$\{\begin{array}{c}\frac{\mid x^2-3x\mid -10}{\mid x\mid }\le 3\\ \frac{\sqrt[3]{x^3+1}-\sqrt{x^2+1}}{x-2}\ge 0\end{array}$

• $\frac{\mid x^2-3x\mid -10}{\mid x\mid }\le 3\Rightarrow \{\begin{array}{c}x\le 0\\ \frac{x^2-3x-10}{-x}\le 3\end{array}\vee \{\begin{array}{c}0\le x\le 3\\ \frac{3x-x^2-10}{x}\le 3\end{array}\vee \{\begin{array}{c}x\ge 3\\ \frac{x^2-3x-10}{x}\le 3\end{array}\Rightarrow$

  $\Rightarrow \{\begin{array}{c}x\le 0\\ \frac{x^2-10}{x}\ge 0\end{array}\vee \{\begin{array}{c}0\le x\le 3\\ \frac{x^2+10}{x}\ge 0\end{array}\vee \{\begin{array}{c}x\ge 3\\ \frac{x^2-6x-10}{x}\le 0\end{array}\Rightarrow \{\begin{array}{c}x\le 0\\ -\sqrt{10}\le x<0\vee x\ge \sqrt{10}\end{array}\vee \{\begin{array}{c}0\le x\le 3\\ x>0\end{array}\vee \{\begin{array}{c}x\ge 3\\ x\le 3-\sqrt{19}\vee 0<x\le 3+\sqrt{19}\end{array}\Rightarrow$

  $\Rightarrow -\sqrt{10}\le x<0\vee 0<x\le 3\vee 3\le x\le 3+\sqrt{19}\Rightarrow$$-\sqrt{10}\le x<0\vee 0<x\le 3+\sqrt{19}$

• $\frac{\sqrt[3]{x^3+1}-\sqrt{x^2+1}}{x-2}\ge 0$
  • $\sqrt[3]{x^3+1}-\sqrt{x^2+1}\ge 0\Rightarrow \sqrt{x^2+1}\le \sqrt[3]{x^3+1}\Rightarrow \{\begin{array}{c}{x}^2+1\ge 0\\ \sqrt[3]{x^3+1}\ge 0\\ {(x^2+1)}^3\le {(x^3+1)}^2\end{array}\Rightarrow$

    $\Rightarrow \{\begin{array}{c}\forall x\in ℝ\\ x\ge -1\\ x^6+3x^2+3x^4+1\le x^6+2x^3+1\end{array}\Rightarrow \{\begin{array}{c}\forall x\in ℝ\\ x\ge -1\\ 3x^4-2x^3+3x^2\le 0\end{array}\Rightarrow \{\begin{array}{c}\forall x\in ℝ\\ x\ge -1\\ x^2(3x^2-2x+3)\le 0\end{array}\Rightarrow \{\begin{array}{c}\forall x\in ℝ\\ x\ge -1\\ x=0\end{array}\Rightarrow$$\Rightarrow x=0$

  • $x>2$

  $x<2$

$-\sqrt{10}<x<0\vee 0<x<2$