Question

Solve the following inequality:
$\frac{x+7}{x-5}+\frac{3x+1}{2}⩾0$

Answer 1

$\frac{x+7}{x-5}+\frac{3x+1}{2}\ge 0$

$\frac{2x+14+(x-5)(3x+1)}{2(x-5)}\ge 0$

$\frac{2x+14+3x^2+x-15x-5}{(x-5)}\ge 0$

$\frac{3x^2-12x+9}{(x-5)}\ge 0$

$\frac{x^2-4x+3}{(x-5)}\ge 0$

$\frac{x^2-3x-x+3}{(x-5)}\ge 0$

$\frac{(x-3)(x-1)}{(x-5)}\ge 0(x=4,\frac{1\times 3}{-1}<0)$

$x\epsilon [1,3]\bigcup [5,\infty )$