In the below image, it has been said that if x≤-4, then both |3x-3| and |2x+8| will be negative.
If x=-4, how can |2x+8| be negative? If x=-4, |2x+8| becomes zero which is a positive vale. So how can they say that if x≤-4, then both |3x-3| and |2x+8| will be negative? They can say it only when x is less than -4.
Similarly, in the following image, we can see it has been said that:
when x≤-5, then |x+5|=-(x+5)
But when x=-5, how can |x+5| be equal to -(x+5)? It should be zero, a positive value, shouldn't it?
Links of the site from which these two screenshots have been collected are below:
Firstly, here is a graph to help visually. As you can see:
Here we can notice that these absolute values can be factored as $$ y=3|x-1|,2|x+4|$$ because the coefficients are positive.
The definition of the absolute value is: $$|f(x)|=-f(x),f(x)\le 0\ \mathrm{or} \ |f(x)|= +f(x), f(x)\ge 0,f(x)\in \Bbb R$$
Clearly, this means that the value is indeed negative, but in terms of simply having the negative sign in front to make the argument positive, like when $|-2|=-(-2)=2$.
In addition, the zero is neither negative nor positive hence the fact that $|0|=\pm 0=0$. This also means that the zero can be included in the negative or positive restriction of the function. Zero is not a positive nor negative value.
Finally, to restate your absolute value question, each absolute value function will be negated for x<-4, and 0 for x=-4 for y=2|x+4| where you can clearly see that the 4 and -4 will cancel after factoring, which does not necessarily mean that they will be negative, but rather in the 2nd quadrant where x<0, y>0. The roots are where each function is equal to 0.
So in conclusion, $sgn(0)=0, sgn(x<0)=-1,sgn(x>0)=1$ and a negative value in the absolute value will never output |x|<0 for $x\in\Bbb R$ because $$|x<0|=|-k,k\in \Bbb N|=-x=-(-k)=x=k$$