## Task B-2.6.

How many integer solutions does the system of inequalities have

$$
\begin{aligned}
& \left|x^{2}-5 x+4\right| \geqslant 2 \\
& |x|<2017
\end{aligned}
$$
## Solution.

Solve the inequality $\left|x^{2}-5 x+4\right| \geqslant 2$. (1)

$$
\begin{aligned}
& \left|x^{2}-5 x+4\right| \geqslant 2 \\
& \Leftrightarrow x^{2}-5 x+4 \leqslant-2 \text { or } x^{2}-5 x+4 \geqslant 2 \\
& \Leftrightarrow x^{2}-5 x+6 \leqslant 0 \text { or } x^{2}-5 x+2 \geqslant 0
\end{aligned}
$$

The solution to the inequality $x^{2}-5 x+6 \leqslant 0$ in the set $\mathbb{R}$ is the interval $[2,3]$.

The solution to the inequality $x^{2}-5 x+2 \geqslant 0$ in the set $\mathbb{R}$ is the union of intervals $\left\langle-\infty, \frac{5-\sqrt{17}}{2}\right] \cup$ $\left[\frac{5+\sqrt{17}}{2}, \infty\right\rangle$.

The integer solutions to inequality (1) are all integers within the set $[2,3]$ and in $\left\langle-\infty, \frac{5-\sqrt{17}}{2}\right] \cup\left[\frac{5+\sqrt{17}}{2}, \infty\right\rangle$, i.e., within the set $A=\{\ldots,-3,-2,-1,0,2,3,5,6,7, \ldots\} . \quad 1$ point

The solution to the inequality $|x|<2017$ in the set $\mathbb{R}$ is the interval $\langle-2017,2017\rangle$.

All integer solutions in this interval are in the set

$$
B=\{-2016,-2015, \ldots,-2,-1,0,1,2,3, \ldots, 2016\}
$$

The final solution to the given system is

$$
A \cap B=\{-2016, \ldots,-3,-2,-1,0,2,3,5,6,7, \ldots, 2016\}
$$

The number of elements in this set is $2016+1+2016-2=4031$.

Note: For solving inequality (1), we can use a graphical representation of the functions $y=$ $\left|x^{2}-5 x+4\right|$ and $y=2$.

![](https://cdn.mathpix.com/cropped/2024_05_30_7713364b1a15726862ddg-10.jpg?height=516&width=705&top_left_y=336&top_left_x=607)

The solution in the set $\mathbb{R}$ must be clearly marked on the $x$-axis.

Integer solutions can be listed or simply counted after this step or after solving the second inequality.

The solution to the second inequality, and the final solution, can be obtained in the same way as in the first method of solving.