1. Chapter 1 Real Numbers and Introduction to Algebra

2. Martin-Gay, Introductory Algebra, 3ed 2 2 1.1 – Tips for Success in Mathematics 1.2 – Symbols and Sets of Numbers 1.3 – Exponents, Order of Operations, and Variable Expressions 1.4 – Adding Real Numbers 1.5 – Subtracting Real Numbers 1.6 – Multiplying and Dividing Real Numbers 1.7 – Properties of Real Numbers 1.8 – Simplifying Expressions Chapter Sections

3. § 1.1 Tips for Success in Mathematics

4. Martin-Gay, Introductory Algebra, 3ed 4 4 Positive Attitude Believe you can succeed. Scheduling Make sure you have time for your classes. Be Prepared Have all the materials you need, like a lab manual, calculator, or other supplies. Getting Ready for This Course

5. Martin-Gay, Introductory Algebra, 3ed 5 5 General Tips for Success TipDetails Get a contact person. Exchange names, phone numbers or e-mail addresses with at least one other person in class. Attend all class periods. Sit near the front of the classroom to make hearing the presentation, and participating easier. Do you homework. The more time you spend solving mathematics, the easier the process becomes. Check your work. Review your steps, fix errors, and compare answers with the selected answers in the back of the book. Learn from your mistakes. Find and understand your errors. Use them to become a better math student. Continued

6. Martin-Gay, Introductory Algebra, 3ed 6 6 General Tips for Success TipDetails Get help if you need it. Ask for help when you don’t understand something. Know when your instructors office hours are, and whether tutoring services are available. Organize class materials. Organize your assignments, quizzes, tests, and notes for use as reference material throughout your course. Read your textbook. Review your section before class to help you understand its ideas more clearly. Ask questions. Speak up when you have a question. Other students may have the same one. Hand in assignments on time. Don’t lose points for being late. Show every step of a problem on your assignment.

7. Martin-Gay, Introductory Algebra, 3ed 7 7 Using This Text ResourceDetails Practice Problems. Try each Practice Problem after you’ve finished its corresponding example. Chapter Test Prep Video CD. Chapter Test exercises are worked out by the author, these are available off of the CD this book contains. Lecture Video CDs. Exercises marked with a CD symbol are worked out by the author on a video CD. Check with your instructor to see if these are available. Symbols before an exercise set. Symbols listed at the beginning of each exercise set will remind you of the available supplements. Objectives. The main section of exercises in an exercise set is referenced by an objective. Use these if you are having trouble with an assigned problem. Continued

8. Martin-Gay, Introductory Algebra, 3ed 8 8 Using This Text ResourceDetails Icons (Symbols). A CD symbol tells you the corresponding exercise may be viewed on a video segment. A pencil symbol means you should answer using complete sentences. Integrated Reviews. Reviews found in the middle of each chapter can be used to practice the previously learned concepts. End of Chapter Opportunities. Use Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews to help you understand chapter concepts. Study Skills Builder. Read and answer questions in the Study Skills Builder to increase your chance of success in this course. The Bigger Picture. This can help you make the transition from thinking “section by section” to thinking about how everything corresponds in the bigger picture.

9. Martin-Gay, Introductory Algebra, 3ed 9 9 Getting Help TipDetails Get help as soon as you need it. Material presented in one section builds on your understanding of the previous section. If you don’t understand a concept covered during a class period, there is a good chance you won’t understand the concepts covered in the next period. For help try your instructor, a tutoring center, or a math lab. A study group can also help increase your understanding of covered materials.

10. Martin-Gay, Introductory Algebra, 3ed 10 Martin-Gay, Introductory Algebra, 3ed 10 Preparing for and Taking an Exam Steps for Preparing for a Test 1. Review previous homework assignments. 2. Review notes from class and section-level quizzes you have taken. 3. Read the Highlights at the end of each chapter to review concepts and definitions. 4. Complete the Chapter Review at the end of each chapter to practice the exercises. 5. Take a sample test in conditions similar to your test conditions. 6. Set aside plenty of time to arrive where you will be taking the exam. Continued

11. Martin-Gay, Introductory Algebra, 3ed 11 Martin-Gay, Introductory Algebra, 3ed 11 Preparing for and Taking an Exam Steps for Taking Your Test 1. Read the directions on the test carefully. 2. Read each problem carefully to make sure that you answer the question asked. 3. Pace yourself so that you have enough time to attempt each problem on the test. 4. Use extra time checking your work and answers. 5. Don’t turn in your test early. Use extra time to double check your work.

12. Martin-Gay, Introductory Algebra, 3ed 12 Martin-Gay, Introductory Algebra, 3ed 12 Managing Your Time Tips for Making a Schedule 1. Make a list of all of your weekly commitments for the term. 2. Estimate the time needed and how often it will be performed, for each item. 3. Block out a typical week on a schedule grid, start with items with fixed time slots. 4. Next, fill in items with flexible time slots. 5. Remember to leave time for eating, sleeping, and relaxing. 6. Make changes to your workload, classload, or other areas to fit your needs.

13. § 1.2 Symbols and Sets of Numbers

14. Martin-Gay, Introductory Algebra, 3ed 14 Martin-Gay, Introductory Algebra, 3ed 14 Set of Numbers Natural numbers – {1, 2, 3, 4, 5, 6...} Whole numbers – {0, 1, 2, 3, 4...} Integers – {... –3, -2, -1, 0, 1, 2, 3...} Rational numbers – the set of all numbers that can be expressed as a quotient of integers, with denominator  0 Irrational numbers – the set of all numbers that can NOT be expressed as a quotient of integers Real numbers – the set of all rational and irrational numbers combined

15. Martin-Gay, Introductory Algebra, 3ed 15 Martin-Gay, Introductory Algebra, 3ed 15 Equality and Inequality Symbols SymbolMeaning a = b a  b a < b a > b a  b a  b a is equal to b. a is not equal to b. a is less than b. a is greater than b. a is less then or equal to b. a is greater than or equal to b.

16. Martin-Gay, Introductory Algebra, 3ed 16 Martin-Gay, Introductory Algebra, 3ed 16 The Number Line A number line is a line on which each point is associated with a number. 2– 201345– 1– 3– 4– 5 Negative numbers Positive numbers – 4.81.5

17. Martin-Gay, Introductory Algebra, 3ed 17 Martin-Gay, Introductory Algebra, 3ed 17 For any two real numbers a and b, a is less than b if a is to the left of b on the number line. a < b means a is to the left of b on a number line. a > b means a is to the right of b on a number line. Order Property for Real Numbers Insert between the following pair of numbers to make a true statement. Example

18. Martin-Gay, Introductory Algebra, 3ed 18 Martin-Gay, Introductory Algebra, 3ed 18 Absolute Value The absolute value of a real number a, denoted by |a|, is the distance between a and 0 on the number line. 2– 201345– 1– 3– 4– 5 | – 4| = 4 Distance of 4 Symbol for absolute value |5| = 5 Distance of 5

19. § 1.3 Exponents, Order of Operations, and Variable Expressions

20. Martin-Gay, Introductory Algebra, 3ed 20 Martin-Gay, Introductory Algebra, 3ed 20 Using Exponential Notation We may use exponential notation to write products in a more compact form. can be written as “three to the fourth power” “three to the third power” or “three cubed” “three to the second power” or “three squared.” In WordsExpression Evaluate 2 6. Example

21. Martin-Gay, Introductory Algebra, 3ed 21 Martin-Gay, Introductory Algebra, 3ed 21 Using the Order of Operations Order of Operations 1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars or square roots. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right.

22. Martin-Gay, Introductory Algebra, 3ed 22 Martin-Gay, Introductory Algebra, 3ed 22 Using the Order of Operations Evaluate: Write 3 2 as 9. Divide 9 by 3. Add 3 to 6. Divide 9 by 9. Example

23. Martin-Gay, Introductory Algebra, 3ed 23 Martin-Gay, Introductory Algebra, 3ed 23 DefinitionExample Variable: A letter to represent all the numbers fitting a pattern. Evaluate: 7 + 3z when z = – 3 Algebraic Expression: A combination of numbers, letters (variables), and operation symbols. Evaluating the Expression: Replacing a variable in an expression by a number and then finding the value of the expression Evaluating Algebraic Expressions

24. Martin-Gay, Introductory Algebra, 3ed 24 Martin-Gay, Introductory Algebra, 3ed 24 Determining Whether a Number is a Solution DefinitionExample Solving: In an equation containing a variable, finding which values of the variable make the equation a true statement. Is -7 a solution of: a + 23 = –16? Solution: In an equation, a value for the variable that makes the equation a true statement. – 7 is not a solution.

25. Martin-Gay, Introductory Algebra, 3ed 25 Martin-Gay, Introductory Algebra, 3ed 25 Translating Phrases Addition (+) Subtraction (–) Multiplication (·) Division (  ) sum plus added to more than increased by total difference minus subtract less than decreased by less product times multiply multiplied by of double/triple quotient divide shared equally among divided by divided into

26. Martin-Gay, Introductory Algebra, 3ed 26 Martin-Gay, Introductory Algebra, 3ed 26 Translating Phrases Write as an algebraic expression. Use x to represent “a number.” In words: a.) 5 decreased by a number b.) The quotient of a number and 12 5 Translate: 5 decreased bya number – x a.) In words: a number Translate: x and12  b.) The quotient of Example

27. § 1.4 Adding Real Numbers

28. Martin-Gay, Introductory Algebra, 3ed 28 Martin-Gay, Introductory Algebra, 3ed 28 Adding Real Numbers Adding 2 numbers with the same sign Add their absolute values. Use common sign as sign of sum. Adding 2 numbers with different signs Take difference of absolute values (smaller subtracted from larger). Use the sign of larger absolute value as sign of sum.

29. Martin-Gay, Introductory Algebra, 3ed 29 Martin-Gay, Introductory Algebra, 3ed 29 Opposites or additive inverses are two numbers the same distance from 0 on the number line, but on opposite sides of 0. The sum of a number and its opposite is 0. If a is a number, – (– a) = a. Example Add the following numbers. ( – 3) + 6 + ( – 5) = –2 Additive Inverses

30. § 1.5 Subtracting Real Numbers

31. Martin-Gay, Introductory Algebra, 3ed 31 Martin-Gay, Introductory Algebra, 3ed 31 Subtracting real numbers Substitute the opposite of the number being subtracted Add. a – b = a + (– b) Example Subtract the following numbers. ( – 5) – 6 – ( – 3) = ( – 5) + ( – 6) + 3 = – 8 Subtracting Real Numbers

32. Martin-Gay, Introductory Algebra, 3ed 32 Martin-Gay, Introductory Algebra, 3ed 32 Complementary angles are two angles whose sum is 90 o. Example Find the measure of the following complementary angles. x 150 – 2x x + 150 – 2x = 90 150 – x = 90 – x = – 60 x = 60° and 150 – 2x = 30° Complementary Angles

33. Martin-Gay, Introductory Algebra, 3ed 33 Martin-Gay, Introductory Algebra, 3ed 33 Supplementary angles are two angles whose sum is 180 o. Example Find the measure of the following supplementary angles. xx + 78 x + x + 78 = 180 2x + 78 = 180 2x = 102 x = 51° and x + 78 = 129° Supplementary Angles

34. § 1.6 Multiplying and Dividing Real Numbers

35. Martin-Gay, Introductory Algebra, 3ed 35 Martin-Gay, Introductory Algebra, 3ed 35 Multiplying or dividing 2 real numbers with same sign Result is a positive number Multiplying or dividing 2 real numbers with different signs Result is a negative number Multiplying or Dividing Real Numbers

36. Martin-Gay, Introductory Algebra, 3ed 36 Martin-Gay, Introductory Algebra, 3ed 36 Example Find each of the following products. 4 · (–2) · 3 = – 24 ( – 4) · ( – 5) =20 Multiplying or Dividing Real Numbers

37. Martin-Gay, Introductory Algebra, 3ed 37 Martin-Gay, Introductory Algebra, 3ed 37 If b is a real number, 0 · b = b · 0 = 0. Multiplicative inverses or reciprocals are two numbers whose product is 1. The quotient of any real number and 0 is undefined. The quotient of 0 and any real number = 0. a  0 Multiplicative Inverses (Reciprocals)

38. Martin-Gay, Introductory Algebra, 3ed 38 Martin-Gay, Introductory Algebra, 3ed 38 If a and b are real numbers, and b  0, Example Simplify the following. Simplifying Real Numbers

39. § 1.7 Properties of Real Numbers

40. Martin-Gay, Introductory Algebra, 3ed 40 Martin-Gay, Introductory Algebra, 3ed 40 Commutative and Associative Property Associative property of addition: (a + b) + c = a + (b + c) of multiplication: (a · b) · c = a · (b · c) Commutative property of addition: a + b = b + a of multiplication: a · b = b · a

41. Martin-Gay, Introductory Algebra, 3ed 41 Martin-Gay, Introductory Algebra, 3ed 41 Distributive property of multiplication over addition a(b + c) = ab + ac Identities for addition: 0 is the identity since a + 0 = a and 0 + a = a. for multiplication: 1 is the identity since a · 1 = a and 1 · a = a. Distributive Property

42. Martin-Gay, Introductory Algebra, 3ed 42 Martin-Gay, Introductory Algebra, 3ed 42 Inverses For addition: a and –a are inverses since a + ( – a) = 0. For multiplication: b and are inverses (b  0) since b · = 1. Inverses

43. § 1.8 Simplifying Expressions

44. Martin-Gay, Introductory Algebra, 3ed 44 Martin-Gay, Introductory Algebra, 3ed 44 Terms A term is a number, or the product of a number and variables raised to powers – (the number is called a coefficient) Examples of Terms 7 (coefficient is 7) 5x 3 (coefficient is 5) 4xy 2 (coefficient is 4) z 2 (coefficient is 1)

45. Martin-Gay, Introductory Algebra, 3ed 45 Martin-Gay, Introductory Algebra, 3ed 45 Like terms contain the same variables raised to the same powers To combine like terms, add or subtract the numerical coefficients (as appropriate), then multiply the result by the common variable factors You can combine like terms by adding or subtracting them (this is not true for unlike terms) Like Terms

46. Martin-Gay, Introductory Algebra, 3ed 46 Martin-Gay, Introductory Algebra, 3ed 46 6x 2 + 7x 2 19xy – 30xy 13xy 2 – 7x 2 y 13x 2 -11xy Can’t be combined (since the terms are not like terms) Examples of Combining Terms Terms Before CombiningAfter Combining Terms Combining Like Terms