Algebra II Practice

iii© Mark Twain Media, Inc., Publishers

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 ...............................100

Practice, Challenge, and Checking

Progress Answer Keys ....................109

Check-up Problems Answer Keys .......123

References .............................................126

Table of Contents

Algebra II Practice

1© Mark Twain Media, Inc., Publishers

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.

Introduction to the Math Practice Books Series

Algebra II Practice

13© Mark Twain Media, Inc., Publishers

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 = 9 + 4 Addition Property of Equalityx + (-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 =

35

7 7

b = 5

Chapter 1: Solving Equations and Problems

Chapter 1: Solving Equations and Problems

Algebra II Practice

14© Mark Twain Media, Inc., Publishers

Name: Date:

Chapter 1: Solving Equations and Problems (cont.)

Chapter 1: Solving Equations and Problems

Practice: Simplifying Expressions and Solving Equations With One Variable

Directions: Simplify the following expressions.

1. 7x – 2 + 3(x – 9) + 5

2. 40 ÷ 4 – 3(9 – 2)

3. 22 – 5(x + 3)

4. 2(x – 5) + 3(x + 11) – x

5. 3x – 6(2 – x ) + 3(x – 2) + 8

6. a – 2a – [3a – (4a – 5)]

7. 33a – 24a + 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 + 3 10. 3(x – 55) = 0

11. 3(x – 55) + 22 = x + 55 12. 5 – 2x + 9 – 3 = 3x + 2

13. If 8x + 5(3 + x ) – a = 15 + 5x, then a = ?

14. 3x + 4(x – 3) = 6x – 9 15. 5t – (4 – t ) = 8 – (2 + t )

Algebra II Practice

15© Mark Twain Media, Inc., Publishers

Name: Date:

Chapter 1: Solving Equations and Problems (cont.)

Chapter 1: Solving Equations and Problems

16. 17(2x + 3) = 15(2x + 3) 17. 3x + 4x = 6(2x – 1)

18. 18 – 42(1 – 3x ) = 98 19. 3 – 5 (x + 9) = 3

20. 12 + [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[52 – 5(14 ÷ 7) + 90 – 8]

4. Simplify and solve. 22x + 9(2 – 3x ) = 3(2 – 3x )

5. Simplify and solve. 43(19 – 7x) + 32(19 – 7x) = 25(7x – 19)

Algebra II Practice

109© Mark Twain Media, Inc., Publishers

Practice, Challenge, and Checking Progress Answer Keys

Practice: Simplifying Expressions and Solving Equations With One Variable (pages 14–15)1. 10x – 242. -113. -5x – 114. 4x + 235. 12x – 106. -57. 20a – 188. 09. a = 4

10. x = 5511. x = 99

12. x = (g, or 1.813. a = 8x14. x = 3

15. t = QjP16. x = -#s

17. x = ^g18. x = 219. x = -9

20. x = - !d

Challenge Problems: Simplifying Expressions and Solving Equations With One Variable (page 15)1. 5x – 82. x + 233. 244. -3

5. QjO

Practice: Changing Words Into Symbols; Problem Solving With Equations (pages 18–20)1. 5x – 9 = 86: x = 192. 22x = 5 – 0.6; x = $0.203. 12 • x = $204; x = $17

4. One equation is x + (x + 6) = 90. 42 boys and 48 girls

5. One equation is x = gR;W;T;. x = 0.085 or 8.5%

6. One equation is x ≈ 2,516,667 km2

7. 1.75x + 2.50 = 14.75; x = 7 8. 0.045x + 4.95 = 22.95; x = 400 9. One equation is 39.95 + x(475 – 350) =

108.7; x = $0.55 per minute.10. If x is the number of sundaes, one equa-

tion 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 21.

12. 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 equationis (x + 54) + x = 438. First period had 192 extra points, and second period had 246 extra points.

16. An equation is 16x + 310 = 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 110′

19. One equation is (0.5x + 5) + x + 2x = 180. Angles have measures of 30°, 50°, and 100°.

20. 0.5(24)(x ) = 48; height is 4 meters

Answer Keys