y=4|2-5x|+4.
(1) When 2-5x≥0, that is 2≥5x, ⅖≥x, same as x≤⅖, y=4(2-5x)+4=8-20x+4=12-20x:
y=12-20x.
(2) When 2-5x<0, that is, 2<5x, ⅖<x, same as x>⅖, y=4(5x-2)+4=20x-8+4=20x-4:
y=20x-4.
When y<8 in (1), 12-20x<8, 4<20x, ⅕<x, so x>⅕, but this is only true if x≤⅖, therefore ⅕<x≤⅖.
When y<8 in (2), 20x-4<8, 20x<12, 5x<3, x<⅗, but this is only true if x>⅖. Therefore ⅖<x<⅗.
If we combine ⅕<x≤⅖ and ⅖<x<⅗ we get ⅕<x<⅗, so this is the required interval (⅕,⅗) and represents the required set of numbers (all numbers exclusively between ⅕ and ⅗): {x∈ℝ| x∈(⅕,⅗)}.
CHECK
Let x=⅖, then y(⅖)=4|2-5(⅖)|+4=4, which is less than 8, so y<8✔️
Let x=⅖+δ where δ is a very small positive value, then:
y(⅖+δ)=4|2-2-5δ|+4=4|-5δ|+4=20δ+4 which is less than 8 so y<8✔️
Let x=⅕+δ, then y(⅕+δ)=4|2-1-5δ|+4=4(1-5δ)+4=8-20δ which is less than 8 so y<8✔️
Let x=⅗-δ, then y(⅗-δ)=4|2-3+5δ|+4=4|-1+5δ|+4=4(1-5δ)+4=8-20δ<8, so y<8✔️