N° 4 SISTEMI DISEQUAZIONI

${\begin{matrix} \frac{\sqrt{6 x - x^{2}}}{3 - 2 x} \geq 1 \\ ∣ \frac{x + 3}{x - 2} ∣ < 4 \end{matrix}$

6 x − x 2 3 − 2 x − 1 ≥ 0 ⇒ 6 x − x 2 − 3 + 2 x 3 − 2 x ≥ 0 ⇒
- $\sqrt{6 x - x^{2}} \geq 3 - 2 x \Rightarrow {\begin{matrix} 3 - 2 x \geq 0 \\ 6 x - x^{2} \geq {(3 - 2 x)}^{2} \end{matrix} \lor {\begin{matrix} 3 - 2 x < 0 \\ 6 x - x^{2} \geq 0 \end{matrix} \Rightarrow {\begin{matrix} x \leq \frac{3}{2} \\ 6 x - x^{2} \geq 9 + 4 x^{2} - 12 x \end{matrix} \lor {\begin{matrix} x > \frac{3}{2} \\ 0 \leq x \leq 6 \end{matrix} \Rightarrow$
  $\Rightarrow {\begin{matrix} x \leq \frac{3}{2} \\ 5 x^{2} - 18 x + 9 \leq 0 \end{matrix} \lor \frac{3}{2} < x \leq 6 \Rightarrow {\begin{matrix} x \leq \frac{3}{2} \\ \frac{3}{5} \leq x \leq 3 \end{matrix} \lor \frac{3}{2} < x \leq 6 \Rightarrow \frac{3}{5} \leq x \leq \frac{3}{2} \lor \frac{3}{2} < x \leq 6 \Rightarrow \frac{3}{5} \leq x \leq 6$
- $3 - 2 x > 0 \Rightarrow x < \frac{3}{2}$
$\frac{3}{5} \leq x < \frac{3}{2}$
$∣ \frac{x + 3}{x - 2} ∣ < 4 \Rightarrow {\begin{matrix} \frac{x + 3}{x - 2} \geq 0 \\ \frac{x + 3}{x - 2} - 4 < 0 \end{matrix} \lor {\begin{matrix} \frac{x + 3}{x - 2} < 0 \\ \frac{x + 3}{2 - x} - 4 < 0 \end{matrix} \Rightarrow {\begin{matrix} x \leq - 3 \lor x > 2 \\ \frac{x + 3 - 4 x + 8}{x - 2} < 0 \end{matrix} \lor {\begin{matrix} - 3 < x < 2 \\ \frac{x + 3 - 8 + 4 x}{2 - x} < 0 \end{matrix} \Rightarrow {\begin{matrix} x \leq - 3 \lor x > 2 \\ \frac{3 x - 11}{x - 2} > 0 \end{matrix} \lor {\begin{matrix} - 3 < x < 2 \\ \frac{5 x - 5}{2 - x} < 0 \end{matrix} \Rightarrow$
$\Rightarrow {\begin{matrix} x \leq - 3 \lor x > 2 \\ x < 2 \lor x > \frac{11}{3} \end{matrix} \lor {\begin{matrix} - 3 < x < 2 \\ x < 1 \lor x > 2 \end{matrix} \Rightarrow x \leq - 3 \lor x > \frac{11}{3} \lor - 3 < x < 1 \Rightarrow$ $x < 1 \lor x > \frac{11}{3}$

$\frac{3}{5} \leq x < 1$