N° 6 SISTEMI DISEQUAZIONI

${\begin{matrix} \sqrt{x^{2} - 5 x + 7} \leq 2 x - 4 \\ ∣ x^{2} - 5 x + 6 ∣ \leq ∣ x - 2 ∣ \end{matrix}$

$\sqrt{x^{2} - 5 x + 7} \leq 2 x - 4 \Rightarrow {\begin{matrix} 2 x - 4 \geq 0 \\ x^{2} - 5 x + 7 \geq 0 \\ x^{2} - 5 x + 7 \leq {(2 x - 4)}^{2} \end{matrix} \Rightarrow {\begin{matrix} x \geq 2 \\ \forall x \in ℝ \\ x^{2} - 5 x + 7 \leq 4 x^{2} + 16 - 16 x \end{matrix} \Rightarrow {\begin{matrix} x \geq 2 \\ \forall x \in ℝ \\ 3 x^{2} - 11 x + 9 \geq 0 \end{matrix} \Rightarrow$
$\Rightarrow {\begin{matrix} x \geq 2 \\ ∀ x \in ℝ \\ x \leq \frac{11 - \sqrt{13}}{6} \lor x \geq \frac{11 + \sqrt{13}}{6} \end{matrix} \Rightarrow$ $x \geq \frac{11 + \sqrt{13}}{6}$
$∣ x^{2} - 5 x + 6 ∣ \leq ∣ x - 2 ∣ \Rightarrow {\begin{matrix} x \leq 2 \\ x^{2} - 5 x + 6 \leq 2 - x \end{matrix} \lor {\begin{matrix} 2 \leq x \leq 3 \\ - x^{2} + 5 x - 6 \leq x - 2 \end{matrix} \lor {\begin{matrix} 3 \leq x \\ x^{2} - 5 x + 6 \leq x - 2 \end{matrix} \Rightarrow$
$\Rightarrow {\begin{matrix} x \leq 2 \\ x^{2} - 4 x + 4 \leq 0 \end{matrix} \lor {\begin{matrix} 2 \leq x < 3 \\ x^{2} - 4 x + 4 \geq 0 \end{matrix} \lor {\begin{matrix} 3 \leq x \\ x^{2} - 6 x + 8 \leq 0 \end{matrix} \Rightarrow {\begin{matrix} x \leq 2 \\ x = 2 \end{matrix} \lor {\begin{matrix} 2 \leq x \leq 3 \\ \forall x \in ℝ \end{matrix} \lor {\begin{matrix} 3 \leq x \\ 2 \leq x \leq 4 \end{matrix} \Rightarrow$
$\Rightarrow x = 2 \lor 2 \leq x \leq 3 \lor 3 \leq x \leq 4 \Rightarrow$ $2≤ x \leq 4$

$\frac{11 + \sqrt{13}}{6} \leq x \leq 4$