## 1380.31 – Find the falsehood

Suppose $h$, $m$ and $k$ are integers and $h < m < k < 0$. Which of these statements is false?

- $mk < hk$,
- $mh < km$,
- $h + k < m + k$,
- $0 < k - h$.

## Solution

Solution. 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:

(a) $mk < hk$,

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.

(b) $mh < km$,

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.

(c) $h + k < m + k$,

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.

(d) $0 < k - h$.

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