Question

Solve the following inequality:
$\frac{\sqrt{2x^2+7x-4}}{x+4}<\frac{1}{2}.$

Answer 1

$\frac{\sqrt{2x^2+7x-4}}{x+4}<\frac{1}{2}$

Squaring on both the sides,

$\Rightarrow 0<\frac{2x^2+7x-4}{(x+4{)}^2}<\frac{1}{4}$

$\Rightarrow 0<\frac{(x-\frac{1}{2})(x+4)}{(x+4{)}^2}<\frac{1}{4}$
$\Rightarrow 0<\frac{x-\frac{1}{2}}{x+4}<\frac{1}{4}$

$\Rightarrow 0<x-\frac{1}{2}<\frac{1}{4}(x+4)$

$\Rightarrow 0<x-\frac{1}{4}x<1+\frac{1}{2}$

$\Rightarrow 0<\frac{3}{4}x<\frac{3}{2}$

$\Rightarrow 0<x<2$

$\Rightarrow x\in (0,2)$