The answer is the second option.
First of all, let's rearrange the inequality by dividing both sides by 4. Since we are dividing by a positive number, we preserve the inequality sign. So, we have
[tex] |x+2| \geq 4 [/tex]
Now, let's focus on what absolute value does in general. The absolute value function [tex] f(x)=|x| [/tex] takes a number as input, and returns the positive version of that input. In other words, if fed with a positive input [tex] x [/tex], the function will return [tex] x [/tex] itself, since it was already positive. On the contrary, a negative output [tex] x [/tex] yields the output [tex] -x [/tex], to make sure that the output is positive.
So, if you want the absolute value of a number to be greater than or equal to 4, you can either take a positive number greater than or equal to 4, or a negative number smaller than or equal to -4. Let me show a couple of example.
If you choose 17, then you have [tex] |17|=17\geq4 [/tex], which proves that a positive number greater than 4 is a good choice.
Similarly, if you choose -6, then you have [tex] |-6|=6\geq4 [/tex], which proves that a negative number smaller than -4 is also a good choice.
Now let's return to our equation: we want
[tex] |x+2| \geq 4 [/tex]
and we just proved that we need the quantity inside the absolute value to be larger than or equal to 4 of less than or equal to -4. In formula, this becomes
[tex] x+2 \geq 4 \implies x \geq 2 [/tex]
or
[tex] x+2 \leq -4 \implies x \leq -6 [/tex]
