Question

Solution set of the inequation $(x-\sqrt{1-\left|x\right|})<0$, is

• A. $[-1,\frac{-1+\sqrt{5}}{2})$
• B. $[-1,1]$
• C. $[-1,\frac{-1+\sqrt{5}}{2}]$
• D. $(-1,\frac{-1+\sqrt{5}}{2})$

Answer 1

Answer: (A)

$[-1,\frac{-1+\sqrt{5}}{2})$

Given,

$x-\sqrt{1+\left|x\right|}<0$

$\Rightarrow x<\sqrt{1-\left|x\right|}\Rightarrow \left|x^2\right|<1$ for $x>0$

$x^2<1-\left|x\right|=1-x.$

$\Rightarrow x^2+x-1<0$

$\Rightarrow x=\frac{-1\pm \sqrt{1-4(-1)}}{2}=\frac{-1+\sqrt{5}}{2}$ ...(1)

And $\sqrt{1-\left|x\right|}$ is defined where

$1-\left|x\right|\ge -\Rightarrow \left|x\right|\ge 1$

$\Rightarrow -1\le x\le 1$

$\therefore x\in [-1,\frac{-1+\sqrt{5}}{2})$