### Combined Inequalities

The text clearly states that

*a* < *b* < *c* means *a* < *b* and *b* < *c*
It is true that we often use the combined inequality form, namely,
*a* < *b* < *c* in our work, and that form is convenient, brief, and
it expresses the notion of *b* being between *a* and *c*.
However, the combined form needs to be used with some amount of
understanding.
In one sense the combined form is actually an incorrect statement.
Consider the parallel construction **a + b + c**. The Associative
Property of Addition tells us that we can do this as
**(a + b) + c** or as **a + (b + c)**, and we will get the same result.
If we look at **(a + b) + c** we can see that adding **a** and **b** will give a new value.
Then we replace **(a + b)**
by that new value, leaving us with the task of adding that new value to **c**.
However, the same analysis for our combined inequality
causes a problem.
In particular, the
*a* < *b* portion is evaluated as either **TRUE** or **FALSE**.
If we replace *a* < *b* by either **TRUE** or **FALSE**
then the problem becomes either **TRUE < ***c* or **FALSE < ***c*
neither of which makes any sense.
Most computer languages do not allow programmers to use combined
inequaltities for exactly this reason.
Nonetheless, we will continue to use
*a* < *b* < *c*
recognizing that it is merely shorthand for
*a* < *b* and *b* < *c*.

An additional problem related to using combined inequalities is that they tempt us to
attempt to use a similar "shorthand" for the situation

*8* < *b* or *b* < *3*
The combined inequality __can not__ be used for this "**or**" condition.
We might be tempted to try to write
(this is incorrect) *8* < *b* < *3* (this is incorrect)
but that is not correct. In fact, it implies that
*8* < *3* which we recognize as wrong.
It is extremely important to recognize that a **combined inequality** is
merely a shorthand notation for the the longer expression, and that the
longer expression uses the "**and**" as a conjuction between the two expressions.

**PRECALCULUS: College Algebra and Trigonometry**

© 2000 Dennis Bila, James Egan, Roger Palay