Question

como se resolve
$\frac{1}x = \frac{3}{2 } - \frac{1}{x-1}$

Answer 1

Usando o MMC na segunda parte da equação, temos:

$\frac{1}x = \frac{3}{2 } - \frac{1}{x-1}\\\\\frac{1}x = \frac{3x-3}{2x-2} - \frac{2}{2x-2}\\\\\frac{1}x = \frac{3x-3-2}{2x-2}}\\\\ \frac{1}x = \frac{3x-5}{2x-2}}$

Multiplicando cruzado:

$2x-2=3x^2-5x\to3x^2-5x-2x+2=0\\\\3x^2-7x+2=0$

Resolvendo a equação do 2º grau:

$3x^2-7x+2=0\\\\\Delta=(-7)^2-4\cdot(3)\cdot(2)\\\Delta=49-24\\\Delta=25\\\\x=\frac{-(-7)\pm\sqrt25}{2\cdot3}\to x=\frac{7\pm5}{6}\\\\x'=\frac{7+5}{6}=2\\\\x''=\frac{7-5}{6}= \frac{2}{6}= \frac{1}{3} \\\\s=\{ \frac{1}{3} ,2\}$