Question

Solve the following inequality:
$\sqrt{-x^2+6x-5}>8-2x.$

Answer 1

$\sqrt{-x^2+6x-5}>8-2x$

Squaring

$-x^2+6x-5>64-32x+4x^2$

$5x^2-38x+69<0$

$5x^2-15x-23x+69<0$

$5x(x-3)-23(x-3)<0$

$(5x-23)(x-3)<0$

$x\in (3,\frac{23}{5})$

$-x^2+6x-5\ge 0$

$x^2-6x+5\le 0$

$(x-5)(x-1)\le 0$

$x\in (1,5)$

$\therefore x\in (3,\frac{23}{5})$