0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – copper.hat Oct 10 '13 at 19:28
  • $\begingroup$ 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 $p\gt k$. $\endgroup$ – André Nicolas Oct 10 '13 at 19:31
  • $\begingroup$ Thanks a lot copper.hat and Andre Nicolas for your nice explanation. It is really helpful for me. $\endgroup$ – Silvia Oct 11 '13 at 5:30
1
$\begingroup$

Here is another approach:

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

enter image description here

So, from this we can see that $\phi(k)> \phi(p)$ iff $k<p$ and $k <0$.

$\endgroup$
  • $\begingroup$ Nice answer! Often not enough attention is paid to the geometry. $\endgroup$ – André Nicolas Oct 10 '13 at 20:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.