The way you typed it means:
(x/2)+3<2x-6 make all terms have a common denominator of 2...
(x+6)/2<(4x-12)/2 which is equal to
x+6<4x-12 subtract 6 from both sides...
x<4x-18 subtract 4x from both sides
-3x<-18 divide both sides by -3 (REVERSE signs when dividing/multiplying by a negative term!!)
x>6
