1. We need to solve

and count the number of real solutions.

2. Since absolute values are present, split the real line according to the sign changes of $x$ and $x+4$:

• $x\ge 0$
• $-4\le x<0$
• $x<-4$

Case 1: $x\ge 0$

Then $|x|=x,\qquad |x+4|=x+4$ So the equation becomes

Expanding,

Thus,

Now in this case $x\ge 0$, so only

is valid. So this case gives 1 solution.

Case 2: $-4\le x<0$

Then $|x|=-x,\qquad |x+4|=x+4$ So the equation becomes

Expand:

Discriminant:

So there is no real solution in this case.

Case 3: $x<-4$

Then

So the equation becomes

Expand:

Thus,

But in this case $x<-4$, so only

is valid. So this case gives 1 solution.

3. Total number of real solutions:

Therefore, the number of elements in the set is

4. Comparing with the stored correct answer:

• Stored correct answer: B
• Our derived answer: 2, which corresponds to B

So the stored answer is correct.