1380.31 – Find the falsehood

We are given that $h < m < k < 0$ and four statements. We could simply pick negative values for $h$, $m$, and $k$ and check the four statements. But, if we like, we can use fancy reasoning. Have a look:

1. $mk < hk$
   
   Let’s cancel $k$, which we are allowed to do because $k$ is not zero. But $k$ is negative, so the cancellation switches the direction of the inequality. We get $m > h$. This we were given. This one is true.

2. $mh < km$
   
   Again let’s cancel the common factor ($m$) and switch the direction of the inequality. We get $h > k$. Whoa Nellie! We were given $h < k$. This must be the false one.

3. $h + k < m + k$
   
   Here we may subtract the common addend ($k$). This does not require we switch the direction of the inequality. We get $h < m$ which we were given. This one is true.

4. $0 < k - h$
   
   Add $h$ to both sides and we get $h < k$, which we were given. This one is true.