1. Find all $x$ for which $[x]+\{2 x\}=2.5$, where $[x]$ is the integer part of the number $x$, and $\{x\}$ is the fractional part of the number $x$, that is, $\{x\}=x-[x]$.
Answer: $\{2.25 ; 2.75\}$.

Solution:

From the equation and definitions, it follows that $[x]=2$, and $\{2 x\}=0.5$.

Consider the equation $\{2 x\}=0.5$:

1) if $0 \leq\{x\}<\frac{1}{2}$, then $\{2 x\}=2\{x\} \Rightarrow\{x\}=0.25$.
2) if $\frac{1}{2} \leq\{x\}<1$, then $\{2 x\}=2\{x\}-1 \Rightarrow\{x\}=0.75$.

Since $x=[x]+\{x\}$, the solutions to the original equation are

$x_{1}=2+0.25=2.25, \quad x_{2}=2+0.75=2.75$.