The value of #|4x+8|# is defined as follows:

if #4x+8 >= 0# then #|4x+8| = 4x+8#

otherwise (that is, id #4x+8 < 0#), #|4x+8| = -(4x+8)#.

Let's consider a point #x=-2#, where term #4x+8# changes from negative (to the left of #x=-2#) to positive (to the right of #x=-2#).

From definition of absolute value mentioned above,

if #x < -2#, 4x+8 < 0 and, therefore, #|4x+8| = -(4x+8)# and our initial inequality looks like this:

#-(4x+8) >= 20# or

#-4x-8 >= 20# or

#-8-20 >= 4x# or

#4x <= -28# or

#x <= -7#

Since #x<=-7# lies inside the interval #x < -2# that we consider, all #x<=-7# are solutions.

From definition of absolute value mentioned above,

if #x >= -2#, 4x+8 >= 0 and, therefore, #|4x+8| = 4x+8# and our initial inequality looks like this:

#4x+8 >= 20# or

#4x >= 12# or

#x >= 3#

Since #x>=3# lies inside the interval #x >= -2# that we consider, all #x>=3# are solutions.

So, we have two separate intervals that represent the solutions to this inequality:

#x <= -7# and #x >= 3#.

Graphically, it looks like this:

graph{|4x+8| [-24, 24, -23.12, 23.12]}

It can be observed that this graph is above the line #y=20# when #x<=-7# or #x >= 3#.