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Solve |x-1|-2|=|x-3|....

Solve `|x-1|-2|=|x-3|`.

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Verified by Experts

The correct Answer is:
`[1,oo)`
`||x-1||-2|=|x-3|`
Case I : `x ge 3`
`:. |x-1-2|=x-3 implies x-3=x-3`
`:.x ge 3`
Case II : If `1 le x lt3`
`:. |x-1-2|=-(x-3)`
`:. -(x-3)=-(x-3)implies1le x lt3`
Case III : `-1 le x lt 1`
`:, |-x+1-2|=-(x-3)`
`:. |-x-1|=-(x-3) implies |x+1|=-x+3`
`x+1=-x+3 implies 2x=2impliesx=1`
Which is not possible
Case IV : IF `x lt -1`
`:. |-x+1-2|=-(x-3)`
`implies |x+1|=-(x-3)implies-(x+1)=-(x-3)`
`implies -x-1=-x+3`
`implies` No solution
Hence `x in [1,oo)`
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