5x-10x^2<0hArr5x(1-2x)<0
Hence sign of x changes around x=0 and 1-2x=0 i.e. x=1/2 and these two points divide real number line in three parts.
Note that on real number line, below x=0, while x<0, 1-2x>0 and therefore 5x-10x^2<0
Between x=0 and x=1/2,
x>0 and 1-2x>0 and therefore 5x-10x^2>0
and beyond x=1/2, while x>0, 1-2x<0 and hence 5x-10x^2>0
Hence, solution is that either x<0 or x>1/2
In terms of sign chart this can be expressed as
color(white)(XXXXXXXXXXX)0color(white)(XXXXXXX)1/2
xcolor(white)(XXXXXX)-ive color(white)(XXXX)+ive color(white)(XXXX)+ive
(1-2x)color(white)(XXX)+ive color(white)(XXX)+ive color(white)(XXXX)-ive
5x-10x^2color(white)(xxX)-ive color(white)(XXX)+ive color(white)(XXXX)-ive
and as we need 5x-10x^2 to be negative i.e. less than 0, solution is either x<0 or x>1/2.
