Question

1)
$- 2(x - 3)(x + 4) > 0$
2)
$\frac{x + 4}{x - 2} \geqslant 0$
3)
${x}^{2} (x + 10)(x - \frac{1}{2} ) \leqslant 0$
4)
$( {x}^{2} - 1)( {x}^{2} + 1) > 0$
​

Answer 1

$1) \ -2(x-3)(x+4)>0 \ \ \ | : (-2)\\(x-3)(x+4) < 0\\(x-3)(x+4)=0\\x=3; \ x = -4 \\x \in (-4; \ 3)$

$2) \ \dfrac{x+4}{x-2} \geqslant 0\\2.1) \ x-2 \neq 0; \ x \neq 2\\2.2) \ x + 4 = 0; \ x = -4\\x \in (-\infty; \ -4] \cup (2; \ +\infty)$

$3) \ x^{2}(x+10)\bigg(x - \dfrac{1}{2} \bigg) \leqslant 0\\x^{2}(x+10)\bigg(x - \dfrac{1}{2} \bigg) = 0\\x = 0; \ x = -10; \ x = \dfrac{1}{2} \\x \in \bigg[-10; \ \dfrac{1}{2} \bigg]$

$4) \ (x^{2} - 1)(x^{2} + 1) > 0\\x^{4} - 1 > 0\\x^{4} - 1 \neq 0\\x^{4} \neq 1\\x \neq \pm 1\\x \in (-\infty; \ -1) \cup (1; \ +\infty)$