First, multiply each side of the inequality by #color(red)(2)# to eliminate the fraction and keep the inequality balanced:

#color(red)(2) xx (1/2x - 9) < color(red)(2) xx 2x#

#(color(red)(2) xx 1/2x) - (color(red)(2) xx 9) < 4x#

#x - 18 < 4x#

Next, we subtract #color(red)(x)# from each side of the equation to isolate the #x# terms on one side of the inequality and the constants on the other side of the inequality while keeping the inequality balanced:

#x - color(red)(x) - 18 < 4x - color(red)(x)#

#0 - 18 < (4 - 1)x#

#-18 < 3x#

Now we can divide each side of the inequality by #color(red)(3)# to solve for #x# and keep the inequality balanced:

#-18/color(red)(3) < (3x)/color(red)(3)#

#-6 < (color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3))#

#-6 < x#

Then, we can reverse or "flip" the inequality to solve in terms of #x#:

#x > -6#