Q.1the set of all solutions of the inequality (1/2)^((x^2)-2x)< (1/4) contains the set (3,infinite). But how?

Answer 1

1/2)^z<(1/2)^2
take the log (base 2 ) of each side

-z<-2
z>2
solve for x (z=x^2-2x)
x^2-2x-2>0

x=(2+-sqrt(4+8))/2=1+-sqrt3
so test these two roots, and you see that only 1+sqrt3 will satisfy the inequality.
x>2.73

Answer 2

Great solution,seen 14 years after this answer has been posted here! Hope you'll be doin well