Question

Solve the inequation $\sqrt[5]{5\left[\frac{3}{x+1}+\frac{7}{x+2}\right]}<\sqrt[5]{5\frac{6}{x-1}}$. The number of non negative integer solutions is

Answer 1

The given inequation is equivalent to
$\frac{3}{x+1}+\frac{7}{x+2}<\frac{6}{x-1}\Rightarrow \frac{{4x}^2-15x-25}{(x+1)(x+2)(x-1)}<0\Rightarrow \frac{(x+5/4)(x-5)}{(x+1)(x+2)(x-1)}<0$
From wavy curve method:
$xϵ(-\infty ,-2)\cup (-\frac{5}{4},1)\cup (1,5)$
for non-negative integral solution
$x\in \left\{3,4,5\right\}$
Therefore, Three non negative integral solutions exist.