You can try

Reduce[{-(1/4) < x <= 0 || 0 < x <= 1/2 || x > 1/2}, x]

x > -(1/4)

Reduce[{x <= -1 || -1 < x < 0}, x]

x < 0

You can try

Reduce[{-(1/4) < x <= 0 || 0 < x <= 1/2 || x > 1/2}, x]

x > -(1/4)

Reduce[{x <= -1 || -1 < x < 0}, x]

x < 0

set1 = Reduce[{-(1/4) < x <= 0 || 0 < x <= 1/2 || x > 1/2}]
set2 = Reduce[{x <= -1 || -1 < x < 0}]
Reduce[set1 && set2, x]

-(1/4) < x < 0