How do you solve #|\frac { 6} { 3z + 6} | = 8#?

#|6/(3z+6)|=8#

#|2/(z+2)|=8#

#2/(z+2)=+8 or 2/(z+2)=-8#

#z+2=2/8 or z+2=-2/8#

#z=2/8-2 or z=-2/8-2#

#z=2/8-16/8 or z=-2/8-16/8#

#z=-14/8 or z=-18/8#

#z=-7/4 or z=9/4#

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

Solution 1:

#6/(3z + 6) = -8#

#6/(3z + 6) = -8/1#

#(3z + 6)/6 = -1/8#

#color(red)(6) xx (3z + 6)/6 = color(red)(6) xx -1/8#

#cancel(color(red)(6)) xx (3z + 6)/color(red)(cancel(color(black)(6))) = -color(red)(6)/8#

#3z + 6 = -3/4#

#3z + 6 - color(red)(6) = -3/4 - color(red)(6)#

#3z + 0 = -3/4 - (4/4 xx color(red)(6))#

#3z = -3/4 - 24/4#

#3z = -27/4#

#(3z) xx 1/color(red)(3) = -27/4 xx 1/color(red)(3)#

#(color(red)(cancel(color(black)(3)))z) xx 1/cancel(color(red)(3)) = -(color(red)(cancel(color(black)(27)))9)/4 xx 1/cancel(color(red)(3))#

#z = -9/4#

Solution 2:

#6/(3z + 6) = 8#

#6/(3z + 6) = 8/1#

#(3z + 6)/6 = 1/8#

#color(red)(6) xx (3z + 6)/6 = color(red)(6) xx 1/8#

#cancel(color(red)(6)) xx (3z + 6)/color(red)(cancel(color(black)(6))) = color(red)(6)/8#

#3z + 6 = 3/4#

#3z + 6 - color(red)(6) = 3/4 - color(red)(6)#

#3z + 0 = 3/4 - (4/4 xx color(red)(6))#

#3z = 3/4 - 24/4#

#3z = 21/4#

#(3z) xx 1/color(red)(3) = -21/4 xx 1/color(red)(3)#

#(color(red)(cancel(color(black)(3)))z) xx 1/cancel(color(red)(3)) = -(color(red)(cancel(color(black)(21)))7)/4 xx 1/cancel(color(red)(3))#

#z = -7/4#

The Solution Is:

#z = {-9/4, -7/4}#