Question

RESOLVA ESTA INEQUAÇÃO, POR FAVOR! 1$\frac{1}{x-1} +\frac{2}{x-2} -\frac{3}{x-3}$ < 0

Answer 1

$\dfrac{1}{x-1}+\dfrac{2}{x-2}+\dfrac{3}{x-3}<0$

$\dfrac{1\cdot(x-2)\cdot(x-3)+2\cdot(x-1)\cdot(x-3)+3\cdot(x-1)\cdot(x-2)}{(x-1)\cdot(x-2)\cdot(x-3)}<0$

$\dfrac{x^2-7x+6+2x^2-8x+6+3x^2-9x+6}{(x-1)\cdot(x-2)\cdot(x-3)}<0$

$\dfrac{6x^2-24x+18}{(x-1)\cdot(x-2)\cdot(x-3)}<0$

Numerador:

$6x^2-24x+18=0$

$x^2-4x+3=0$

$\Delta=(-4)^2-4\cdot1\cdot3$

$\Delta=16-12$

$\Delta=4$

$x=\dfrac{-(-4)\pm\sqrt{4}}{2\cdot1}=\dfrac{4\pm2}{2}$

$x'=\dfrac{4+2}{2}~\longrightarrow~x'=\dfrac{6}{2}~\longrightarrow~x=3$

$x"=\dfrac{4-2}{2}~\longrightarrow~x"=\dfrac{2}{2}~\longrightarrow~x"=1$

$S=\{x\in\mathbb{R}~|~x<1~\text{ou}~1 < x < 2\}$

Logo, $1 < x < 2$ ou $x < 1$