very difficult (or do not have) solutions?
The answer is, because we have a few known facts (Mathematicians call them "axioms") and we have to prove any results from them.
Let us change your mentioned example. The inverse of the functions $g(x)=x^5$ and $h(x)=-x$ can be easily found. So, why can we not find the inverse of the following function easily:$$f(x)=g(x)+h(x)=x^5-x$$(I only added two "elementary functions")?
Please note that there are some facts written in less than ten words but their proofs have hundred pages; there are also some facts which most people can understand but a few mathematicians can understand their proofs.
Mathematics is an axiomatic theory. It does not ask people to find easy proofs for its facts; it only wants them to prove results from only a few axioms.
Best Answer
If you increase $x$, the left-hand sides of both equations increase. If you increase $y$, the left-hand side of the first equation increases, while the left-hand side of the second equation decreases.
That means that if there is some other solution $(a,b)$ and, let's say $a>3$, then the first equation says that $b<2$ while the second equation says that $b>2$.
So there are no more solutions (at least not among the positive reals).