How do you solve #abs(5x + 5) - 7 = 18#?

Recall the definition of the absolute value of a real number:
If #N >= 0# then #|N|=N#
If #N < 0# then #|N|=-N#.

Since absolute value of a number is evaluated differently, depending on this number's sign, let's consider two different cases:

Case A is when the absolute value is taken of the positive number or zero (and in this case absolute value of a positive number or zero is this number itself);

Case B is when the absolute value is taken of the negative number (and in this case absolute value of a negative number is its negation).

Here is the solution in each of these cases.

Case A: #5x+5 >= 0#,
which is equivalent to
#5x >= -5# or
#x >= -1#
In this case #|5x+5|=5x+5# and our equation looks like
#5x+5-7=18# or
#5x = 20# or
#x = 4# (which satisfies the initial condition of #x >= -1# and, therefore, is the real solution.
Check: #|5*5+5|-7=25-7=18# OK

Case B: #5x+5 < 0#,
which is equivalent to
#5x < -5# or
#x < -1#
In this case #|5x+5|=-(5x+5)# and our equation looks like
#-(5x+5)-7=18# or
#-5x = 30# or
#x = -6# (which satisfies the initial condition of #x < -1# and, therefore, is the real solution.
Check: #|-6*5+5|-7=|-25|-7=25-7=18# OK