I solved it on a number line.

                a               b                c
—-A——–Y——–Z——–B—

Question- a< b+c?

sol- 1)  a+b< c

a< c-b

a< c-b < c < c+b (suff)

2)  a < c

hence a < c+b (suff)

D is the answer.

Remember : Modulus(Absolute value) is nothing but distance between two points.

Best way to solve this question is drawing the number line,

Shashank has given the best explanation.

I will just use different words to add little more clarity on that.

Rephrasing the question:

We are given 4 distinct points on the number line, A, Y, Z, B, from left to right (increasing order).

Question is: is the distance between A and Y smaller than the distance between Y and B?

Statement I is sufficient:

A—Y–Z——B would visualize a typical situation, distance between A and Z less than the distance between Z and B.

Since Y is between A and Z, the answer to the question “is the distance between A and Y smaller than the distance between Y and B” is definitely YES.

So, sufficient.

Statement II is sufficient:

A—Y-Z—-B this would be like, distance between A and Y, less than the distance between Z and B.

Now Z is between Y and B, so again, the distance between A and Y is smaller than the distance between Y and B.

Sufficient.

So the answer is D.

Hope this helps.