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