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.

Solution 1)

#4/5x + 5/6 = -1/3#

#4/5x + 5/6 - color(red)(5/6) = -1/3 - color(red)(5/6)#

#4/5x + 0 = (2/2 xx -1/3) - color(red)(5/6)#

#4/5x = -2/6 - color(red)(5/6)#

#4/5x = -7/6#

#color(red)(5)/color(blue)(4) xx 4/5x = color(red)(5)/color(blue)(4) xx -7/6#

#cancel(color(red)(5))/cancel(color(blue)(4)) xx color(blue)(cancel(color(black)(4)))/color(red)(cancel(color(black)(5)))x = -35/24#

#x = -35/24#

Solution 2)

#4/5x + 5/6 = 1/3#

#4/5x + 5/6 - color(red)(5/6) = 1/3 - color(red)(5/6)#

#4/5x + 0 = (2/2 xx 1/3) - color(red)(5/6)#

#4/5x = 2/6 - color(red)(5/6)#

#4/5x = -3/6#

#color(red)(5)/color(blue)(4) xx 4/5x = color(red)(5)/color(blue)(4) xx -1/2#

#cancel(color(red)(5))/cancel(color(blue)(4)) xx color(blue)(cancel(color(black)(4)))/color(red)(cancel(color(black)(5)))x = -5/8#

#x = -5/8#

The solutions are: #x = -35/24# and #x = -5/8#