Question about solving string of inequalities

Question

Suppose I have a string of inequalities, i.e. $$ f(x,y,z)<g(x,y,z)<h(x,y,z) $$ Clearly, if I find when $g(x,y,z)<h(x,y,z)$ and $f<g$, then $f<h$ when both these conditions hold.

Is it ever useful to compare $f(x,y,z)<h(x,y,z)$ in addition to $f<g$ and $g<h$?

Or is it useful to compare the conditions (assuming I can get them in a comparable form) from $g<h$ and $f<h$ to each other? Maybe comparing the two can give a single, simplified condition for $f<g<h$ to hold?

More generally, I get confused when I have systems of many $\geq 3$ inequalities about when i can substitute the results from comparing two of the inequalities into another inequality (or the results from comparing 2 other inequalities)

Answer 1

Note that

$$f(x,y,z)<g(x,y,z)<h(x,y,z)$$

is equivalent to the following system

• $f(x,y,z)<g(x,y,z)$
• $g(x,y,z)<h(x,y,z)$

and the condition $f(x,y,z)<h(x,y,z)$ is implicitely assumed by the system.