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Algebraic Inequalities: New Vistas

Titu Andreescu and Mark Saul
Publication Date: 
Number of Pages: 
MSRI Mathematical Circles Library
Problem Book
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Tom Schulte
, on

This slim book moves economically, effectively, and elegantly from basic arithmetic to sophisticated inequality topics. The reader of this self-contained text eventually arrives at the Cauchy-Schwarz inequality and Chebyshev's sum inequality. Part of what makes the presentation effective, despite requiring no more than high school algebra, are the detailed solutions for the problem sets. Typically, more than one approach to a solution is detailed. The solutions can be several pages longer than the chapter and exercises. The problems are even restated with the solutions, which is a generous and seldom-seen convenience.

In a first-year college algebra text, concepts typically evolve over a textbook chapter from introduction through expressions, use in functions, equations, and finally inequalities. In this way, consideration of inequalities is spread out and considered piecemeal. This focused presentation, containing enough material for about four weeks of a 15-week course, builds on algebraic considerations to study the arithmetic, geometric, and harmonic means. Students at this level can benefit from these topics while rarely being challenged with them. In my experience, a case in point is the uniform motion problem. Generally, this is treated as a linear equation around the hoary \(d=rt\) formulation for objects moving at a constant or average rate of speed. For several years now, I have augmented every textbook I have taught from around this section with about an hour of lecture and examples. The three pages here developing an application of the harmonic mean give the most cogent and lucid approach to word problems of this type for this level of student that I have encountered or developed. I will be trying it in the classroom this upcoming semester.

The chapter on the Cauchy-Schwarz inequality includes an introduction to vectors and concludes the journey that began with, for example, Problem 1.3: For \(a\geq0\), prove that \[a+1\geq2\sqrt{a}\] and brings even the independent reader to Problem 9.12: For positive \(a, b, c\) such that \(a+b+c= 1\), show that \[\sqrt{9a+1}+\sqrt{9b+1}+\sqrt{9c+1}\leq 6.\]

This compact and concentrated work will be a useful service to students and teachers in fundamental mathematics.


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Tom Schulte looks forward to teaching intermediate algebra to students at Oakland Community College in Michigan this summer, as he has for several years.

See the table of contents in the publisher's webpage.