Let's approach this by solving the inequality (as opposed to ruling out answers that were given).
To solve an absolute value inequality, you first need the abs. val. by itself. That is already done in this exercise.
The next step depends if the abs. val. is greater than or less than a positive number.
If k is a positive number and if you have the |x| > k, then this splits into
x > k or x < -k
If k is a positive number and if you have the |x| < k, then this becomes
-k < x < k
Essentially -k and k become the ends or the intervals and you have to decide if you have the numbers between k and -k (the inside) or the numbers outside -k and k.
In your exercise, you have | 10 + 4x | ≤ 14. So this splits apart into
-14 ≤ 10+4x ≤ 14
because it's < and not >. The < vs ≤ only changes if the end number will be a solid or open circle.
Solving -14 ≤ 10+4x ≤ 14 would then go like this:
-14 ≤ 10+4x ≤ 14
-24 ≤ 4x ≤ 4 by subtracting 10
-6 ≤ x ≤ 1 by dividing by 4
So that's the inequality and the graph will be the one with closed (solid) circles at -6 and 1 and shading in the middle.