8.4. Find all solutions to the equation $4[x]=25\{x\}-4.5$. Here $[x]$ denotes the integer part of the number $x$, that is, the greatest integer not exceeding the number $x$, and $\{x\}$ is the fractional part of the number $x$, which by definition is equal to $x-[x]$.
Answer: $x=k+(8 k+9) / 50$ for $k=-1,0, \ldots, 5-$ a total of 7 fractions.

Solution: multiply the equation by 2 and transform it. We get the equation $8[x]+9=50\{x\}$. Notice that the right side is a non-negative number, and it is no more than 50, that is, $0 \leq 8[x]+9 \leq 50$, from which $-9 / 8 \leq[x] \leq 41 / 8$. But $[x]$ is an integer, so it can only be equal to $-1,0,1,2,3,4$ or 5. Therefore, $x=[x]+\{x\}=[x]+(8[x]+9) / 50$, so the answer can be written in the form $x=k+(8 k+9) / 50$ for $k=-1,0, \ldots, 5$.

Comment: the case $k=-1$ works, as the fraction $-49 / 50$ has an integer part of -1 by definition, and the fractional part, also by definition, is $-49/50 - (-1)=1 / 50$.

Criteria: only the inequality $-9 / 8 \leq[x] \leq 41 / 8$ is obtained - 2 points.

The answer is not written explicitly, left in the form of a formula as in the solution - do not deduct points.

The case $k=-1$ is lost - minus 1 point.

Any other case is lost - minus 2 points.

Two any cases for $k>0$ are lost - no more than 3 points should be given for the solution.