Krage17 wrote:

I am neither of two you asked to help, but I am retaking the GMAT soon, so here is how I approached this problem:

Assuming that P and k and both sides have the absolute value, the only way we will have an inequality is when when one or more numbers have negative signs, and thus change their value when an absolute value is taken. We already know that P on the left side equals |P| on the right side because the left side of the inequality is larger (and therefore, there is no way P is negative). So, what truly shifts the equation here is the sign flip when we take an absolute value of k. In other words, k must be negative, otherwise |k| would be equal to k and P+|k| would be equal to |P|+k. Hence, P (which is positive) must be larger than k (which is negative).

I know it's somewhat conceptual and may not be easy to follow, but that's how I looked at it.

This approach is interesting, and i think the most efficient way to approach this problem. But your assumption here is wrong my friend-

Here's how P can be negative:

p = -1, k = -2

p + |k| > |p| + k

-1 + |-2| > |-1| + (-2)

1 > -1

This is true for any negative value of p when k<p.

So we cannot conclude that p is non-negative. However, what we can certainly conclude from the given inequality is-

k has to be negitive|Any number| \(\geq\) That numberNow, the given inequality:

p + |k| > |p| + k

The |k| on LHS can be greater than or equal to k on RHS

Case 1: |k| = k, which means that k is positive

in this case, to hold the ineq. true p > |p|. But this cant be true. Hence K is negative.

Case 2:|k| > k, which means that k is negative

in this case, to hold the ineq. true,

p on the LHS can be equal to (p is positive) or less than (p is negative) |p| on RHS. So we can safely conclude from the given ineq. that

p can be +ve or -ve, but

k will always be -ve
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