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

Mathematics Asked on January 5, 2022

If $a>b$ and $b<c$, and $a$ and $c$ are positive, under what conditions is $a<c$? I am just curious to know. I know that the following are true $frac ab>1$, $frac bc<1$, and $frac ac<1$. Any ideas on what next?

3 Answers

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 $1$.

Answered by zwim on January 5, 2022

This may be what you are looking for.

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

Answered by user548054 on January 5, 2022

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

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

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

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

Answered by QC_QAOA on January 5, 2022

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