This companion book to Thinkwell's Algebra 1 Summer Edition course includes: • Illustrated lesson notes • Lesson worksheets • Practice exercises • Review worksheets • Suggested pacing guide • Formula and resource worksheets Volume 1 covers: Ch 1 - Foundations for Algebra Ch 2 - Equations, Proportions, and Percent Ch 3 - Inequalities Ch 4 - Functions Ch 5 - Linear Functions

A companion book for Thinkwell's Algebra 1 – Summer Edition online video course. Volume #1 covers Chapters 1 – 5.

Copyright © 2023 by Thinkwell Corporation. All rights reserved.

No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system without prior written permission of Thinkwell Corporation unless such copying is expressly permitted by federal copyright law. Address inquiries to: Permissions, Thinkwell Corp., 505 E. Huntland Drive, Suite 150, Austin, TX 78752

to Algebra 1 welcome Summer Edition

Before we get too far into the fun, I wanted to personally introduce myself: I'm your virtual instructor, Professor Burger. Welcome to Thinkwell's wonderful world of Algebra 1!

take notes highlight key concepts and earmark ideas you want to remember doodle, sketch, and visualize the math ideas presented in each topic. About This Book This is a companion book to Thinkwell's Algebra 1 – Summer Edition – online video course. Use this book as a complement to the online materials. I always say that to learn math you must DO MATH! I encourage keep this book open as you view the video lessons, put pencil to paper and Together, you and I will explore the ideas that are foundational to Algebra. I hope you ’ ll let me help you in MAKING MEANING of the math ideas we ’ ll see together — there ’ s no need to memorize if you focus on deep understanding. I invite (urge) you to take the time to truly think through the math we ’ ll explore together and to mindfully practice the skills behind the ideas every day. If you do, you ’ ll not only succeed in Algebra 1, but you ’ ll also be on solid ground for all the other math in your future!

In a nutshell, make this book your own and keep it by your side as you study the concepts in this course.

This book is divided into chapters. Within each chapter are a series of Algebra 1 topics. Every topic online contains my Video Lessons along with an electronic version of the Notes, Practice questions and Worksheet questions (although I wish we ’ d call them “ Funsheets ” , but that ’ s another story). How To Use This Book Use this book alongside the online course and TAKE NOTES here while watching my video lessons. Your own notes are a key to your own success — I promise. Since this book summarizes the concepts, vocabulary, and key examples presented in the Video Lessons, it is a great tool to help you navigate the videos — but this companion book is not intended as a shortcut to replace the Video Lessons. To get the most out of this learning experience, I urge you to watch (and think through) all of the online Video Lessons. Maybe even watch some twice! The online Algebra 1 course offers lots of opportunities to practice the skills you ’ ll need for success in Algebra 1. Each topic ’ s Worksheet Practice and Interactive Practice is a collection of questions connected to the content presented in the Video Lessons. I've included those questions here in this book, so you can explore them offline and spend time really thinking through each question. I always say that the best way to learn math is to DO MATH. So, take advantage of all the opportunities to practice what you've learned!

lET'S GO! I look forward to our Algebra 1 journey together! Remember to make meaning and focus on deep understanding … and also remember that YOU CAN DO IT! Have fun! If you have any questions, please reach out to my friends at Thinkwell. Email them at support@thinkwell.com. Also, I ’ m on Twitter @ebb663, if you want to say, “ hello ” .

I wish you all the best in your Algebra success!

Your virtual teacher,

— Prof. B.

Optional: Optional:

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Use this book to help you stay organized. tips for success

Check out the suggested pacing guide in this book and create a study schedule for yourself. Your schedule will be your plan for Algebra success! The Summer goes by fast--don't let it get away from you!

Be an active learner. Before you begin studying, collect the tools you'll need: a pencil, scratch paper, highlighters, or graph paper are great things to have on-hand.

As you watch the Video Lessons online, work out the examples along with Prof. Burger on the Lesson Notes here (or on your own paper). Highlight important points in the Lesson Notes, and earmark topics you want to go back to review before a Test.

Practice as you go.

After each Video example, complete the Worksheet questions for that example. Once you've watched all the video lessons and answered all the Worksheet questions, check your understanding by completing the Practice question set. Go online to check your answers and to see answer feedback with step- by-step explanations.

Work smarter, not harder.

If you're pressed for time, do the online Chapter Pre- Tests first to identify topics where you need to focus. Only complete the topics you need and skip the others.

Reach out if you need help!

Have questions? Need help? Reach out to us at support@thinkwell.com. We're here to help!

Pacing guide Suggested

This pacing guide follows an 8-week plan to sequentially progress through Thinkwell's Algebra 1 Summer Edition online course. Since the course is self-paced, feel free to go as quickly or as slowly through the material as you need to – this guide is just a suggestion. Use the online Chapter Pre-Tests and Pre-Test Guides to help you determine where you need to focus most. The list below corresponds with Thinkwell's Algebra 1 Summer Edition online course scope and sequence. WEEK 1: – Chapter 1 Foundations for Algebra WEEK 2: – Chapter 2 Equations, Proportions, and Percent WEEK 3: – Chapters 3 Inequalities & 4 Functions WEEK 4: – Chapter 5 Linear Functions WEEK 5: – Chapter 6 Systems of Equations and Inequalities & Subchapter 7.1 Exponents WEEK 6: – Subchapter 7.2 Polynomials & Chapter 8 Factoring Polynomials WEEK 7: – Chapter 9 Quadratic Functions and Equations WEEK 8: – Chapter 10 Data Analysis and Probability

Chapter 1 Foundations for Algebra

5

1.1

The Language of Algebra

7

1.1.1 Variables and Expressions

Key Objectives • Translate between words and algebraic expressions. • Evaluate algebraic expressions. • Write algebraic expressions. Key Terms • A variable is a letter or symbol that represents a quantity that can change. • A constant is a quantity that does not change. • An algebraic expression contains one or more variables and may contain operation symbols. • To evaluate an expression means to find its value. Expressions are math statements that can include constants (numbers), variables, and operations. Operations in expressions are represented symbolically. Four operations commonly used in expressions are addition ( + ), subtraction ( − ), multiplication ( × or ·), and division ( ÷ or /). Note that multiplication is the understood operation when a number is written next to a variable, with no symbol between them. For example, 2 n means “2 times n .” Example 1 Translating from Algebra to Words To express a mathematical statement (expression) as words, first translate the meaning of each operation. Because operations can be described using many different phrases, there are often multiple correct ways to write an expression as words. For example, the operation addition ( + ) can be described using the words “increased by”, “added to”, or with “sum of”.

9

1.1.1 Variables and Expressions (continued)

Example 2 Translating from Words to Algebra

It is often helpful to translate ideas from words to a mathematical expression in order to solve real-world problems using algebra. The key to writing an expression that correctly represents an idea is to identify the operation described by the idea.

10

1.1.1 Variables and Expressions (continued)

Example 3 Evaluating Algebraic Expressions

To evaluate an expression, find the value of that expression by substituting a number for the variable (or variables) and then simplifying the resulting numerical expression, as demonstrated by Prof. Burger in Example 3.

Example 4 Recycling Application In Example 4, Prof. Burger demonstrates translating words into an algebraic expression that is evaluated for specific values of the variable to answer real-world questions.

11

1.1.1 Variables and Expressions − Worksheet

Example 1:

Give two ways to write each algebraic expression in words. 1. n − 5 2. f 3 3. c + 15

4. 9 − y

5. x

6. t + 12

7. 8 x

8. x − 3

12

Example 2:

9. George drives at 45 mi/h. Write an expression for the number of miles George travels in h hours.

10. The length of a rectangle is 4 units greater than its width w . Write an expression for the length of the rectangle.

Example 3:

Evaluate each expression for a = 3, b = 4, and c = 2. 11. a − c 12. ab

13. b ÷ c

14. ac

Example 4:

15. Brianna practices the piano 30 minutes each day. a. Write an expression for the number of hours she practices in d days.

b. Find the number of hours Brianna practices in 2, 4, and 10 days.

12

1.1.1 Variables and Expressions − Practice

1. Give two ways to write the algebraic expression a + b in words.

2. Give two ways to write the algebraic expression p ÷ 29 in words.

3. Ramona wrote 14 letters to friends each month for n months in a row. Write an expression to show how many total letters Ramona wrote.

4. Mary is 3 years older than James. If James is x years old, write an expression for Mary’s age.

5. Joan practices table tennis for 6 hours every 2 days. Write an expression for the number of hours she practices in x days.

6. Evaluate the expression m + p for m = 9 and p = 7.

7. Evaluate the expression a/b for a = 22 and b = 12.

8. Steve practices piano for 5 hours each day. Find the number of hours he practices in 1, 10, and 29 day(s).

9. Leah scored 34 points in the first half of the basketball game, and she scored y points in the second half of the game. Write an expression to determine the number of points she scored in all. Then, find the number of points she scored in all if she scored 15 points in the second half of the game.

13

1.1.2 Adding and Subtracting Real Numbers

Key Objectives • Add and subtract numbers on a number line. • Add and subtract real numbers. Key Terms • Real numbers are all numbers on a number line. • Absolute value is the distance of a number from zero on a number line; shown by | |. • On a number line, opposites are two numbers the same distance from 0 but on different sides of 0. Two numbers are opposites if their sum is 0. The process for adding two signed numbers depends on whether the two numbers have the same sign or different signs. • To find the sum of two numbers with different signs (i.e., one number is positive and the other number is negative), find the difference of their absolute values and then apply the sign of the number with the greater absolute value. • To find the sum of two numbers with the same sign (i.e., two positive numbers or two negative numbers), find the sum of their absolute values and then apply the common sign.

Example 1 Adding and Subtracting Numbers on a Number Line

A number line can be used to model addition and subtraction of signed numbers. To model the addition of a positive number, move that number of units to the right along the number line. To model the addition of a negative number, move that number of units to the left along the number line.

14

1.1.2 Adding and Subtracting Real Numbers (continued)

Example 2 Adding Real Numbers In Example 2, Prof. Burger demonstrates addition of negative numbers and evaluating an expression that contains numbers with different signs.

15

1.1.2 Adding and Subtracting Real Numbers (continued) Example 3 Subtracting Real Numbers To subtract a number • change the operation from subtraction to addition,

• change the number to its opposite, and then • follow the rules for adding signed numbers.

16

1.1.2 Adding and Subtracting Real Numbers (continued) Example 4 Consumer Economics Application There are many real-world examples where the addition and subtraction of signed numbers can be applied. In Example 4, Prof. Burger uses money (and money management) to demonstrate this skill.

17

1.1.2 Adding and Subtracting Real Numbers − Worksheet

Example 1:

Add or subtract using a number line.

− − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 3 6 1 4

1. − 4 + 7

2. − 3.5 − 5

3. 5.6 − 9.2

4.

Example 2:

Add.

+ − ⎛ ⎝ ⎜

⎞ ⎠ ⎟

3 4

3 4

5. 91 + ( − 11)

6.

7. 15.6 + x for x = − 17.9

4

3

Example 3:

Subtract.

1 5

4 5

8. 23 − 36

9. 4.3 − 8.4

10.

x

x

2

for

1

−

=

Example 4:

11. The Dow Jones Industrial Average (DJIA) reports the average prices of stocks for 30 companies. Use the table to determine the total decrease in the DJIA for the two days.

DJIA 1987

Friday, Oct. 16 − 108.35 Monday, Oct. 19 − 507.99

18

1.1.2 Adding and Subtracting Real Numbers − Practice

1. Add using a number line. − 4 + 1

2. Add using a number line. 1 + ( − 5)

3. Subtract − 8 − ( − 3) using a number line.

4. Add.

− 3 + 13

5. Add.

6. Add.

34 + ( − 21)

− 15 + ( − 13)

7. Subtract. 8 − ( − 7)

8. Subtract. − 5 − ( − 8)

9. The highest temperature recorded in the town of Westgate this summer was 101°F. Last winter, the lowest temperature recorded was − 9°F. Find the difference between these extremes.

19

1.1.3 Multiplying and Dividing Real Numbers Key Objectives • Multiply and divide signed real numbers. • Multiply and divide with fractions. • Multiply and divide with zero. Key Terms • Two numbers are reciprocals if their product is 1. For example, the reciprocal of 2/3 is 3/2.

When two numbers are multiplied or divided, the signs of the numbers being multiplied or divided determine whether the result is positive or negative, as summarized in the following table.

Example 1 Multiplying Signed Numbers To multiply two signed numbers, multiply their absolute values and then apply a sign to the product according to the following rules. • If their signs are the same (either both positive or both negative), then the product is positive. • If their signs are different (one is positive and the other is negative), then their product is negative.

20

1.1.3 Multiplying and Dividing Real Numbers (continued) Example 2 Dividing Signed Numbers Recall that to divide by a fraction, multiply by its reciprocal. To divide by a mixed number, first change the mixed number into an improper fraction, and then multiply by its reciprocal. The rules for dividing signed numbers are the same as the rules for multiplying signed numbers. • If their signs are the same (either both positive or both negative), then the quotient is positive. • If their signs are different (one is positive and the other is negative), then their quotient is negative.

21

Example 3 Multiplying and Dividing with Zero The product of any number and 0 is 0. Zero divided by any nonzero number is also 0, but any number divided by zero is undefined. When multiplying or dividing with 0, it does not matter whether the nonzero number is positive or negative. The rules remain the same regardless of the sign of the nonzero number. 1.1.3 Multiplying and Dividing Real Numbers (continued)

Example 4 Athletics Application Distance is found by multiplying a rate and a time. In Example 4, Prof. Burger finds an unknown distance given a rate and a time.

22

1.1.3 Multiplying and Dividing Real Numbers − Worksheet

Example 1:

Find the value of each expression.

1. − 72 ÷ ( − 9)

2. 11( − 11)

3. − 7.2 ÷ x for x = 3.6

Example 2:

Divide.

5 7

÷ − ⎛ ⎝ ⎜

⎞ ⎠ ⎟

÷ − ⎛ ⎝ ⎜

⎞ ⎠ ⎟

2 3

1 3

4 5

8 5

16 25

4 5

4. ÷ 5

5.

6.

7.

÷

Example 3:

Multiply or divide if possible.

2 3

7 8

8. 3.8 ÷ 0

9. 0( − 27)

10.

11.

÷ 0

0

÷

Example 4:

12. It is estimated that 7 million people saw off-Broadway shows in 2002. Assume that the average price of a ticket was $30. How much money was spent on tickets for off-Broadway shows in 2002?

23

1.1.3 Multiplying and Dividing Real Numbers − Practice

1. Multiply. − 10 ∙ 2

2. Find the value of the expression 28/( − 7).

3. Find the value of 4 b for b = 1/2.

4. Multiply. 0 ∙ 1.035

5. Divide.

6. Divide. ÷ 3

14.807 ÷ 0

3 5

1 6

7. Divide. − ÷ 7 4

8. Carina hiked at Yosemite National Park for 1.75 hours. Her average speed was 3.5 mi/h. How many miles did she hike?

1 2

9. A car traveled on a straight road for 2 1 4 hours at a speed of 60 miles per hour. How many miles did the car travel?

24

1.1.4 Powers and Exponents

Key Objectives • Write powers for geometric models. • Evaluate powers. • Write powers. Key Terms • A power is an expression written with a base and an exponent. • An exponent indicates how many times a number (the base of a power) is used as a factor. • The base of a power is the number that is used as a factor in the power. The expression 3 2 (read as “3 to the 2nd power” or “3 squared”) is a power, where 3 is the base and 2 is the exponent. The exponent in a power indicates the number of times the base is used as a factor. In the power 3 2 , the exponent 2 indicates that the base 3 is used as a factor two times. So, to simplify the power 3 2 , multiply 3 by itself 2 times: 3 2 = 3 · 3 = 9. Example 1 Writing Powers for Geometric Models In Example 1, Prof. Burger writes powers represented by geometric models to illustrate the efficiency of using powers to express the total number of items in a set.

25

1.1.4 Powers and Exponents (continued) Example 2 Evaluating Powers In a power such as a b , the base is a and the exponent is b , which indicates that a is used as a factor b times.

Example 3 Writing Powers

There are many ways to write a number as an equivalent numeric expression using operations. For example, 16 can be written as 13 + 3 or 8 · 2. Some numbers can be written as a power. For example, 16 can be written as 4 2 because 16 = 4 · 4 = 4 2 .

26

1.1.4 Powers and Exponents (continued) Example 4 Culinary Application In Example 4, Prof. Burger demonstrates using a power to relate categories that are scaled using a constant multiple.

27

1.1.4 Powers and Exponents − Worksheet

Example 1:

Write the power represented by each geometric model.

1.

2.

3.

9

9

Example 2:

Simplify each expression.

4

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1 2

4. 7 2

5. ( − 2) 4

6. ( − 2) 5

7. −

Example 3:

Write each number as a power of the given base.

8. 81; base 9

9. 100,000; base 10

10. − 64; base − 4

11. 10; base 10

12. 81; base 3

13. 36; base − 6

Example 4:

14. Jan wants to predict the number of hits she will get on her Web page. Her Web page received 3 hits during the first week it was posted. If the number of hits triples every week, how many hits will the Web page receive during the 5th week?

28

1.1.4 Powers and Exponents − Practice

1. Identify the power represented by the geometric model.

2. Evaluate the expression. 2 5

3. Evaluate the expression. ( − 1) 3

4. Evaluate the expression. ( − 1/2) 3

5. Simplify − 3 4 .

6. Write 729 as a power of the base 9.

7. Express 256 as a power of 16.

8. Express 10,000 as a power of 10.

9. Suppose you have developed a scale that indicates the brightness of sunlight. Each category in the table is 8 times brighter than the next lower category. For example, a day that is dazzling is 8 times brighter than a day that is radiant. How many times brighter is a dazzling day than a dim day? Sunlight Intensity Category Brightness Dim 2 Illuminated 3 Radiant 4 Dazzling 5

29

1.1.5 Square Roots and Real Numbers

Key Objectives • Find square roots of perfect squares and of fractions. • Classify real numbers. • Approximate square roots. Key Terms • The square root of a number is one of the two equal factors of the number. • A real number is a rational or irrational number. Every point on the number line represents a real number. • An irrational number is a real number that cannot be written as a ratio of integers. • A rational number is a number that can be written as a fraction with integers for its numerator and denominator (denominator cannot be 0). • A terminating decimal is a rational number in decimal form with a finite number of decimal places. • A repeating decimal is a rational number in decimal form that has a block of one or more digits that repeat continuously. • Integers are the set of all whole numbers and their opposites.

• Whole numbers are the set of natural numbers and 0. • Natural numbers are the set of counting numbers. Example 1 Finding Square Roots of Perfect Squares

The product of a whole number and itself is called a perfect square. For example, 25 is a perfect square because 5 · 5 = 25, and 49 is a perfect square because 7 · 7 = 49. The first 11 perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. The square root of a perfect square is the number that when multiplied by itself is equal to that perfect square. For example, the square root of 49 is 7 because 7 · 7 = 49. However, note that − 7 · − 7 = 49 as well. Therefore, there are two square roots of 49, 7 and − 7. In fact, every positive real number has two square roots that are opposites of each other. The symbol is used to represent the square root of a number. The nonnegative square root of a real number n is represented by n . The negative square root of a real number n is represented by n − .

30

1.1.5 Square Roots and Real Numbers (continued) Example 2 Finding Roots of Fractions To find the square root of a fraction, find the fraction such that its square is equal to that original fraction.

Example 3 Problem-Solving Application The number under the square root symbol, called the radicand, is not always a perfect square. In this case, the square root can be found by using a calculator, or it can be approximated by estimating.

31

1.1.5 Square Roots and Real Numbers (continued) Example 4 Classifying Real Numbers

Real numbers are all of the numbers on a number line. Every real number is either a rational number or an irrational number. A rational number can be written as a fraction where the numerator is an integer and the denominator is a nonzero integer. An irrational number cannot be written as such a fraction. Rational numbers that can be written with denominator of 1 are called integers, such as − 17, 0, and 4. The nonnegative integers are called whole numbers, and integers that are positive are called natural numbers. So, every natural number is also a whole number, an integer, and a rational number.

32

1.1.5 Square Roots and Real Numbers − Worksheet

Example 1:

Find each square root.

1. 64

2. 225

3.

4. 169

1 −

Example 2:

Find each square root.

1 9

9 64

1 16

6. 4 9

5.

7.

8.

−

1 100

4 81

1 36

1 64

9.

10.

11.

12.

−

Example 3:

13. A contractor is told that a potential client’s kitchen floor is in the shape of a square. The area of the floor is 45 ft 2 . Find the side length of the floor to the nearest tenth.

Example 4: Write all classifications that apply to each real number.

15. 1 6

14. − 27

16. 12

17. − 6.8

33

1.1.5 Square Roots and Real Numbers − Practice

1. Find the square root. 64

2. Find the square root. − 144

4 49

16 25

4.

3. Find

− Find

.

.

6. The area of a square window pane is 83 in 2 . How long is each side of the window pane? Round your answer to the nearest tenth of an inch.

5. The area of a square garden is 87 square meters. Estimate the side length of the garden.

8. Write all classifications that apply to the real number 81 4 .

7. Write all classifications that apply to the real number 4.

9. Write all classifications that apply to the real number. 3 2

34

1.1 Review Worksheet, Part 1

1.1.1 Variables and Expressions Give two ways to write each algebraic expression in words.

1. 5 p

2. 4 − y

3. 3 + x

4. 3 y

5. − 3 s

6. r ÷ 5

7. 14 − t

8. x + 0.5

9. Friday’s temperature was 20° warmer than Monday’s temperature t . Write an expression for Friday’s temperature.

10. Ann sleeps 8 hours per night. Write an expression for the number of hours Ann sleeps in n nights.

Evaluate each expression for r = 6, s = 5, and t = 3. 11. r − s 12. s + t

13. r ÷ t

14. sr

15. Jim is paid for overtime when he works more than 40 hours per week. a. Write an expression for the number of hours he works overtime when he works h hours.

b. Find the number of hours Jim works overtime when he works 40, 44, 48, and 52 hours.

1.1.2 Adding and Subtracting Real Numbers Add or subtract using a number line. 16. − 2 + 6 17. 6 + ( − 2) 18. −

19. − + 2 5

1 4

6

12

35

1.1 Review Worksheet, Part 1 (continued)

Add.

20. − 18 + ( − 12)

21. − 2.3 + 3.5

22. x + 29 for x = − 15

Subtract.

24. − − − ⎛ ⎝ ⎜ 3 4

⎞ ⎠ ⎟

1 4

23. 12 − 22

25. 38 − x for x = 24.6

26. A meteorologist reported that the day’s high temperature was 17°F and the low temperature was − 6°F. What was the difference between the day’s high and low temperatures?

1.1.3 Multiplying and Dividing Real Numbers Find the value of each expression.

27. − 30 ÷ ( − 6)

28. 8( − 4)

29. x ( − 12) for x = − 25

Divide.

÷ − ⎛ ⎝ ⎜

÷ − ⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎞ ⎠ ⎟

÷ 4 1 2

3 4

1 2

3 20

1 4

9 14

15 28

1 2

30.

31.

32.

33.

1

2

1

÷

Multiply or divide if possible.

0 1

34. 0 · 15

35. − 0.25 ÷ 0

36. 0 ÷ 1

37.

3

÷

38. A cold front changes the temperature by − 3°F each day. If the temperature started at 0°F, what will the temperature be after 5 days?

36

1.1 Review Worksheet, Part 2

1.1.4 Powers and Exponents Write the power represented by each geometric model. 1. 2. 3. 5

3

5

3

3

Simplify each expression.

2

7. − ⎛ ⎝ ⎜

⎞ ⎠ ⎟

3 5

4. 3 3

5. ( − 4) 2

6. − 4 2

Write each number as a power of the given base.

8. 49; base 7

9. 1000; base 10

10. − 8; base − 2

11. 1,000,000; base 10

12. 64; base 4

13. 343; base 7

14. Protozoa are single-celled organisms. Paramecium aurelia is one type of protozoan. The number of Paramecium aurelia protozoa doubles every 1.25 days. There was one protozoan on a slide 5 days ago. How many protozoa are on the slide now?

37

1.1 Review Worksheet, Part 2 (continued)

1.1.5 Square Roots and Real Numbers Find each square root. 15. 121 16. 9

17. − 100

18. 400

25 36

1 25

1 16

21. 1 4

19.

20.

22. −

23. Mr. and Mrs. Phillips are going to build a new home with a foundation that is in the shape of a square. The house will cover 222 square yards. Find the length of the side of the house to the nearest tenth of a yard.

Write all classifications that apply to each real number.

24. 5 12

25. 49

26. − 3

27. 18

38

1.2

Tools of Algebra

39

1.2.1 Set Theory

Key Objectives • Find the union and intersection of sets. • Make Venn diagrams and find complements of sets. • Determine relationships between sets. • Find the cross product of sets. Key Terms • A set is a collection of items called elements. • An element is an item in a set. • The intersection of two sets is the set of all elements that are found in both sets. The intersection of sets is denoted by ∩ . • The union of two sets is the set of all elements that are found in either set. The union of sets is denoted by ∪ . • The empty set is the set that contains no elements. The empty set is denoted by ∅ or {}. • The universe (universal set) for a particular situation is the set that contains every element relating to the situation. • The complement of set A in universe U is the set of all elements in U that are not in set A . • A subset is a set that is contained entirely within another set. Set B is a subset of set A if every element of B is contained in A , denoted B ⊆ A . • The cross product of two sets A and B is the set of ordered pair elements ( a , b ), where a is an element of A and b is an element of B . The cross product of sets A and B is denoted by A × B . Example 1 Finding the Union and Intersection of Sets Sets are collections of items, called elements, are commonly named with uppercase letters, such as A and B . The union of two sets A and B , denoted A ∪ B , is the set of all elements that are in either set A or in set B . The intersection of two sets A and B , denoted A ∩ B , is the set of all elements common to both sets A and B . If two sets have no elements in common, then their intersection is the empty set, denoted ∅ or {}.

41

1.2.1 Set Theory (continued)

Example 2 Making a Venn Diagram and Finding the Complement of a Set

A Venn diagram shows the relationship between sets. Each set in a Venn diagram is represented by a closed figure, usually an oval. Within the oval, or closed figure, is the name of the set and the elements of the set. If two sets have elements in common (that is, if their intersection is not the empty set), then their ovals are drawn with an overlapping region. The elements common to both sets are included in this region. In other words, the overlapping region represents the intersection of the two sets. The totality of the two ovals represents the union of the two sets.

42

1.2.1 Set Theory (continued)

Example 3 Determining Relationships Between Sets If B ⊆ A (read as “set B is a subset of set A ”), then every element of set B is contained in set A .

43

1.2.1 Set Theory (continued)

Example 4 Consumer Application

44

1.2.1 Set Theory − Worksheet

Example 1:

Find the union and intersection of each pair of sets.

1. A = {10, 12, 14, 16}; B = {9, 10, 11, 12}

2. A is the set of positive prime numbers less than 10; B is the set of whole-number factors of 10.

Example 2:

Find the complement of set A in universe U . 3. U = { − 5, − 4, − 3, − 2, − 1, 0, 1, 2}; A = { − 1, 0, 1}

4. U is the set of whole numbers less than 10; A is the set of perfect squares less than 10.

Example 3:

Determine whether each statement about the sets is true or false. Use a Venn diagram to support your answer. 5. A is the set of whole-number factors of 9, and B is the set of whole-number factors of 16. Statement: A ∩ B = ∅

6. A is the set of perfect squares, and B is the set of whole numbers. Statement: A ⊆ B

Example 4: 7. The set W = {30, 32, 34, 36, 38, 40, 42} represents the waist sizes in inches of men’s jeans sold at a clothing store. The set I = {28, 30, 32} represents the possible inseam lengths in inches of the jeans. Find W × I to determine all size of jeans sold at the store.

45

1.2.1 Set Theory − Practice

2. If P = { − 1, 0, 1, 2, 4} and Q = {4, 5, 6}, find P ∪ Q .

1. If A = {2, 5} and B = {1, 2, 3, 4, 5}, find A ∪ B .

4. If P = {1, 3, 6, 7, 9} and Q = 1, 2, 4, 5, 6, 7, 8}, find P ∩ Q .

3. If A = {1, 2, 4} and B = {2, 4, 6, 8, 10}, find A ∩ B .

6. Find the complement of set A in universe U . U is the set of whole numbers less than 10. A is the set of factors of 9. 8. If A is the set of all natural numbers, choose the set B that will make the following statement true . B ⊆ A ◦ B = {0, 1, 2, 3, 4} ◦ B = {−1, 1}

5. Find the complement of set B in universe U . U is the set of natural numbers less than or equal to 8 B is the set of even factors of 8 7. If A is the set of all integers, choose the set B that will make the following statement false . B ⊆ A ◦ B = {1, 2, 3} ◦ B = {2.5, 3.5, 4.5} ◦ B = {−4, −1, 0, 5, 10} ◦ B = {0} 9. Which of the following statements correctly describes the information given in the Venn diagram?

◦ B = {−1, 0, 1} ◦ B = {1, 2, 3, 4}

◦ R ⊆ S ◦ S is the complement of R

◦ S ∩ R ◦ S ∪ R

46

1.2.2 Order of Operations

Key Objectives • Simplify numerical expressions. • Evaluate algebraic expressions. Key Terms • The order of operations is a rule for simplifying expressions that contain more than one operation. The order of operations establishes the order in which those operations must be performed. When simplifying an expression that contains two or more operations, the operations must be completed in a specific order, as defined by the order of operations. By the order of operations, perform any operations within parentheses (or other grouping symbols) first, then compute exponents (powers and roots), then perform all multiplication and division from left to right , and then perform all addition and subtraction from left to right . The phrase “Please Excuse My Dear Aunt Sally” or PEMDAS can be a helpful mnemonic device for remembering the order of operations.

Example 1 Simplifying Numerical Expressions Always use the order of operations to simplify any expression with multiple operations.

47

1.2.2 Order of Operations (continued) Example 2 Evaluating Algebraic Expressions In Example 2, Prof. Burger uses the order of operations to evaluate an expression.

Example 3 Simplifying Expressions with Other Grouping Symbols

According to the order of operations, if an expression contains grouping symbols, such as parentheses or brackets, then the operations within those grouping symbols must be performed before any operations outside of those grouping symbols. Absolute-value symbols and fraction bars (when the numerator or denominator of that fraction contain at least one operation) are two additional types of grouping symbols.

48

1.2.2 Order of Operations (continued) Example 4 Translating from Words to Math In Example 4, a phrase that refers to multiple operations is translated into an algebraic expression.

Example 5 Problem Solving Application In Example 5, Prof. Burger writes and simplifies an expression using the order of operations.

49

1.2.2 Order of Operations − Worksheet

Example 1:

Simplify each expression.

1. 5 − 12 ÷ ( − 2)

2. 30 − 5 ∙ 3

3. 50 − 6 + 8

4. 12 ÷ ( − 4)(3)

5. (5 − 8)(3 − 9)

6. 16 + [5 − (3 + 2 2 )]

Example 2:

Evaluate each expression for the given value of the variable.

7. 5 + 2 x − 9 for x = 4

8. 30 ÷ 2 − d for d = 14

9. 51 − 91 + g for g = 20

10. 2(3 + n ) for n = 4

11. 4( b − 4) 2 for b = 5

12. 12 + [20(5 − k )] for k = 1

Example 3:

Simplify each expression.

13. 24 ÷ |4 − 10|

14.

15. 5(2) + 16 ÷ | − 4|

− 4.5 2(4.5)

− + 0 24 6 2

+ 2 3(6) 2 2

16.

17.

18. − ÷ ÷ 44 12 3

Example 4: Translate each word phrase into a numerical or algebraic expression. 19. 5 times the absolute value of the sum of s and − 2

50

1.2.2 Order of Operations − Worksheet (continued)

20. the product of 12 and sum of − 2 and 6

21. 14 divided by the sum of 52 and − 3

Example 5:

22. The surface area of a cylinder can be found using the expression 2 πr ( h + r ). Find the surface area of the cylinder shown. (Use 3.14 for π and give your final answer rounded to the nearest tenth.)

r = 3 ft

h = 7 ft

51

1.2.2 Order of Operations − Practice

1. Evaluate 5 + x − 6 · 8 for x = 8.

2. Translate the word phrase, the product of 9.5 and the difference of − 2 and 2, into a numerical expression.

3. Translate the phrase “5 times the sum of − 2 and z ” into an algebraic expression.

4. Translate the word phrase into a numerical expression. the quotient of the difference of 29 and 9 and the square root of 100

5. The surface area of a cylinder can be found using the expression 2 πr ( r + h ). Find the surface area of the cylinder if h = 6 feet and r = 2 feet. Use 3.14 for π and round to the nearest tenth if needed.

6. Simplify the expression 2 × 4 − 3 + 4 ÷ 2 2 .

7. Simplify 8 + 3[3 − (1) 6 ].

8. Simplify. + 10 5 5 2

|9 11|

+ −

9. Anna has coins in pennies, nickels, dimes, and quarters. The total amount of money she has in dollars can be found using the expression ( P + 5 N + 10 D + 25 Q ) ÷ 100. Use the table to find how much money Anna has. P N D Q 36 14 9 3

52

1.2.3 Simplifying Expressions

Key Objectives • Use the Commutative and Associative Properties. • Use the Distributive Property with mental math. • Combine like terms to simplify algebraic expressions • Use properties to justify simplification steps. Key Terms • The Commutative Property is the property that states that two or more values can be added or multiplied together in any order without changing the result. a + b = b + a a · b = b · a • The Associative Property is the property that states that values being added or multiplied together can be grouped in any order without changing the result. ( a + b ) + c = a + ( b + c ) ( a · b ) · c = a · ( b · c ) • The Distributive Property is the property that states that the product of a value and a sum (or difference) is equal to that value multiplied by each of the values being added (or subtracted). a ( b + c ) = ab + ac a ( b − c ) = ab − ac • A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by plus or minus signs. • Like terms are two or more terms that have the same variable raised to the same exponent. Example 1 Using the Commutative and Associative Properties By the Commutative Property, two or more numbers can be added in any order, and two or more numbers can be multiplied in any order, without changing the result. For example, 8 + 1 = 1 + 8 and 8 · 1 = 1 · 8. By the Associative Property, three or more numbers being added can be grouped in any way, and three or more numbers being multiplied can be grouped in any way, without changing the result. For example, 2 + (3 + 5) = (2 + 3) + 5 and 2 · (3 · 5) = (2 · 3) · 5. In Example 1, Prof. Burger demonstrates using the Commutative and Associative Properties to reorder and regroup parts of an expression, resulting in calculations that are easy to perform using mental math. COMMON ERROR ALERT The Commutative and Associative Properties cannot be applied to subtraction or division. For example, 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4, while (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1. Therefore, the Associative Property does not hold true for division. The same is true for subtraction.

53

1.2.3 Simplifying Expressions (continued) Example 2 Using the Distributive Property with Mental Math By the Distributive Property, the product of a value and a sum (or difference) is equal to that value multiplied by each of the values being added (or subtracted). • Distributive Property (addition): a ( b + c ) = ab + ac • Distributive Property (subtraction): a ( b − c ) = ab − ac The Distributive Property can be used to write an expression in a way that makes mental math easier.

Example 3 Combining Like Terms

Terms (the parts of the expression separated by addition or subtraction) can be a number, a variable, or a product of a number and a variable. In the product of a number and a variable, the number is called the coefficient. For example, 3 x + 5 + x has three terms, 3 x , 5, and x . The coefficient of 3 x is 3. The coefficient of x is 1, because x = 1 x . Terms with variables are like terms if they have exactly the same variable (or variables) to exactly the same power. Algebraic expressions can be simplified by combining (adding or subtracting) the like terms. Like terms with variables are combined by adding (or subtracting) their coefficients. Terms without variables are called constants. Constants are always like terms.

54

1.2.3 Simplifying Expressions (continued)

Example 4 Simplifying Algebraic Expressions The Distributive Property is used in Example 4 to simplify an algebraic expression.

55

1.2.3 Simplifying Expressions − Worksheet

Example 1:

Simplify each expression.

+ + + 16 2 1 2

1 2

1. − 12 + 67 + 12 + 23 2.

3. 27 + 98 + 73

4 1

1 3

4.

5. 2 ∙ 38 ∙ 50

6. 50 ∙ 118 ∙ 20

8 21

⋅ ⋅

Example 2:

Write each product using the Distributive Property. Then simplify.

7. 14(1002)

8. 16(19)

9. 9(38)

10. 8(57)

11. 12(112)

12. 7(109)

Example 3:

Simplify each expression by combining like terms.

13. 6 x + 10 x

14. 35 x − 15 x

15. − 3 a + 9 a

16. − 8 r − r

17. 17 x 2 + x

18. 3.2 x + 4.7 x

Example 4: Simplify each expression. Justify each step with an operation or property. 19. 5( x + 3) − 7 x 20. 9( a − 3) − 4 21. 5 x 2 − 2( x − 3 x 2 )

22. 6 x − x − 3 x 2 + 2 x 23. 12 x + 8 x + t − 7 x 24. 4 a − 2( a − 1)

56

1.2.3 Simplifying Expressions − Practice

1. Simplify the expression. 3 5 (12)(5)

2. Simplify the expression. + + 3 4 4

4 7

2 9

3 7

4. Simplify the expression 205 a − a by combining like terms.

3. Using the Distributive Property, find the product of the expression 5(17) and then simplify.

6. Simplify.

5. Simplify the expression 11 a 3 − 3 a by combining like terms.

2 3

5 3

+ x x 2

2

7. Simplify the expression 12 x + y 2 + 6 x + 3 y 2 by combining like terms.

8. Simplify by combining like terms. 7 a 3 + 5 t + 4 a 3 + 4 t + 3 a 2

9. Simplify.

5 + 2(3 x − 4) − x

57

1.2.4 Introduction to Functions

Key Objectives • Graph points in the coordinate plane. • Locate points in the coordinate plane. • Generate and graph ordered pairs. Key Terms • A coordinate plane is formed by the intersection of two perpendicular number lines (axes) in a plane. The point of intersection is the zero on each number line. • The two perpendicular number lines on a coordinate plane are called the axes . • The x - and y -axes divide the coordinate plane into four regions. Each region is called a quadrant . • An ordered pair is a pair of numbers that can be used to locate a point on a coordinate plane. The first number in an ordered pair is called the x -coordinate and the second number in an ordered pair is called the y -coordinate . • A function can be represented by an equation with two variables where the value of y (the output ) is determined by the value of x (the input ). Example 1 Graphing Points in the Coordinate Plane The coordinate plane is formed by the intersection of two perpendicular number lines, called axes, that intersect at 0 on each number line. This point of intersection is called the origin. The horizontal number line is called the x -axis, and the vertical number line is called the y -axis. • The horizontal axis on a coordinate plane is called the x -axis . • The vertical axis on a coordinate plane is called the y -axis . • The point where the axes intersect in a coordinate plane is called the origin . Each point or position on a coordinate plane can be described by an an ordered pair, written as ( x, y ). The x -value in an ordered pair is called the x -coordinate, and the y-value in an ordered pair is called the y -coordinate. Points on a coordinate plane are named with a capital letter. To graph an ordered pair ( x, y ) on a coordinate plane, begin at the origin and then move x -units to the left or right (depending on whether the x -coordinate is positive or negative) along the x -axis. Then, move y -units up or down (depending on whether the y -coordinate is positive or negative) from that position on the x -axis. The resulting location is the position of the point ( x, y ).

58

1.2.4 Introduction to Functions (continued) Example 2 Locating Points in the Coordinate Plane The x - and y -axes divide the coordinate plane into four regions called quadrants.

All points in quadrant I have positive x - and y -coordinates. All points in quadrant II have a negative x -coordinate and a positive y -coordinate. All points in quadrant III have negative x - and y -coordinates. All points in quadrant IV have a positive x -coordinate and a negative y -coordinate. If a point is not in a quadrant, then it must be on either the x -axis, the y -axis, or at the origin.

Example 3 Art Application

An equation that contains two variables can be used as a rule to generate ordered pairs. Substituting any value for x will generate a corresponding y -value, thus generating the ordered pair ( x , y ). For example, consider the equation y = 2 x + 1. If x = 5, then y = 2(5) + 1 = 11, or the ordered pair (5, 11).

59

1.2.4 Introduction to Functions (continued)

Example 4 Generating and Graphing Ordered Pairs Each of the given equations in Example 4 is a function. In a function, the y -value (the output) is determined by the x -value (the input).

60

1.2.4 Introduction to Functions (continued)

61

1.2.4 Introduction to Functions − Worksheet

Example 1:

Graph each point.

1. J (4, 5)

2. K ( − 3, 2)

3. L (6, 0)

4. M (1, − 7)

Example 2:

y

Name the quadrant in which each point lies.

F

4

5. A

6. B

7. C

A

2

x

D B

8. D

9. E

10. F

2

4

–4 –2

–2

C

E

–4

Example 3:

11. The number of counselors at a summer camp must be equal to 1 4 the number of campers. Write a rule for the number of counselors that must be at the camp. Write ordered pairs for the number of counselors when there are 76, 100, 120, and 168 campers.

Example 4:

Generate ordered pairs for each function for x = − 2, − 1, 0, 1, and 2. Graph the ordered pairs and describe the pattern.

= y x 1 2

12. y = x + 2

13. y = − x

14. y = − 2| x |

15.

2

62

1.2.4 Introduction to Functions − Practice

1. Name the quadrant in which the point ( − 7, 3) lies.

2. Name the quadrant in which the point ( − 5, − 3) lies.

3. Name the quadrant in which the point (2, − 3) lies.

4. Graph the point (1, 4).

63

1.2.4 Introduction to Functions − Practice (continued)

5. Graph the point (3, − 1).

6. Name the axis or quadrant in which the point ( − 3, 2) lies.

8. An advertising agency increased their fees to an $80 setup fee plus $4 for every advertisement. Write a rule for the new fee. Write ordered pairs for the fees for 10, 20, 30, and 40 advertisements.

7. Name the axis or quadrant in which the point (4, 0) lies.

9. Generate ordered pairs for the function y = 2 x 2 − 1 using x = − 2, − 1, 0, 1, 2. Graph the ordered pairs and describe the pattern.

64

1.2 Review Worksheet, Part 1

1.2.1 Set Theory Find the union and intersection of each pair of sets.

1. A = {4, 5, 6}; B = {l, 2, 3, 4, 5, 6}

2. A is the set of whole-number factors of 45; B is the set of whole-number factors of 15.

Find the complement of set A in universe U .

3. U = {0, 3, 6, 9, 12, 15, 18, 2l}; A = {0, 3, 6, 9, 12, 15, 18, 21}

4. U is the set of positive multiples of 5; A is the set of positive multiples of 10.

Determine whether each statement about the sets is true or false. Use a Venn diagram to support your answer. 5. A is the set of integers, and B is the set of positive integers. Statement: A ∩ B = B

6. A is the set of powers of 2 with natural-number exponents, and B is the set of positive even numbers. Statement: A ⊆ B

7. The set D = {52 mm, 53 mm, 54 mm, 55 mm} represents the diameters of skateboard wheels often recommended for teens. The set H = {80A, 85A, 90A, 95A, 100A} represents the hardness ratings of typically available wheels. Find D × H to determine all possible types of skateboard wheels typically available for teens.

65

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