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Q.
The set of all real numbers $x$ for which $x^2-|x+2|+x > 0$ is
IIT JEEIIT JEE 2002Complex Numbers and Quadratic Equations
Solution:
Given, $x^2-|x+2|+x > 0 ...(i)$
Case I When $x+2\ge0$
$\therefore x^2-x-2+x>0 \Rightarrow x^2-2>0$
$\Rightarrow x < -\sqrt2$ or $ x > \sqrt2$
$\Rightarrow x\in(-2,-\sqrt2)\cup(\sqrt2,\infty) ...(ii)$
Case II When $x+2 < 0$
$\therefore x^2+x+2+x>0$
$\Rightarrow x^2+2x+2>0$
$\Rightarrow (x+1)^2+1>0$
which is true for all $x$.
$\therefore x\le-2$ or $ x\in(-\infty,-2) ...(iii)$
From Eqs. (ii) and (iii), we get
$x\in(-\infty-\sqrt 2)\cup(\sqrt 2,\infty )$