Algebra Know-It-ALL

Part One 165 Answer 8-5 For any positive integer n, we call the nth root of a number the (1/n)th power of that number. Then we i ...

166 Review Questions and Answers Question 8-10 Suppose we have a nonzero number x, an integer p, and a nonzero integer q. We wan ...

Part One 167 Question 9-4 In this book, here’s how we symbolize various sets of numbers. The set of natural numbers is N. The s ...

168 Review Questions and Answers Question 9-5 One of the following statements is false. Which one? Some real numbers are ration ...

Part One 169 Question 9-10 What is meant by the cardinality of a set? What is the cardinality of the set N? The set Z? The setQ? ...

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PART 2 Linear Equations and Relations Copyright © 2008 by The McGraw-Hill Companies, Inc. Click here for terms of use. ...

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173 CHAPTER 11 Equations and Inequalities You’ve learned how numbers and variables can be combined to form equations. Now it’s t ...

174 Equations and Inequalities Now suppose we have some variable or quantity. We don’t know its numerical value, but we can find ...

If we multiply through by 8, we get (5/2)× 8 = (20/8) × 8 which simplifies to 40/2 = 160/8, and further to 20 = 20. Dividing thr ...

176 Equations and Inequalities Are you confused? To avoid mistakes like the one just described, we must never divide an equation ...

or an arrowhead pointing to the right (>). When you use this symbol, remember that “larger” means “more positive” or “less ne ...

178 Equations and Inequalities number such as 2. Once you remember this, it’s easy to see that if x is any negative number, then ...

larger! What about 0? If we divide 0 by 10, we end up with 0 again, so 0 does not “qualify” either. We want the number to get st ...

180 Equations and Inequalities We can use parentheses, braces, and brackets in logical statements, just as we use them in ordi- ...

Behavior of the < relation The “strictly smaller” relation fails the reflexive and symmetric tests, just as does the “strictl ...

182 Equations and Inequalities Are you confused? Once you know what all these symbols mean, you can read complicated logical sta ...

We can’t reverse either of these implications and still end up with a valid statement. If I’m thinking of a rational number, I’m ...

184 Equations and Inequalities Here’s a summary of how we can morph “not equal” statements. Can we reverse the order? Yes. Can ...