First, multiply both sides of the equation by #color(red)(24)# to eliminate the fractions while keeping the equation balanced #color(red)(24)# is the Lowest Common Denominator of the 4 fractions:

#color(red)(24)(2/3x + 5/6) = color(red)(24)(7/8x - 1/2)#

#(color(red)(24) * 2/3x) + (color(red)(24) * 5/6) = (color(red)(24) * 7/8x) - (color(red)(24) * 1/2)#

#(cancel(color(red)(24)) 8 * 2/color(red)(cancel(color(black)(3)))x) + (cancel(color(red)(24)) 4 * 5/color(red)(cancel(color(black)(6)))) = (cancel(color(red)(24)) 3 * 7/color(red)(cancel(color(black)(8)))x) - (cancel(color(red)(24)) 12 * 1/color(red)(cancel(color(black)(2))))#

#16x + 20 = 21x - 12#

Next, subtract #color(red)(16x)# and add #color(blue)(12)# to each side of the equation to isolate the #x# term while keeping the equation balanced:

#-color(red)(16x) + 16x + 20 + color(blue)(12) = -color(red)(16x) + 21x - 12 + color(blue)(12)#

#0 + 32 = (-color(red)(16) + 21)x - 0#

#32 = 5x#

Now, divide each side of the equation by #color(red)(5)# to solve for #x# while keeping the equation balanced:

#32/color(red)(5) = (5x)/color(red)(5)#

#32/5 = (color(red)(cancel(color(black)(5)))x)/cancel(color(red)(5))#

#32/5 = x#

#x = 32/5#