Question

Решите неравенство

|х+1| / х(х+2)^3 <= 0

Answer 1

$frac{|x+1|}{x(x+2)^3} leq 0$
$1) begin{cases} x+1 textless 0 \ frac{x+1}{x(x+2)^3} geq 0 end{cases} = textgreater begin{cases} x textless -1 \ x(x+2)^3 textless 0 end{cases} = textgreater begin{cases} x textless -1 \ x(x+2) textless 0 end{cases} = textgreater \ begin{cases} x textless -1 \ -2 textless x textless 0 end{cases} = textgreater -2 textless x textless -1.$
$2) begin{cases} x+1 geq 0 \ frac{x+1}{x(x+2)^3} leq 0 end{cases} = textgreater begin{cases} x geq -1 \ x(x+2)^3 textless 0 end{cases} = textgreater begin{cases} x geq -1 \ x(x+2) textless 0 end{cases} = textgreater \ begin{cases} x geq -1 \ -2 textless x textless 0 end{cases} = textgreater -1 leq x textless 0.$
Объединяем решения 1) и 2) случаев: -2 < x <0.
Ответ: (-2; 0).