=\frac{1}{x+2}\\\\ 21x^2+39x-6=0\\ 7x^2+13x-2=0\\ D=13^2-4*7*-2=225\\ x=\frac{-13+15}{14}=\frac{1}{7}\\ x=\frac{-13-15}{14}=-2 (7x-1)(x+2)<0\\ x \ \in \ (-2;\frac{1}{7})\\\\ \frac{1}{x}+\frac{1}{x+1} \geq \frac{1}{x+2}\\ \frac{2x+1}{x(x+1)} \geq \frac{1}{x+2}\\ \frac{(2x+1)(x+2)}{x(x+1)(x+2)} \geq \frac{x(x+1)}{x(x+1)(x+2)}\\ (2x+1)(x+2) \geq x(x+1)\\ 2x^2+5x+2 \geq x^2+x\\ x^2+4x+2 \geq 0\\ D=16-4*1*2=8\\ x=\frac{-4+\sqrt{8}}{2}=-2+\sqrt{2}\\ " alt=" 21x^2+39x-6<0\\ \frac{1}{x}+\frac{1}{x+1}>=\frac{1}{x+2}\\\\ 21x^2+39x-6=0\\ 7x^2+13x-2=0\\ D=13^2-4*7*-2=225\\ x=\frac{-13+15}{14}=\frac{1}{7}\\ x=\frac{-13-15}{14}=-2 (7x-1)(x+2)<0\\ x \ \in \ (-2;\frac{1}{7})\\\\ \frac{1}{x}+\frac{1}{x+1} \geq \frac{1}{x+2}\\ \frac{2x+1}{x(x+1)} \geq \frac{1}{x+2}\\ \frac{(2x+1)(x+2)}{x(x+1)(x+2)} \geq \frac{x(x+1)}{x(x+1)(x+2)}\\ (2x+1)(x+2) \geq x(x+1)\\ 2x^2+5x+2 \geq x^2+x\\ x^2+4x+2 \geq 0\\ D=16-4*1*2=8\\ x=\frac{-4+\sqrt{8}}{2}=-2+\sqrt{2}\\ " align="absmiddle" class="latex-formula">
![x=\frac{-4-\sqrt{8}}{2}=-2-\sqrt{2}\\ x \in [-\sqrt{2}-2;-2)\ \cup \ (-1;\sqrt{2}-2] \ \cup \ (0;\infty)](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-4-%5Csqrt%7B8%7D%7D%7B2%7D%3D-2-%5Csqrt%7B2%7D%5C%5C+x+%5Cin+%5B-%5Csqrt%7B2%7D-2%3B-2%29%5C+%5Ccup+%5C+%28-1%3B%5Csqrt%7B2%7D-2%5D+%5C+%5Ccup+%5C+%280%3B%5Cinfty%29 "x=\frac{-4-\sqrt{8}}{2}=-2-\sqrt{2}\\ x \in [-\sqrt{2}-2;-2)\ \cup \ (-1;\sqrt{2}-2] \ \cup \ (0;\infty)")
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![x\in (0;\frac{1}{7}) \ \cup \ (-1;\sqrt{2}-2]](https://tex.z-dn.net/?f=x%5Cin+%280%3B%5Cfrac%7B1%7D%7B7%7D%29+%5C+%5Ccup+%5C+%28-1%3B%5Csqrt%7B2%7D-2%5D "x\in (0;\frac{1}{7}) \ \cup \ (-1;\sqrt{2}-2]")