Find the set of all solutions of the equation `2^|y| -|2^(y -1) -1| = 2^(y -1) +1 `

Answer

`y in{-1}uu[1,oo)`

Explanation

Given, ` 2^(|y|)-|2^(y-1)-1|=2^(y-1)+1`
Case I When `y in (-oo, 0]`
`therefore 2^(-y)+(2^(y-1)-1)2^(y-1)+1`
`implies 2^(-y)=2`
`impliesy=-1 in (-oo,0]" "...(i)`
Case II When `y in (0,1]`
`therefore2^(y)+(2^(y-1)-1)=2^(y-1)+1`
`implies 2^(y)=2`
`impliesy=1 in (0,1] " "...(ii)`
Case III When `y in (1, oo)`
`therefore2^(y)-2^(y-1)+1=2^(y-1)+1`
`implies 2^(y)-2.2^(y-1)=0`
implies 2^(y)-2^(y)=0 "true for all" y gt 1 " "...(iii)`
From Eqs. (i), (ii), (iii), we get
`y in{-1}uu[1,oo).`