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Introduction
One-thousand-one algebra problems: That’s a lot of algebra problems.
It will take you seven days to do all of them, if you do 143 each day. Whew! It will take you 91 days to do all of them, if you manage to do 11 each day. And, of course, it will take you 1,001 days to do all the problems if you do just one each day. Whatever your game plan, this is still a lot of problems. You may want to start at the beginning and do each problem in turn, or you may want to jump around and do the problems in an order that suits you best. Either plan is doable. Either plan is fine. Just watch out for topics that build on one another — you may need the information from one skill to succeed in another.
Practice makes perfect. Unlike other subjects where you can just read or listen and absorb the information sufficiently, mathematics takes practice. The only way to figure out how the different algebraic rules work and interact with one another is to get into the problems — get your hands dirty, so to speak. Many problems appear to be the same, on the surface, but different aspects and challenges have been inserted to make the different problems unique. The concepts become more set in your mind when you work with the problems and have the properties confirmed with your solutions.
Yes, one-thousand-one algebra problems are a lot of problems. But you may find that this just whets your appetite for more. Enjoy!
What You’ll Find
This book has 1,001 algebra problems divided up among 23 chapters. Each chapter has many different sets of questions. The sets of questions are sometimes in a logical, sequential order, going from one part of a topic to the next and then to the next. Other times the sets of questions represent the different ways a topic can be presented. In any case, you’re given instructions on doing the problems. And sometimes you’re given a particular formula or format to use.
Instead of just having answers to each of the problems, you find a worked-out solution for each and every one. Flip to the back of the book for the step-by-step process needed to solve the problems. The solutions include verbal explanations inserted in the work where necessary. Sometimes an alternate procedure may be offered. Not everyone does algebra exactly the same way, but this book tries to provide the most understandable and success-promoting process to use when solving the algebra problems presented.
How This Workbook Is Organized
This workbook is divided into two main parts: questions and answers. But you probably figured that out already.
Part 1: Questions
The questions chapters cover many different topics:
Basic operations: The first six chapters cover the types of numbers and the types of operations on those numbers that are essential to working in algebra. The natural numbers and whole numbers are fine for elementary arithmetic, but you need to broaden your horizons with signed numbers and decimals and fractions and exponential expressions. All these types of numbers are added, subtracted, multiplied, and divided. The rules for the different types of numbers have similarities and differences. The problems can help you come to grips with these situations and recognize what’s the same and what’s different.Also important in algebra are the operations involving radicals, absolute value, and factorial. And, tying together all the numbers and operations are the rules on how to deal with them: the order in which you perform the operations, and then the effect of grouping symbols on the whole process.
Algebraic expressions: An algebraic expression can consist of one or more terms — separated by addition and subtraction — or it can be in factored form. The factored form has everything connected by multiplication and division. Each of these forms is useful in some process or another, so it’s important to be able to change from one form to another and back again. Multiply out the factors if you want a listing of terms from highest exponent to lowest. Or, factor many terms to make them all just one if you want to solve for a root or reduce some fraction.You’ll find techniques for multiplying by one term or two — or more. There are some helpful tricks for raising binomials to higher powers. And then you find the factoring techniques — from rules of divisibility to factoring by grouping. One of the challenges of factoring expressions is deciding which technique to use. You find lots of practice to help you make those decisions.
Solving equations: What is the point of learning all those algebra basics and then going through the factoring process? One of the favorite and most common goals for all that practice is to use the techniques to solve an equation. Solving an equation means identifying the number or numbers you can replace the variable with to make a true statement.You’ll find factoring and the multiplication property of zero to be your first approach, and then you’ll also have the quadratic formula to use on some of the more challenging second-degree equations. Polynomials can be solved using synthetic division to help with the factoring. And then you have radical and absolute value equations — with their particular challenges. Finish the section off with inequalities, and you’ll have run the gamut of solving for what variables can represent.
Applications: Mention the words story problem, and you’ll see either a shudder or a brightening smile. People either love them or they don’t. But story problems (practical applications) are a main goal of learning to use algebra effectively.The practical applications found in this section of the workbook are broken into many different types. You find some that are based on an established formula: area, perimeter, simple interest, and so on. Other applications have to do with relationships between numbers or sizes of objects. The trick to doing those applications is understanding the wording, which is why you come armed with all the basics under your belt. Get to work on the work problems before you age too much with the age problems. Just write yourself a simple algebraic equation, and you’re almost finished.
Graphing: Most of us are very visual — we understand things better when a picture is drawn. I usually draw pictures when working on word problems; it helps me focus on what type of equation to write. But the pictures in this section are a bit more structured. The pictures here involve the Cartesian coordinate system, which involves placing points, segments, and lines in their proper positions. Graphing lines is often used when solving systems of equations. And graphing is found in pretty much all the mathematics that follows algebra. This is where you can get a good start on the topic.
Part 2: Answers
This part provides not only the answers to all the questions but explanations of the answers as well. So you get the solution, and you see how to arrive at that solution.
