The absolute value function takes any negative or positive term and transforms it to its positive form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent. However, because #-0# equals #0# we can just solve the term within the absolute value function once for #0#:
#1/2(1 - 2/3x) = 0#
First, multiply each side of the equation by #color(red)(2)# to eliminate the need for parenthesis while keeping the equation balanced:
#color(red)(2) xx 1/2(1 - 2/3x) = color(red)(2) xx 0#
#cancel(color(red)(2)) xx 1/color(red)(cancel(color(black)(2)))(1 - 2/3x) = 0#
#1 - 2/3x = 0#
Next, subtract #color(red)(1)# from each side of the equation to isolate the #x# term while keeping the equation balanced:
#-color(red)(1) + 1 - 2/3x = -color(red)(1) + 0#
#0 - 2/3x = -1#
#(-2)/3x = -1#
Now, multiply each side of the equation by #color(red)(3)/color(blue)(-2)# to solve for #x# while keeping the equation balanced:
#color(red)(3)/color(blue)(-2) xx (-2)/3x = color(red)(3)/color(blue)(-2) xx -1#
#cancel(color(red)(3))/cancel(color(blue)(-2)) xx color(blue)(cancel(color(black)(-2)))/color(red)(cancel(color(black)(3)))x = -3/2#
#x = 3/2#
