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
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