Solution set of `x-sqrt(1-|x|)lt0`, is

Options

A. `[-1,(-1+sqrt(5))/2)`
B. `[-1,1]`
C. `[-1,(-1+sqrt(5))/2]`
D. `(-1,(-1+sqrt(5))/2)`

Answer

A

Explanation

We have `x-sqrt(1-|x|)lt0`…i
which is defined only when
`1-|x|gt0`
`implies|x|le1`
`implies x epsilon [-1,1]`
Now, from Eq. (i) we get
`x lt sqrt(1-|x|)`
Case I If`x ge0` i.e. `0lexle1`
`x-sqrt((1-|x|))lt0`
`impliesx lt sqrt((1-x))`
On squaring both sides we get
`x^(2)+x-1lt0`
`implies(-1-sqrt(5))/2lt xlt (-1+sqrt(5))/2`
But `xge0`
`:.x epsilon[0,(-1+sqrt(5))/2)`
Case II If `x lt 0` i.e. `-1lexlt0`
`x-sqrt((1+x))lt0`
`impliesx lt sqrt(1+x)` [always true]
` x epsilon [-1,0)`
Combining both cases we get
`x epsilon [-1,(-1=sqrt(5))/2)`