# 2. Solve the inequality:

$$
\{x\}([x]-1)<x-2,
$$

where $[x]$ and $\{x\}$ are the integer and fractional parts of the number $\boldsymbol{x}$, respectively (9 points).

#
# Solution:

Let $\boldsymbol{a}=[\boldsymbol{x}], \boldsymbol{b}=\{\boldsymbol{x}\}$, then $\boldsymbol{x}=\boldsymbol{a}+\boldsymbol{b}$. Considering this, we get:

$$
\begin{gathered}
b(a-1)0, \boldsymbol{a}>2,[x]>2$. The last inequality is equivalent to $x \geq 3$. This will be the desired solution.

Answer: $\mathrm{x} \geq 3$.

| Points | Criteria for evaluating the performance of task № 2 |
| :---: | :--- |
| $\mathbf{9}$ | A correct and justified sequence of all steps of the solution is provided. All transformations and calculations are performed correctly. The correct answer is obtained. |
| $\mathbf{5}$ | The problem is not solved, but its solution is significantly advanced, i.e.:  - a significant part of the solution is performed correctly, possibly inaccurately (for example, factorization is performed);  - another part is either not performed or performed incorrectly, possibly even with logical errors (for example, the signs of the factors are not analyzed).  The solution may be incomplete. |
| $\mathbf{0}$ | All cases of the solution that do not meet the criteria for scoring 5 and 9 points. |