Solutions for all numbers x ∈ R for the following equation?

We have two modulus monomials added to be equal #16.#

It means that for every single monomial we will have two options :
when the expression inside is positive and when it is negative.

It means that overall we will have four different cases:

1. When #x+3>0 and 5+4x >0#
   so in this case, x has to be :# x> -3 and x> -5/4#

What this means is that x should be x > -5/4

when you solve the equation for this these conditions, you get
#x+3 + 5 + 4x = 16# where x=5/8 , which agrees with your condition that x has to be greater than #-5/4.#

You do in all cases the same process.

1. (The second case ) you have #x+3>0 and 5+4x <0#

#x> -3 and x <-5/4,# so x should be between -3 and -5/4

#-3 < x < -5/4#

when you solve #x+3 -(5+4x) = 16# you get that x = -6

#-6# is not between #-3 and -5/4# , so in the second case there is no solution

Two more cases you do on the same way.
You will get:
#3. x+3<0 and 5+4x <0#
#x=-24/5 #

#4. x+3<0 and 5+4x >0#
no solution

So the possible solutions are only #x=5/8 and x = -24/5#

This can be done using graphical method also, but I prefer this one.