First, multiply both sides of the equation by color(red)(12) (the lowest common denominator of all the fractions) to eliminate the fractions while keeping the equation balanced:
color(red)(12)(1/2x - 5/3) = color(red)(12)(-1/2 x + 19/4)
(color(red)(12) xx 1/2x) - (color(red)(12) xx 5/3) = (color(red)(12) xx -1/2x) + (color(red)(12) xx 19/4)
(6 xx 1x) - (4 xx 5) = (6 xx -1x) + (3 xx 19)
6x - 20 = -6x + 57
Next, add color(red)(6x) and color(blue)(20) to each side of the equation to isolate the x term on the left side of the equation:
6x - 20 + color(red)(6x) + color(blue)(20) = -6x + 57 + color(red)(6x) + color(blue)(20)
6x + color(red)(6x) - 20 + color(blue)(20) = -6x + color(red)(6x) + 57 + color(blue)(20)
12x - 0 = 0 + 77
12x = 77
Now, divide each side by color(red)(12) to solve for x while keeping the equation balanced:
(12x)/color(red)(12) = 77/color(red)(12)
(color(red)(cancel(color(black)(12)))x)/cancel(color(red)(12)) = 77/12
x = 77/12
