Table of Contents. Introduction to the Series...1. Chapter 5: Rational Expressions...55 Rational Expressions...55 Scientific Notation...

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2 Table of Contents Table of Contents Introduction to the Series...1 Common Mathematics Symbols and Terms...2 Algebra Rules and Laws...12 Chapter 1: Solving Equations and Problems...13 Simplifying Expressions and Solving Equations With One Variable...13 Changing Words Into Symbols; Problem Solving With Equations...16 Chapter 2: Inequalities...23 Inequalities...23 Graphing Inequalities...24 Solving Inequalities...24 Working With Absolute Values...26 Chapter 3: Linear Equations and Inequalities...32 Linear Equations and Graphs...32 Linear Inequalities...38 Linear Systems...40 Chapter 4: Polynomial Products and Factors...46 Simplifying Polynomials...46 Laws of Exponents...46 Multiplying and Factoring Polynomials.49 Solving Polynomial Equations...52 Chapter 5: Rational Expressions...55 Rational Expressions...55 Scientific Notation...58 Chapter 6: Roots, Radicals, and Complex Numbers...64 Simplifying Radicals, Products, Quotients, Sums, Differences...64 Simplifying Binomials With Radicals and Solving Radical Equations...70 Decimal Representation and Complex Numbers...75 Chapter 7: Quadratic Equations and Functions...81 Solving Quadratic Equations...81 Quadratic Functions and Graphs...89 Chapter 8: Variation...95 Direct Variation, Proportion, Inverse Variation, Joint Variation...95 Algebra II Check-up Practice, Challenge, and Checking Progress Answer Keys Check-up Problems Answer Keys References iii

3 Introduction to the Math Practice Books Series Introduction to the Math Practice Books Series The Math Practice Books Series will introduce students in middle school and high school to the course topics of Pre-algebra, Algebra, Algebra II, and Geometry. All of the practice books are aligned with the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics. (NCTM 2000) This series is written for classroom teachers, parents, families, and students. The practice books in this series can be used as a full unit of study or as individual lessons to supplement textbooks or curriculum programs. Parents and students can use this series as an enhancement to what is being done in the classroom or as a tutorial at home. Students will be given a basic overview of the concepts, examples, practice problems, and challenge problems using the concepts introduced in the section. At the end of each section, there will be a set of problems to check progress on the concepts and a challenge set of problems over the whole section. At the end of the book, there will be problems for each section, which could be used for assessment. According to the Mathematics Education Trust and NCTM, new technologies require that the fundamentals of algebra and algebraic thinking should be a part of the background for all citizens. These technologies also provide opportunities to generate numerical examples, graph data, analyze patterns, and make generalizations. An understanding of algebra is also important because business and industry require higher levels of thinking and problem solving. NCTM also suggests that understanding geometry, including the characteristics and properties of two and three-dimensional shapes, spatial relationships, symmetry, and the use of visualization and spatial reasoning, can also be used in solving problems. The NCTM Standards suggest that content and vocabulary are necessary but of equal importance are the processes of mathematics. The process skills described in the Standards include: problem solving, reasoning, communication, and connections. The practice books in this series will address both the content and processes of algebra and algebraic thinking and geometry. This worktext, Algebra II Practice, will help students transition from Algebra to Algebra II. 1

4 Name: Date: Basic Overview: Simplifying Expressions and Solving Equations With One Variable Algebraic expressions can be simplified by applying the Order of Operations, particularly when the expressions contain multiple operations. (1) First evaluate within parentheses from innermost to outermost using rules 2, 3, and 4 in order; (2) Evaluate all exponents; (3) Multiply and/or divide from left to right; and then (4) Add and/or subtract, also from left to right. Equations can sometimes be solved by the guess-and-check method, but more often their solutions follow directly from using the principles of algebra. Equations can be systematically solved, as follows: (1) Use the distributive property to remove parentheses and simplify each side of the equation; (2) Apply the addition property of equality to variables on one side of the equation and constants on the other side; and (3) Apply the multiplication property of equality to isolate the variable. Examples of Simplifying Expressions and Solving Equations With One Variable Example of Simplifying Algebraic Expressions: 3 (4 + 5) = 3 (9) = 27 Order of Operations Example of Simplifying Equations With One Variable by Addition and Subtraction: x 4 = 9 (x 4) + 4 = Addition Property of Equality x + (-4 + 4) = 13 Associative Property of Addition x = 13 Identity Property of Addition Example of Simplifying Equations With One Variable by Multiplication and Division: 7b = 5(6 + 1) 7b = b = 5 13

5 Name: Date: (cont.) Practice: Simplifying Expressions and Solving Equations With One Variable Directions: Simplify the following expressions. 1. 7x 2 + 3(x 9) (9 2) (x + 3) 4. 2(x 5) + 3(x + 11) x 5. 3x 6(2 x) + 3(x 2) a 2a [3a (4a 5)] a 2 4 a + 7a 2(9 a) 8. 13(5 t) (5 t) 12(5 t) Directions: Simplify and solve the following equations. Work the problem on your own paper, if you need more room. 9. 2a + 3a 13 = a (x 55) = (x 55) + 22 = x x = 3x If 8x + 5(3 + x) a = x, then a =? 14. 3x + 4(x 3) = 6x t (4 t) = 8 (2 + t) 14

6 Name: Date: (cont.) (2x + 3) = 15(2x + 3) 17. 3x + 4x = 6(2x 1) (1 3x) = (x + 9) = [5 (9 + 2x)] = 7 5x Challenge Problems: Simplifying Expressions and Solving Equations With One Variable Directions: Work the problems on your own paper, and write the answers on the lines below. 1. Simplify. 2x {2 [2(x 2) (2 x)]} 2. If 3x + 5(x 2) + a = 9x + 13, then a =? 3. Simplify. 3[5 2 5(14 7) ] 4. Simplify and solve. 22x + 9(2 3x) = 3(2 3x) 5. Simplify and solve. 43(19 7x) + 32(19 7x) = 25(7x 19) 15

7 Answer Keys Practice, Challenge, and Checking Progress Answer Keys Practice: Simplifying Expressions and Solving Equations With One Variable (pages 14 15) 1. 10x x x x a a = x = x = x = (g, or a = 8x 14. x = t = QjP 16. x = -#s 17. x = ^g 18. x = x = x = -!d Challenge Problems: Simplifying Expressions and Solving Equations With One Variable (page 15) 1. 5x 8 2. x QjO Practice: Changing Words Into Symbols; Problem Solving With Equations (pages 18 20) 1. 5x 9 = 86: x = x = 5 0.6; x = $ x = $204; x = $17 4. One equation is x + (x + 6) = boys and 48 girls 5. One equation is x = gr;w;t;. x = or 8.5% 6. One equation is x 2,516,667 km x = 14.75; x = x = 22.95; x = One equation is x( ) = 108.7; x = $0.55 per minute. 10. If x is the number of sundaes, one equation is 5x 4 = 96. x = 20, the number of sundaes. 11. One equation is 2(2x + x) = 2((x + 12 ) + (x 3)). One rectangle is 9 x 18 and the other is 6 x If x is the number of trees sold in March, one equation is 2x + 3 = 71; 34 trees. 13. If x is the number of Swedish coins, one equation is x + (3x 26) = 998. He has 256 coins from Sweden and 742 coins from Norway. 14. If x is the number of votes Larry received, one equation is x + 2x + (2x + 7) = 327; Larry, Moe, and Curly received 64, 128, and 135 votes, respectively. 15. If x is the number of extra points that the second-period class had, one equation is (x + 54) + x = 438. First period had 192 extra points, and second period had 246 extra points. 16. An equation is 16x = 790; 30 hours 17. One equation is x(8)(9 1.35) = 918; 15 days 18. One equation is 2(x + (x + 50)) = 340. Dimensions are 60 x One equation is (0.5x + 5) + x + 2x = 180. Angles have measures of 30, 50, and (24)(x) = 48; height is 4 meters 109