The algebra book must be no more than 400-500 pages in length and should contain end-of-lesson/chapter exercises.
(and even more if the book has it...)
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I teach a college algebra class at a state university in New York and have found that my students have great success with this book. http://www.amazon.com/Intermediate-Algebra-College-Students-Developmental/dp/0321620917
I'm sure you can find a cheap used copy, perhaps an older edition. I have been using this book since the fourth edition, it's basically unchanged.
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I am taking a Pre-Calc Coursera course and they use http://stitz-zeager.com/szprecalculus07042013.pdf, which I like thus far and I think it meets your criteria, except for length. I think Exercises and the included answers to those exercises preclude a book in the 500 page range.
Gelfand's Algebra book is fantastic. It is written for the student to enjoy.
Lang's Basic Mathematics also sounds like a good choice. It's between 400 and 500 pages, but contains more than you asked for.
The Art of Problem Solving algebra and precalculus books are very good, if you're looking for books with a contest prep focus (they're still great, if not).
Alternatively, you could work through Khan Academy's knowledge map.