If $ \frac{a}{b} < \frac{c}{d} $, then $ \frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d} $
Solution 1:
Simplest way is to transform, in this case. Your original inequality is equivalent to $ad<bc$.
The first inequality you try to prove is equivalent to $a(b+d)<b(a+c)$. This in turn is equivalent to $ab+ad<ab+bc$, which is true since $ad<bc$ by assumption.