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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) ?

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    $\begingroup$ The sign is reversed. Example, $3\leq 5$ but $(3\cdot (-1))\leq (5\cdot (-1))$ is false. Rather, $(3\cdot (-1))\color{red}{\geq}(5\cdot (-1))$. Multiplication by a negative "flips" the inequality sign. $\endgroup$ – JMoravitz Oct 22 at 14:46
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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 $$-5 \times -2 = 10 \not\leq -15 = -5 \times 3.$$

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No, it is reversed. It should read

  1. 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.
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