Bất phương trình 09

Question

$(x+5)(x-2)+3\sqrt{x(x+3)}>0$
$(x+1)(x+4)<5\sqrt{x^{2}+5x+28}$

score 5 · Answer 1

b. Bất phương trình tương đương với:
$(x+1)(x+4)<5\sqrt{x^2+5x+28}$
$\Leftrightarrow \left[\begin{array}{l}(x+1)(x+4)\le0\\\left\{\begin{array}{l}(x+1)(x+4)>0\\(x+1)^2(x+4)^2<25(x^2+5x+28)\end{array}\right.\end{array}\right.$
$\Leftrightarrow \left[\begin{array}{l}-4\le x\le-1\\\left\{\begin{array}{l}(x+1)(x+4)>0\\x^4+10x^3+8x^2-85x-684<0\end{array}\right.\end{array}\right.$
$\Leftrightarrow -9<x<4$

score 5 · Answer 2

a.
Điều kiện: $x(x+3)\ge0 \Leftrightarrow \left[\begin{array}{l}x\ge0\\x\le-3\end{array}\right.$
Bất phương trình tương đương với:
$3\sqrt{x(x+3)}>(x+5)(2-x)$
$\Leftrightarrow \left[\begin{array}{l}(x+5)(2-x)\le0\\\left\{ \begin{array}{l}(x+5)(2-x)>0\\9x(x+3)>(x+5)^2(2-x)^2\end{array} \right.\end{array}\right.$
$\Leftrightarrow \left[\begin{array}{l}x\ge2\\x\le-5\\\left\{\begin{array}{l}(x+5)(2-x)>0\\x^4+6x^3-20x^2-87x+100<0\end{array}\right.\end{array}\right.$
$\Leftrightarrow \left[\begin{array}{l}x\ge2\\x\le-5\\\left\{\begin{array}{l}(x+5)(2-x)>0\\(x-1)(x+4)(x^2+3x-25)<0\end{array}\right.\end{array}\right.$
$\Leftrightarrow \left[\begin{array}{l}x<-4\\x>1\end{array}\right.$