The solution set of the inequation `|x|-1 lt 1-x,` is

Options

A. `(-1, 1)`
B. `(0, oo)`
C. `(-1, oo)`
D. none of these

Answer

D

Explanation

Let us consider the following cases:
CASE I When `x ge 0`
In this case, we have
`|x| = x`
`therefore ||x|-1|lt |1-x|`
`rArr |x-1| lt |x-1|`, which is not true for any `x ge 0`
So, the inequation is not valid for any `x ge 0`
CASE II When `x lt 0`:
In this case, we have
`|x| = -x`
`therefore ||x|-1|lt |1-x|`
`|-x-1|lt |1-x| " " [therefore ||x-1 lt |1-x|]`
`rArr |x+1| lt |x-1|`
If `x lt -1, "then "|x+1| = -(x+1) " and " |x-1| = - (x-1)`
`therefore |x+1| lt |x-1|`
`rArr -(x+1) lt -(x-1)`
`rArr -1 lt 1`, which is always true.
Thus, `(-oo, -1)` is the solution set.
`"If"-1 lt x lt 0, " then " |x+1| = x + 1, |x-1| = -(x-1)`
`therefore |x+1| lt |x-1|`
`rArr x + 1 lt -(x-1) rArr 2x lt 0 rArr x lt 0`
But, `-1 lt x lt 0`. Therefore `x in (-1, 0).`
Hence, `(-oo, 0)` is the solution set.
ALITER Draw graphs of `y=||x|-1| " and " y = |x-1|` to get the solution.