Question About A Problem Involving Inequalities

Question

I'm currently stuck on this inequality problem:

So far,  I tried to prove it by using a proof by contrapositive, but that doesn't seem to work. Could I get a hint on how to proceed?

Macavity · Accepted Answer

Not really just a hint, however you can stop reading after the line below and consider completing yourself.
$$frac{bc-ad}{bd} = frac{c}d-frac{a}b = left(frac{c}d-frac{p}q right) + left(frac{p}q-frac{a}b right)geqslant frac1{dq}+frac1{bq}$$
as each term in the brackets is positive.

BTW, the "lowest" such fraction is called the mediant $= dfrac{a+c}{b+d}$.

Question About A Problem Involving Inequalities

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