N° 12 SISTEMI DISEQUAZIONI

${\begin{matrix} \frac{3 x - \sqrt{x^{2} - 1}}{∣ x ∣ + \sqrt{x + 5}} \geq 0 \\ \sqrt{- 3 x^{2} + 2 x + 1} \geq 0 \end{matrix}$

3 x − x 2 − 1 ∣ x ∣ + x + 5 ≥ 0
- $3 x - \sqrt{x^{2} - 1} \geq 0 \Rightarrow \sqrt{x^{2} - 1} \leq 3 x \Rightarrow {\begin{matrix} 3 x \geq 0 \\ x^{2} - 1 \geq 0 \\ x^{2} - 1 \leq 9 x^{2} \end{matrix} \Rightarrow {\begin{matrix} x \geq 0 \\ x \leq - 1 \lor x \geq 1 \\ 8 x^{2} + 1 \geq 0 \end{matrix} \Rightarrow x \geq 1$
- $∣ x ∣ + \sqrt{x + 5} > 0 \Rightarrow x \geq - 5$
  $x \geq 1$
$\sqrt{- 3 x^{2} + 2 x + 1} \geq 0 \Rightarrow - 3 x^{2} + 2 x + 1 \geq 0 \Rightarrow 3 x^{2} - 2 x - 1 \leq 0 \Rightarrow$ $- \frac{1}{3} \leq x \leq 1$

$x = 1$