# Inequality and absolute value: $p + |k| \gt |p| + k$

Here is the problem I am confused about. The given relation is:

$p + |k| \gt |p| + k$

It is not mentioned whether $p$ and $k$ are integers.

I need to determine whether $p$ and $k$ are equal or one is greater than other. If the second is true which one is greater.

To solve this inequality problem I was trying to consider the values of $p$ and $k$ with all combinations of signs i.e. $p$ positive and $k$ negative; $p$ negative and $k$ positive; and $p$ and $k$ both negative and positive.

When I assume both $p$ and $k$ are negative, I could not find the solution. The solution is in the book too. But I failed to understand the solution. Can anyone please help me to solve the problem? Thanks.

• Split the problem into 4 parts. $p <0, k<0$, $p \ge 0, k<0$, etc, and look for solutions in each quadrant. Then piece them together. – copper.hat Oct 10 '13 at 19:28
• Rewrite the inequality as $p-|p|\gt k-|k|$. Let $p$ and $k$ be negative. Then $|p|=-p$, $|k|=-k$. So our inequality says $2p\gt 2k$, and therefore $p\gt k$. – André Nicolas Oct 10 '13 at 19:31
• Thanks a lot copper.hat and Andre Nicolas for your nice explanation. It is really helpful for me. – Silvia Oct 11 '13 at 5:30

Let $\phi(x) = |x|-x$, and note that the problem is equivalent to determining $p,k$ such that $\phi(k) > \phi(p)$.
Now plot $\phi$ to get
So, from this we can see that $\phi(k)> \phi(p)$ iff $k<p$ and $k <0$.