exponential equation

Question

81^x - 4*9^x+1/2+27>0

Answer 1

You may rewrite your inequality as (9^x)^2-12(9^x)+27>0 and by setting y=9^x you obtain the quadratic inequality y^2-12y+27>0. This has two roots, 3 and 9. Since you want the quadratic to be positive we must have either y>9 or y<3. But since y=9^x we must have either 9^x>9 (which is the same as x>1) or 9^x=3^(2x)<3 (which is the same as 2x<1 and hence x<1/2).