Does cross multiply always work for inequalities if both denominators are both positive or negative

Question

Is it true that if$\frac{A}{B}<\frac{C}{D}$is true then $A*D<C*B$ must be true if B and D are both positive or negative?

Answer 1

If $B,D$ are nonzero of same sign, then $BD>0,$ so multiplying by it keeps the inequality direction the same. So I'd say yes, true.

Answer 2

Our basic axiom is: For $a > 0$; and $m < n$ then $am < an$ and when we cross multiply we multiply by denominators.

If $B, D$ are both positive then $BD$ is positive. If $B, D$ are both negative then $BD$ is positive.

Either way:

$\frac AB > \frac CD \implies$

$\frac AB (BD) > \frac CD (BD) \implies$

$AD > CB$ so ... yes, it is true.

If $B$ and $D$ are opposite signs then $BD < 0$ So $BD - BD < 0 - BD$ and $0 < -BD$ so

(Our other basic axiom is: For all $a$ if $m < n$ then $m +a < n+a$.)

$\frac AB(-BD) > \frac CD(-BD) \implies$

$-AD > -CB \implies$

$-AD + (AD + CB) > -CB + (AD+CB) \implies$

$CB > AD$ so the exact opposite is true if the denominators are different signs.

......

P.S. Because of this it is common practice when writing a rational number as a fraction $\frac {a}{b}$ is often assumed that $a$ is an integer; negative, positive or zero; and $b$ is a natural number always positive. Not all texts do this but if $r = \frac {a}{b}$ where $a$ and $b$ are integers, then 1) we know $b \ne 0$. And 2) if $b < 0$ we could just replace $b$ with $|b|=-b$ and replace $a$ with $-a$ and get the same results. Assuming that $b > 0$ just makes things more consistent and easier.