Inequality with two absolute value

Question

How can you tackle an inequality problem that has two absolute values?

Example is the following

$p + |k| > |p| + k$

and the questions is a quantitative comparison between

A) $p $

B) $k$

The final answer is $p > k$ in all cases.

however when I apply negative for both

$$\begin{align}-p+k&>p-k\\ 2k&>2p\\ k&>p\end{align}$$

Answer 1

Equivalently, $$|k| - k > |p| - p.$$ If $k,p \ge 0$ the inequality cannot hold, since both sides are equal to zero. Also $k \ge 0$ and $p<0$ is not a case, since it holds $0>-2p$. Thus, the only option for $k$ is to be less than zero. Let's check what it holds:

• $k<0 \;\&\;p \ge 0:$

$$-2k > 0, $$ thus in that case $\boxed{k<0\le p.}$

• $k<0 \, \& \, p < 0 :$

$$-2k > -2p\implies k<p,$$ thus in that case $\boxed{k<p<0.}$

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Notice that $|x| - x = \begin{cases} 0,& x \ge 0\\[2ex] -2x,& x < 0. \end{cases}$