Warning: This is a long solution, but there are some key properties/strategies that some students may find useful

Useful properties:
#1: If x is POSITIVE, then |x| = x
#2: If x is NEGATIVE, then |x| = -x

For example, if x = 3, then |x| = |3| = 3 = x
Conversely, if x = -3, then |x| = |-3| = 3 = -(-3) = -x

Okay, first notice that p and k cannot be equal.
IF it were the case that p = k, then we can replace p with k to get: \(k + |k| > |k| + k\)
This makes no sense. So, we can be sure that p and k are not equal

Now let's examine 4 possible cases:

case i: p is POSITIVE and k is POSITIVE
Applying property #1, we get: \(p + k > p + k\)
This makes no sense. \(p + k = p + k\)
So, case i is impossible.

case ii: p is POSITIVE and k is NEGATIVE
Applying properties #1 and 2, we get: \(p + (-k) > p + k\)
Subtract p from both sides of the inequality to get: \(-k > k\)
Add k to both sides to get: \(0 > 2k\)
Since k is NEGATIVE in this case, the inequality \(0 > 2k\) is true.
So, case ii is possible

case iii: p is NEGATIVE and k is POSITIVE
Applying properties #1 and 2, we get: \(p + k > (-p) + k\)
Subtract k from both sides of the inequality to get: \(p > -p\)
Add p to both sides to get: \(2p > 0\)
Since p is NEGATIVE in this case, the inequality \(2p > 0\) is NOT true.
So, case iii is impossible.

case iv: p is NEGATIVE and k is NEGATIVE
Applying property #2, we get: \(p + (-k) > (-p) + k\)
Add p to both sides to get: \(2p - k > k\)
Add k to both sides to get: \(2p > 2k\)
Divide both sides by 2 to get: \(p > k\)
This tells us that, if p is NEGATIVE and k is NEGATIVE, then \(p > k\)
case iv is possible

At this point, we can see that there are only two possible cases, and for each case, we can be certain that \(p > k\)

Answer: A

Cheers,
Brent