Positive Real Number Inequalities can be Multiplied/Disproof for Negative Parameters

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Theorem

Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$.

Let $b > 0$ and $d > 0$.


Then $a c > b d$.


If $b < 0$ or $d < 0$ the inequality does not hold.


Proof

Proof by Counterexample:

Let $a = c = -1, b = d = -2$.

Then $a c = 1$ but $b d = 2$.

$\blacksquare$


Sources