STRATEGY: Upon reading any GRE Multiple Choice question, we should always ask, Can I use the answer choices to my advantage?
In this case, we can easily test x-values that are included in the shaded interval.
Now let's give ourselves up to 20 seconds to identify a faster approach.
Another approach is to solve each answer choice for x until we find one whose solution matches the given shaded interval.
I'm pretty sure testing values is going to be a lot faster and much easier

When we check the number line, we can see that x = -8 is included in the shaded interval, which means x = -8 must also be a solution to the correct answer choice.

Now plug x = -8 into each answer choice to get:
A) |-8| ≤ 4. This simplifies to 8 ≤ 4, which is not true. ELIMINATE.
B) |-8| ≤ 8. This simplifies to 8 ≤ 8, which is true. KEEP.
C) |(-8) - 2| ≤ 4. This simplifies to 10 ≤ 4, which is not true. ELIMINATE.
D) |(-8) - 2| ≤ 6. This simplifies to 10 ≤ 6, which is not true. ELIMINATE.
E) |(-8) + 2| ≤ 6. This simplifies to 6 ≤ 6, which is true. KEEP.
We're already down to just two answer choices!

Let's test another extreme x-value...
We can see that x = 4 is also included in the shaded interval, which means x = 4 must also be a solution to the correct answer choice.
Plug x = 4 into the two remaining choices to get:
B) |4| ≤ 8. This simplifies to 4 ≤ 8, which is true. KEEP.
E) |4 + 2| ≤ 6. This simplifies to 6 ≤ 6, which is true. KEEP.
That was no help!

When we examine the two remaining answer choices (B and E), we can see that choice B tells us x CAN equal 8 (since x = 8 satisfies the inequality |x| ≤ 8) , whereas choice E tells us x CAN'T equal 8 (since x = 8 does not satisfy the inequality |x + 2| ≤ 6).

When we check the number line, we can see that x = 8 is NOT included in the shaded region of the number line.
In other words x = 8 cannot be a solution, which means we can eliminate choice B, because it tells us that x CAN equal 8.

By the process of elimination, the correct answer is E.