Đáp án:
$\left[\begin{array}{l} x<-1\\ -1<x<0 \\ x>1 \end{array} \right.$
Giải thích các bước giải:
$|x^3+1|>x+1(*)\\ \circledast x^3+1=0 \Leftrightarrow x^3 = -1 \Leftrightarrow x = -1 \Rightarrow (*) \Leftrightarrow 0>0(L)\\ \circledast x^3+1 > 0 \Leftrightarrow x> -1\\ (*)\Leftrightarrow x^3+1>x+1\\ \Leftrightarrow x^3-x>0\\ \Leftrightarrow x(x^2-1)>0\\ \Leftrightarrow x(x-1)(x+1)>0\\ \Leftrightarrow x(x-1)>0(Do \ x+1 \ge 0 \ \forall \ x \ge -1)\\ \Leftrightarrow \left[\begin{array}{l} \left\{\begin{array}{l} x>0\\ x-1>0 \end{array} \right.\\ \left\{\begin{array}{l} x<0\\ x-1<0 \end{array} \right.\end{array} \right.\\ \Leftrightarrow \left[\begin{array}{l} \left\{\begin{array}{l} x>0\\ x>1 \end{array} \right.\\ \left\{\begin{array}{l} x<0\\ x<1 \end{array} \right.\end{array} \right.\\ \Leftrightarrow \left[\begin{array}{l} x>1 \\x<0 \end{array} \right.$
Kết hợp điều kiện $\Rightarrow \left[\begin{array}{l} x>1 \\-1 < x<0 \end{array} \right.$
$\circledast x< -1\\ (*)\Leftrightarrow -x^3-1>x+1\\ \Leftrightarrow -x^3-x-2>0\\ \Leftrightarrow x^3+x+2<0\\ \Leftrightarrow x^3+x^2-x^2-x+2x+2<0\\ \Leftrightarrow x^2(x+1)-x(x+1)+2(x+1)<0\\ \Leftrightarrow (x^2-x+2)(x+1)<0\\ \Leftrightarrow \left(\left(x-\dfrac{1}{2}\right)^2+\dfrac{7}{4}\right)(x+1)<0$
Đúng với $\forall \ x< -1$
Kết hợp 2 trường hợp
$\Rightarrow \left[\begin{array}{l} x<-1\\ -1<x<0 \\ x>1 \end{array} \right.$
