#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#.
