Question

If $|x^2+x-4|=|x^2-4|+|x|$, then $x\in$

• A. $[0,2]$
• B. $(0,2)$
• C. $[-2,0]\cup [2,\infty )$
• D. $[-2,\infty )$

Answer 1

The correct option is C $[-2,0]\cup [2,\infty )$

$|x^2+x-4|=|x^2-4|+|x|$
This is of the form $|a|+|b|=|a+b|$
$\Rightarrow ab\ge 0\left(a\text{and}b\text{should have same sign}\right)$
$\Rightarrow x(x^2-4)\ge 0$
$\Rightarrow x(x-2)(x+2)\ge 0$
$\Rightarrow x\in [-2,0]\cup [2,\infty )$