Question

Solve the inequality: $({\log }_{2}x{)}^2-|({\log }_{2}x)-2|\ge 0$.

Answer 1

$({\log }_{2}x{)}^2-\left|\right({\log }_{2}x)-2|\ge 0$

Let ${\log }_{2}x$ be $t$ , $t>0$

$t^2-|t-2|\ge 0$

$|t-2|\le t^2$

When $t\le 2$

$t-2\le t^2$

$t^2-t+2\ge 0$

as $△<0$

so $t^2-t+2>0$ for all $t$

$\therefore t\in (2,\infty )$

Where $t<2$

$-(t-2)\le t^2$

$t^2+t-2\ge 0$

$(t-2)(t+1)\ge 0$

$t\in (-\infty ,-1)\cup (2,\infty )$

$\therefore t\in (-\infty ,-1)$

$\therefore t\in (2,\infty )$

${\log }_{2}x\in (2,\infty )$

$\therefore x\in (4,\infty )$