[latex]q=dfrac{2x+1}{x+2}\ |q| extless 1\ left|dfrac{2x+1}{x+2} ight| extless 1\ dfrac{2x+1}{x+2} extless 1 wedge dfrac{2x+1}{x+2} extgreater -1\ (2x+1)(x+2) extless (x+2)^2 wedge (2x+1)(x+2) extgreater -(x+2)^2\ 2x^2+4x+x+2 extless x^2+4x+4 wedge 2x^2+4x+x+2 extgreater -x^2-4x-4\ x^2+x-2 extless 0 wedge 3x^2+9x+6 extgreater 0\ x^2-x+2x-2 extless 0 wedge x^2+3x+2 extgreater 0\ x(x-1)+2(x-1) extless 0 wedge x^2+x+2x+2 extgreater 0\ (x+2)(x-1) extless 0 wedge x(x+1)+2(x+1) extgreater 0\ xin(-2,1) wedge (x+2)(x+1) extgreater 0\ xin(-2,1) wedge xin(-infty,-2)cup(-1,infty)\ xin(-1,1) [/latex] [latex] dfrac{1}{x+2}+ dfrac{2x+1}{(x+2)^{2}} + dfrac{(2x+1)^{2}}{(x+2)^3}+ldots geq 3 \\ dfrac{ dfrac{1}{x+2}}{1- dfrac{2x+1}{x+2}}geq3\\ dfrac{ dfrac{1}{x+2}}{dfrac{x+2}{x+2}- dfrac{2x+1}{x+2}}geq3\\ dfrac{ dfrac{1}{x+2}}{dfrac{x+2-2x-1}{x+2}}geq3\\ dfrac{ dfrac{1}{x+2}}{dfrac{-3x+1}{x+2}}geq3\\ dfrac{1}{-3x+1}geq3\\ dfrac{1}{3x-1}leq-3\\ 3x-1leq-3(3x-1)^2\ 3x-1leq-3(9x^2-6x+1)\ 3x-1leq-27x^2+18x-3\ 27x^2-15x+2leq0\ 27x^2-9x-6x+2leq0 [/latex] [latex]9x(3x-1)-2(3x-1)leq0\ (9x-2)(3x-1)leq0\ xinleftlangledfrac{2}{9},dfrac{1}{3} ight angle\\ xinleftlangledfrac{2}{9},dfrac{1}{3} ight angle wedge xin(-1,1)\ oxed{xinleftlangledfrac{2}{9},dfrac{1}{3} ight angle}[/latex]
