Conditions on inequalities $a>b$ and $b<c$ to deduce $a<c.$

Maybe transcribing in everyday life situation would help.

If a child $ b $ is smaller than his mother $ a $ (i.e $a>b$) and his father $ c $ (i.e $b<c$)

Then how can you deduce the mother is smaller than the father ($a<b$) ?

There is not enough information to know. What could work would be to know "how much smaller is the child compared to his parents", i.e. knowing the value of the fractions $frac ba$ and $frac bc$ we can calculate $frac ac$ and compare it to $1$.

This may be what you are looking for.

$$begin{cases} a>b \ b<c \ a=min left{a,cright} \ a≠c end{cases} Longrightarrow a<c$$

You can not say anything about $a$ and $c$. Try the following examples:

$$b=1,a=3,c=2Rightarrow a>c$$

$$b=1,a=2,c=3Rightarrow a<c$$

$$b=1,a=2,c=2Rightarrow a=c$$