Question

1. (1/x-1) + 2/x > (1/x+1)
2. (x+1)(x+2)(x+3)/(x-1)(x-2)(x-3)>1

Answer 1

$1); ; frac{1}{x-1}+frac{2}{x}>frac{1}{x+1}\\frac{x(x+1)+2(x-1)(x+1)-x(x-1)}{x(x-1)(x+1)}>0\\frac{x^2+x+2x^2-2-x^2+x}{x(x-1)(x+1)}>0\\frac{2(x^2+x-1)}{x(x-1)(x+1)} >0\\x^2+x-1=0; ,; ; D=1+4=5; ,; x_{1,2}=frac{-1pm sqrt5}{2}\\x_1approx -1,62; ,; ; x_2approx 0,62\\---( -1,62)+++(-1)---(0)+++(0,62)---(1)+++\\xin (frac{-1-sqrt5}{2},-1)cup (0,frac{-1+sqrt5}{2})cup (1,+infty )$

$2); ; frac{(x+1)(x+2)(x+3)}{(x-1)(x-2)(x-3)}>1\\frac{(x+1)(x+2)(x+3)-(x-1)(x-2)(x-3)}{(x-1)(x-2)(x-3)}>0\\frac{(x^2+3x+2)(x+3)-(x^2-3x+2)(x-3)}{(x-1)(x-2)(x-3)}>0\\frac{x^3+3x^2+3x^2+9x+2x+6-(x^3-3x^2-3x^2+9x+2x-6)}{(x-1)(x-2)(x-3)}>0\\frac{12x^2+12}{(x-1)(x-2)(x-3)}>0; ,; ; frac{12(x^2+1)}{(x-1)(x-2)(x-3)}>0\\12(x^2+1)>0; ; pri; ; xin (-infty ,+infty ); ; Rightarrow (x-1)(x-2)(x-3)>0\\---(1)+++(2)---(3)+++\\xin (1,2)cup (3,+infty )$