N° 3 SISTEMI DISEQUAZIONI

${\begin{matrix} ∣ \frac{x^{2}}{x - 1} ∣ + x > 3 \\ \sqrt{x^{2} - 5 x + 6} > x - 1 \end{matrix}$

$∣ \frac{x^{2}}{x - 1} ∣ + x > 3 \Rightarrow {\begin{matrix} \frac{x^{2}}{x - 1} \geq 0 \\ \frac{x^{2}}{x - 1} + x > 3 \end{matrix} \lor {\begin{matrix} \frac{x^{2}}{x - 1} < 0 \\ \frac{x^{2}}{1 - x} + x > 3 \end{matrix} \Rightarrow {\begin{matrix} x > 1 \\ \frac{x^{2} + x^{2} - x - 3 x + 3}{x - 1} > 0 \end{matrix} \lor {\begin{matrix} x < 1 \\ \frac{x^{2} + {x - x}^{2} - 3 + 3 x}{1 - x} > 0 \end{matrix} \Rightarrow$
$\Rightarrow {\begin{matrix} x > 1 \\ \frac{2 x^{2} - 4 x + 3}{x - 1} > 0 \end{matrix} \lor {\begin{matrix} x < 1 \\ \frac{4 x - 3}{1 - x} > 0 \end{matrix} \Rightarrow {\begin{matrix} x > 1 \\ x > 1 \end{matrix} \lor {\begin{matrix} x < 1 \\ \frac{3}{4} < x < 1 \end{matrix} \Rightarrow$ $x \in] \frac{3}{4}, 1 [\cup] 1, + \infty [$
$\sqrt{x^{2} - 5 x + 6} > x - 1 \Rightarrow {\begin{matrix} x - 1 \geq 0 \\ x^{2} - 5 x + 6 > {(x - 1)}^{2} \end{matrix} \lor {\begin{matrix} x - 1 < 0 \\ x^{2} - 5 x + 6 \geq 0 \end{matrix} \Rightarrow {\begin{matrix} x \geq 1 \\ x^{2} - 5 x + 6 > x^{2} - 2 x + 1 \end{matrix} \lor {\begin{matrix} x < 1 \\ x \leq 2 \lor x \geq 3 \end{matrix} \Rightarrow$
$\Rightarrow {\begin{matrix} x \geq 1 \\ x < \frac{5}{3} \end{matrix} \lor {\begin{matrix} x < 1 \\ x \leq 2 \lor x \geq 3 \end{matrix} \Rightarrow 1 \leq x < \frac{5}{3} \lor x < 1 \Rightarrow$ $x \in] - \infty, \frac{5}{3} [$

$x \in] \frac{3}{4}, 1 [\cup] 1, \frac{5}{3} [$