Question

Solve the following inequalities.
$x-3\sqrt{x-3}-1>0.$

Answer 1

$x-3\sqrt{x-3}-1>0$

$\frac{x-1}{3}>\sqrt{x-3}$

$\sqrt{x-3}<\frac{x-1}{3}$

$(x-3)>0$

$x>3x\epsilon (3,\infty )-\left(i\right)$

$x-3<{\left(\frac{x-1}{3}\right)}^2$

$x-3<\frac{x^2+1-2x}{9}$

$9x-27<x^2-2x+1$

$x^2-11x+28>0$

$x^2-7x-4x+28>0$

$(x-7)(x-4)>0$

$x\epsilon (-\infty ,4)\bigcup (7,\infty )-\left(ii\right)$

$from\left(i\right)\&\left(ii\right)$

$x\epsilon (3,4)\bigcup (7,\infty )$