1. Let $[x]$ and $\{x\}$ be the integer and fractional parts of the number $x$. The integer part of the number $x$ is the greatest integer not exceeding $x$, and $\{x\}=x-[x]$. Find $x$ if $2 x+3[x]-5\{x\}=4$.
Solution.

$x=[x]+\{x\} \rightarrow 2([x]+\{x\})+3[x]-5\{x\}=4 \rightarrow 5[x]-3\{x\}=4 \rightarrow\{x\}=\frac{5[x]-4}{3} \in[0 ; 1)$

$[x] \in\left[\frac{4}{5} ; \frac{7}{5}\right) \rightarrow[x]=1 \rightarrow\{x\}=\frac{1}{3} \rightarrow x=[x]+\{x\}=1+\frac{1}{3}=\frac{4}{3}$

Answer: $x=\frac{4}{3}$.