Compatibility in Real numbers?

Question

I'm reading Good Math by Mark Chu-Carroll and it says,

• "$\le$" is compatiable with "+" and "*":

  1. ...
  2. if x $\le$ y, then for all z where 0 $\le$ z, (x * z) $\le$ (y * z).
  3. if x $\le$ y, then for all z where z $\le$ 0, (x * z) $\le$ (y * z).

Is that 3. true?

When

x = -2, y = -1, and z = -1

Then

(-2 * -1) $\le$ (-1 * -1) ?

Answer 1

No, it is not. Either there is a typo in the book, or there is a typo in your transcription.

For example, $-2 \leq 3$, and $-5 \leq 0$, but $$-5 \times -2 = 10 \not\leq -15 = -5 \times 3.$$

Answer 2

No, it is reversed. It should read

3. if x $\le$ y, then for all z where z $\le$ 0, (y * z) $\le$ (x * z). Alternately, you can reverse the $\le$ to $\ge$ and keep $x$ and $y$ where they are.