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Big Ideas Math Integrated Mathematics Table of Contents

BIG IDEAS

MATH

®

Integrated Mathematics I Chapter 1

Solving Linear Equations

Chapter 7

Data Analysis and Displays

Chapter 2

Solving Linear Inequalities

Chapter 8

Basics of Geometry

Chapter 3

Graphing Linear Functions

Chapter 9

Reasoning and Proofs

Chapter 4

Writing Linear Functions

Chapter 10

Parallel and Perpendicular Lines

Chapter 5

Solving Systems of Linear Equations

Chapter 11

Transformations

Chapter 6

Exponential Functions and Sequences

Chapter 12

Congruent Triangles

Integrated Mathematics II Chapter 1

Functions and Exponents

Chapter 7

Quadrilaterals and Other Polygons

Chapter 2

Polynomial Equations and Factoring

Chapter 8

Similarity

Chapter 3

Graphing Quadratic Functions

Chapter 9

Right Triangles and Trigonometry

Chapter 4

Solving Quadratic Equations

Chapter 10

Circles

Chapter 5

Probability

Chapter 11

Circumference, Area, and Volume

Chapter 6

Relationships Within Triangles

Integrated Mathematics III Chapter 1

Geometric Modeling

Chapter 6

Rational Functions

Chapter 2

Linear and Quadratic Functions

Chapter 7

Sequences and Series

Chapter 3

Polynomial Functions

Chapter 8

Chapter 4

Rational Exponents and Radical Functions

Trigonometric Ratios and Functions

Chapter 9

Trigonometric Identities

Chapter 10

Data Analysis and Statistics

Chapter 5

Exponential and Logarithmic Functions

Ron Larson Laurie Boswell An A American Company

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INTRODUCING

®

INTEGRATED MATHEMATICS by Ron Larson and Laurie Boswell Big Ideas Learning is pleased to introduce a new high school program, Big Ideas Math Integrated Mathematics I, II, and III. The program was written by renowned authors Ron Larson and Laurie Boswell and was developed using the consistent, dependable learning and instructional theory that have become synonymous with Big Ideas Math. Students will gain a deeper understanding of mathematics by narrowing their focus to fewer topics at each grade level. They will also master content through inductive reasoning opportunities, engaging explorations, concise stepped-out examples, and rich thought-provoking exercises. The Big Ideas Math Integrated Mathematics research-based curriculum features a continual development of concepts that have been previously taught while integrating algebra, geometry, probability, and statistics topics throughout each course. In Integrated Mathematics I, students will study linear and exponential equations and functions. Students will use linear regression and perform data analysis. They will also learn about geometry topics such as simple proofs, congruence, and transformations. Integrated Mathematics II expands into quadratic, absolute value, and other functions. Students will also explore polynomial equations and factoring, and probability and its applications. Coverage of geometry topics extends to polygon relationships, proofs, similarity, trigonometry, circles, and three-dimensional figures. In Integrated Mathematics III, students will expand their understanding of area and volume with geometric modeling, which students will apply throughout the course as they learn new types of functions. Students will study polynomial, radical, logarithmic, rational, and trigonometric functions. They will also learn how visual displays and statistics relate to different types of data and probability distributions.

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About the Authors

Ron Larson, Ph.D., is well known as the lead author of a comprehensive program for mathematics that spans middle school, high school, and college courses. He holds the distinction of Professor Emeritus from Penn State Erie, The Behrend College, where he taught for nearly 40 years. He received his Ph.D. in mathematics from the University of Colorado. Dr. Larson’s numerous professional activities keep him actively involved in the mathematics education community and allow him to fully understand the needs of students, teachers, supervisors, and administrators.

Laurie Boswell, Ed.D., is the Head of School and a mathematics teacher at the Riverside School in Lyndonville, Vermont. Dr. Boswell is a recipient of the Presidential Award for Excellence in Mathematics Teaching and has taught mathematics to students at all levels, from elementary through college. Dr. Boswell was a Tandy Technology Scholar and served on the NCTM Board of Directors from 2002 to 2005. She currently serves on the board of NCSM and is a popular national speaker.

Dr. Ron Larson and Dr. Laurie Boswell began writing together in 1992. Since that time, they have authored over two dozen textbooks. In their collaboration, Ron is primarily responsible for the student edition while Laurie is primarily responsible for the teaching edition.

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Program Resources Print

Technology

Student Edition

Student Edition

Designed to the UDL Guidelines

Teaching Edition Laurie's Notes

Student Journal Available in English and Spanish

With complete English and Spanish audio

Dynamic eBook App Dynamic Solutions Tool Dynamic Investigations Lesson Tutorial Videos

Dynamic Classroom Vocabulary Flash Cards

Resources by Chapter Start Thinking Warm Up Cumulative Review Warm Up Practice A and B Enrichment and Extension Puzzle Time Family Communication Letters Available in English and Spanish

Assessment Book Performance Tasks Prerequisite Skills Test with Item Analysis Quarterly Standards Based Tests Quizzes Chapter Tests Alternative Assessments with Scoring Rubrics Pre-Course Test with Item Analysis Post Course Test with Item Analysis

Worked-Out Solutions Extra Examples Warm Up and Closure Activities

Dynamic Teaching Tools Interactive Whiteboard Lesson Library • Compatible with SMART®, Promethean®, and Mimio® technology

Real-Life STEM Videos Editable Online Resources • • • •

Lesson Plans Assessment Book Resources by Chapter Differentiating the Lesson

Answer Presentation Tool

Dynamic Assessment and Progress Monitoring Tool Assessment Creation and Delivery Progress Monitoring

Multilingual Glossary Key mathematical vocabulary terms in 14 languages

Copyright © Big Ideas Learning, LLC. All rights reserved.

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Program Overview Program Philosophy: Rigor and Balance with Real-Life Applications Essential Question

How can you use a formula for one measurement to write a formula for a different measurement?

The Big Ideas Math® program balances conceptual understanding with procedural fluency. Real-life applications help turn mathematical learning into an engaging and meaningful way to see and explore the real world.

Using an Area Formula Work with a partner. a. Write a formula for the area A of a parallelogram.

h = 5 in.

b. Substitute the given values into the formula. Then solve the equation for b. Justify each step.

Explorations and guiding Essential Questions encourage conceptual understanding.

1.5 Lesson

A = 30 in.2

b

c. Solve the formula in part (a) for b without first substituting values into the formula. Justify each step. d. Compare how you solved the equations in parts (b) and (c). How are the processes similar? How are they different?

What You Will Learn Rewrite literal equations.

Core Vocabul Vocabulary larry literal equation, p. 36 formula, p. 37 Previous surface area

Rewrite and use formulas for area. Rewrite and use other common formulas.

Rewriting Literal Equations An equation that has two or more variables is called a literal equation. To rewrite a literal equation, solve for one variable in terms of the other variable(s).

Rewriting a Literal Equation Solve the literal equation 3y + 4x = 9 for y.

SOLUTION 3y + 4x = 9

Write the equation.

3y + 4x − 4x = 9 − 4x

Subtract 4x from each side.

3y = 9 − 4x

Simplify.

3y 9 − 4x —=— 3 3

Divide each side by 3.

4 y = 3 − —x 3

Real-life applications provide students with opportunities to connect classroom lessons to realistic scenarios.

Simplify.

4 The rewritten literal equation is y = 3 − — x. 3

Direct instruction lessons allow for procedural fluency and provide the opportunity to use clear, precise mathematical language.

Using the Formula for Simple Interest You deposit $5000 in an account that earns simple interest. After 6 months, the account earns $162.50 in interest. What is the annual interest rate?

COMMON ERROR The unit of t is years. Be sure to convert months to years. 1 yr 12 mo



⋅ 6 mo = 0.5 yr

SOLUTION To find the annual interest rate, solve the simple interest formula for r. I = Prt I —=r Pt 162.50 (5000)(0.5)

—=r

0.065 = r

Write the simple interest formula. Divide each side by Pt to solve for r. Substitute 162.50 for I, 5000 for P, and 0.5 for t. Simplify.

The annual interest rate is 0.065, or 6.5%.

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Dynamic Technology Package The Big Ideas Math program includes a comprehensive technology package that enhances the curriculum and allows students to engage with the underlying mathematics in the text.

Student Edition The Dynamic Student Edition gives students access to the complete textbook and robust embedded digital resources. Interactive investigations, direct links to remediation, and additional resources are linked at point-of-use. Students can customize their Dynamic Student Editions through note taking and bookmarking, and it can also be accessed offline after it has been downloaded to a device. Audio support and Lesson Tutorial Videos are also included in English and Spanish.

Investigations Dynamic Investigations are powered by Desmos® and GeoGebra®. These interactivities expand on the Explorations in the program and allow students to learn mathematics through a hands-on approach. Teachers and students can integrate these investigations into their discovery learning.

Real-Life STEM Videos Every chapter in the program contains a Real-Life STEM Video that ties directly to a Performance Task. Students can explore topics like the speed of light, natural disasters, and wind power while applying their knowledge to a comprehensive project or task.

Classroom The Dynamic Classroom is an online interactive version of the Student Edition that can be used as a lesson presentation tool. Teachers can present their lessons and have point-of-use access to all of the online resources available that supplement every section of the program.

Teaching Tools These tools include an Interactive Whiteboard Lesson Library that includes customizable lessons and templates for every section in the program. Lessons are compatible with SMART®, Promethean®, and Mimio® whiteboards. The Answer Presentation Tool can be used to display worked-out solutions to homework and test problems from Big Ideas Math content.

Assessment and Progress Monitoring Tool This online tool allows teachers to create tests by standard or by Big Ideas Math content. Teachers can assign any exercise from the student textbook, problems from the program’s ancillary pieces, or additional items from the item bank, and students can complete their assignments within the tool’s interface. Copyright © Big Ideas Learning, LLC. All rights reserved.

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Big Ideas Math Integrated Mathematics Table of Contents Integrated Mathematics I Chapter 1

Solving Linear Equations

Chapter 7

Data Analysis and Displays

Chapter 2

Solving Linear Inequalities

Chapter 8

Basics of Geometry

Chapter 3

Graphing Linear Functions

Chapter 9

Reasoning and Proofs

Chapter 4

Writing Linear Functions

Chapter 10

Parallel and Perpendicular Lines

Chapter 5

Solving Systems of Linear Equations

Chapter 11

Transformations

Chapter 6

Exponential Functions and Sequences

Chapter 12

Congruent Triangles

Integrated Mathematics II Chapter 1

Functions and Exponents

Chapter 7

Quadrilaterals and Other Polygons

Chapter 2

Polynomial Equations and Factoring

Chapter 8

Similarity

Chapter 3

Graphing Quadratic Functions

Chapter 9

Right Triangles and Trigonometry

Chapter 4

Solving Quadratic Equations

Chapter 10

Circles

Chapter 5

Probability

Chapter 11

Circumference, Area, and Volume

Chapter 6

Relationships Within Triangles

Integrated Mathematics III Chapter 1

Geometric Modeling

Chapter 6

Rational Functions

Chapter 2

Linear and Quadratic Functions

Chapter 7

Sequences and Series

Chapter 3

Polynomial Functions

Chapter 8

Chapter 4

Rational Exponents and Radical Functions

Trigonometric Ratios and Functions

Chapter 9

Trigonometric Identities

Chapter 10

Data Analysis and Statistics

Chapter 5

Exponential and Logarithmic Functions

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Integrated Mathematics I Student Edition Chapter 1 Solving Linear Equations Copyright © Big Ideas Learning, LLC. All rights reserved.

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1 1.1 1.2 1.3 1.4 1.5

Solving Linear Equations Solving Simple Equations Solving Multi-Step Equations Solving Equations with Variables on Both Sides Solving Absolute Value Equations Rewriting Equations and Formulas

Density of Pyrite (p. 41)

SEE the Big Idea Cheerleading Competition (p. 29)

Boat (p. 22)

Biking (p. 14)

Average Speed (p. 6)

Copyright © Big Ideas Learning, LLC. All rights reserved.

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Maintaining Mathematical Proficiency Adding and Subtracting Integers Example 1

(Grade 7)

Evaluate 4 + (−12). ∣ −12 ∣ > ∣ 4 ∣. So, subtract ∣ 4 ∣ from ∣ −12 ∣.

4 + (−12) = −8 Use the sign of −12.

Example 2 Evaluate −7 − (−16). −7 − (−16) = −7 + 16

Add the opposite of −16.

=9

Add.

Add or subtract. 1. −5 + (−2)

2. 0 + (−13)

3. −6 + 14

4. 19 − (−13)

5. −1 − 6

6. −5 − (−7)

7. 17 + 5

8. 8 + (−3)

9. 11 − 15

Multiplying and Dividing Integers Example 3

(Grade 7)



Evaluate −3 (−5). The integers have the same sign.



−3 (−5) = 15 The product is positive.

Example 4

Evaluate 15 ÷ (−3). The integers have different signs.

15 ÷ (−3) = −5 The quotient is negative.

Multiply or divide.





10. −3 (8)

11. −7 (−9)

13. −24 ÷ (−6)

14. —

15. 12 ÷ (−3)

17. 36 ÷ 6

18. −3(−4)



16. 6 8

−16 2

12. 4 (−7)

19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers,

(c) multiplying integers, and (d) dividing integers. Give an example of each.

Dynamic Solutions available at BigIdeasMath.com

1

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Mathematical Practices

Mathematically proficient students carefully specify units of measure.

Specifying Units of Measure

Core Concept Operations and Unit Analysis Addition and Subtraction

When you add or subtract quantities, they must have the same units of measure. The sum or difference will have the same unit of measure. Example

Perimeter of rectangle = (3 ft) + (5 ft) + (3 ft) + (5 ft)

3 ft

= 16 feet

5 ft

When you add feet, you get feet.

Multiplication and Division

When you multiply or divide quantities, the product or quotient will have a different unit of measure. Example

Area of rectangle = (3 ft) × (5 ft) = 15 square feet

When you multiply feet, you get feet squared, or square feet.

Specifying Units of Measure You work 8 hours and earn $72. What is your hourly wage?

SOLUTION dollars per hour

dollars per hour

Hourly wage = $72 ÷ 8 h ($ per h) = $9 per hour

The units on each side of the equation balance. Both are specified in dollars per hour.

Your hourly wage is $9 per hour.

Monitoring Progress Solve the problem and specify the units of measure. 1. The population of the United States was about 280 million in 2000 and about

310 million in 2010. What was the annual rate of change in population from 2000 to 2010? 2. You drive 240 miles and use 8 gallons of gasoline. What was your car’s gas mileage

(in miles per gallon)? 3. A bathtub is in the shape of a rectangular prism. Its dimensions are 5 feet by 3 feet by

18 inches. The bathtub is three-fourths full of water and drains at a rate of 1 cubic foot per minute. About how long does it take for all the water to drain?

2

Chapter 1

Solving Linear Equations Copyright © Big Ideas Learning, LLC. All rights reserved.

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1.1

Solving Simple Equations Essential Question

How can you use simple equations to solve

real-life problems? Measuring Angles Work with a partner. Use a protractor to measure the angles of each quadrilateral. Copy and complete the table to organize your results. (The notation m∠ A denotes the measure of angle A.) How precise are your measurements? a.

A

b.

B

B

c. A

A B

C

D

UNDERSTANDING MATHEMATICAL TERMS

Quadrilateral

A conjecture is an a. unproven statement about a general b. mathematical concept. c. After the statement is proven, it is called a rule or a theorem. You will learn more about these terms in Chapter 9.

D

m∠A (degrees)

m∠B (degrees)

D

C

m∠C (degrees)

C

m∠A + m∠B + m∠C + m∠D

m∠D (degrees)

Making a Conjecture Work with a partner. Use the completed table in Exploration 1 to write a conjecture about the sum of the angle measures of a quadrilateral. Draw three quadrilaterals that are different from those in Exploration 1 and use them to justify your conjecture.

Applying Your Conjecture

CONNECTIONS TO GEOMETRY You will learn more about angle measures of quadrilaterals in a future course.

Work with a partner. Use the conjecture you wrote in Exploration 2 to write an equation for each quadrilateral. Then solve the equation to find the value of x. Use a protractor to check the reasonableness of your answer. a.

b.

85°

c. 78°

30°

100° x° 90° 80°



72°

x° 60°

90°

Communicate Your Answer 4. How can you use simple equations to solve real-life problems? 5. Draw your own quadrilateral and cut it out. Tear off the four corners of

the quadrilateral and rearrange them to affirm the conjecture you wrote in Exploration 2. Explain how this affirms the conjecture. Section 1.1

Solving Simple Equations

3

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1.1

Lesson

What You Will Learn Solve linear equations using addition and subtraction. Solve linear equations using multiplication and division.

Core Vocabul Vocabulary larry

Use linear equations to solve real-life problems.

conjecture, p. 3 rule, p. 3 theorem, p. 3 equation, p. 4 linear equation in one variable, p. 4 solution, p. 4 inverse operations, p. 4 equivalent equations, p. 4 Previous expression

Solving Linear Equations by Adding or Subtracting An equation is a statement that two expressions are equal. A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a and b are constants and a ≠ 0. When you solve an equation, you use properties of real numbers to find a solution, which is a value that makes the equation true. Inverse operations are two operations that undo each other, such as addition and subtraction. When you perform the same inverse operation on each side of an equation, you produce an equivalent equation. Equivalent equations are equations that have the same solution(s).

Core Concept Addition, Subtraction, and Substitution Properties of Equality

CONNECTIONS TO GEOMETRY

Let a, b, and c be real numbers.

Segment lengths and angle measures are real numbers. You will use these properties later in the book to write logical arguments about geometric figures.

Addition Property of Equality

If a = b, then a + c = b + c.

Subtraction Property of Equality

If a = b, then a − c = b − c.

Substitution Property of Equality

If a = b, then a can be substituted for b (or b for a) in any equation or expression.

Solving Equations by Addition or Subtraction Solve each equation. Justify each step. Check your answer. a. x − 3 = −5

b. 0.9 = y + 2.8

SOLUTION a. x − 3 = −5 +3

Addition Property of Equality

Write the equation.

+3

Check x − 3 = −5 ? − 2 − 3 = −5

Add 3 to each side.

x = −2

Simplify.

−5 = −5

The solution is x = −2.

b. Subtraction Property of Equality

0.9 = y + 2.8 − 2.8

− 2.8

−1.9 = y

Write the equation.

Check

Subtract 2.8 from each side.

0.9 = y + 2.8 ? 0.9 = −1.9 + 2.8

Simplify.

The solution is y = −1.9.

Monitoring Progress



0.9 = 0.9



Help in English and Spanish at BigIdeasMath.com

Solve the equation. Justify each step. Check your solution. 1. n + 3 = −7

4

Chapter 1

1

2

2. g − —3 = −—3

3. −6.5 = p + 3.9

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Solving Linear Equations by Multiplying or Dividing Just as addition and subtraction are inverse operations, multiplication and division are also inverse operations.

Core Concept Multiplication and Division Properties of Equality Let a, b, and c be real numbers. Multiplication Property of Equality Division Property of Equality





If a = b, then a c = b c, c ≠ 0. a b If a = b, then — = — , c ≠ 0. c c

Solving Equations by Multiplication or Division Solve each equation. Justify each step. Check your answer. n a. −— = −3 5

b. π x = −2π

c. 1.3z = 5.2

SOLUTION n −— = −3 5

a. Multiplication Property of Equality

−5

⋅( )

Write the equation.



n − — = −5 (− 3) 5

Check

n −— = −3 5 15 ? −— = −3 5 −3 = −3

Multiply each side by − 5.

n = 15

Simplify.

The solution is n = 15.

b. π x = −2π Division Property of Equality

πx π

Write the equation.

−2π π

—=—

Divide each side by π.

x = −2



Check π x = −2π ? π (−2) = −2π − 2π = −2π

Simplify.

The solution is x = −2.

c. 1.3z = 5.2 Division Property of Equality

1.3z 5.2 —=— 1.3 1.3 z=4

Write the equation.

Check 1.3z = 5.2 ? 1.3(4) = 5.2

Divide each side by 1.3. Simplify.

5.2 = 5.2

The solution is z = 4.

Monitoring Progress





Help in English and Spanish at BigIdeasMath.com

Solve the equation. Justify each step. Check your solution. y 3

4. — = −6

5. 9π = π x

Section 1.1

6. 0.05w = 1.4

Solving Simple Equations

5

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Solving Real-Life Problems MODELING WITH MATHEMATICS Mathematically proficient students routinely check that their solutions make sense in the context of a real-life problem.

Core Concept Four-Step Approach to Problem Solving 1.

Understand the Problem What is the unknown? What information is being given? What is being asked?

2.

Make a Plan This plan might involve one or more of the problem-solving strategies shown on the next page.

3.

Solve the Problem Carry out your plan. Check that each step is correct.

4.

Look Back Examine your solution. Check that your solution makes sense in the original statement of the problem.

Modeling with Mathematics In the 2012 Olympics, Usain Bolt won the 200-meter dash with a time of 19.32 seconds. Write and solve an equation to find his average speed to the nearest hundredth of a meter per second.

REMEMBER

SOLUTION

The formula that relates distance d, rate or speed r, and time t is d = rt.

1. Understand the Problem You know the winning time and the distance of the race. You are asked to find the average speed to the nearest hundredth of a meter per second. 2. Make a Plan Use the Distance Formula to write an equation that represents the problem. Then solve the equation. 3. Solve the Problem

⋅ 200 = r ⋅ 19.32 d=r t

REMEMBER The symbol ≈ means “approximately equal to.”

200 19.32

19.32r 19.32

Write the Distance Formula. Substitute 200 for d and 19.32 for t.

—=—

Divide each side by 19.32.

10.35 ≈ r

Simplify.

Bolt’s average speed was about 10.35 meters per second. 4. Look Back Round Bolt’s average speed to 10 meters per second. At this speed, it would take 200 m 10 m/sec

— = 20 seconds

to run 200 meters. Because 20 is close to 19.32, your solution is reasonable.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

7. Suppose Usain Bolt ran 400 meters at the same average speed that he ran the

200 meters. How long would it take him to run 400 meters? Round your answer to the nearest hundredth of a second.

6

Chapter 1

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Core Concept Common Problem-Solving Strategies Use a verbal model.

Guess, check, and revise.

Draw a diagram.

Sketch a graph or number line.

Write an equation.

Make a table.

Look for a pattern.

Make a list.

Work backward.

Break the problem into parts.

Modeling with Mathematics On January 22, 1943, the temperature in Spearfish, South Dakota, fell from 54°F at 9:00 a.m. to − 4°F at 9:27 a.m. How many degrees did the temperature fall?

SOLUTION 1. Understand the Problem You know the temperature before and after the temperature fell. You are asked to find how many degrees the temperature fell. 2. Make a Plan Use a verbal model to write an equation that represents the problem. Then solve the equation. 3. Solve the Problem Words

Number of degrees Temperature Temperature = − the temperature fell at 9:27 a.m. at 9:00 a.m.

Variable

Let T be the number of degrees the temperature fell.

Equation

−4

=



54

−4 = 54 − T

T Write the equation.

−4 − 54 = 54 − 54 − T −58 = − T

Subtract 54 from each side. Simplify.

58 = T

Divide each side by − 1.

The temperature fell 58°F.

REMEMBER The distance between two points on a number line is always positive.

4. Look Back The temperature fell from 54 degrees above 0 to 4 degrees below 0. You can use a number line to check that your solution is reasonable. 58 −8 −4

0

4

8 12 16 20 24 28 32 36 40 44 48 52 56 60

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

8. You thought the balance in your checking account was $68. When your bank

statement arrives, you realize that you forgot to record a check. The bank statement lists your balance as $26. Write and solve an equation to find the amount of the check that you forgot to record.

Section 1.1

Solving Simple Equations

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1.1

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY Which of the operations +, −, ×, and ÷ are inverses of each other? 2. VOCABULARY Are the equations − 2x = 10 and −5x = 25 equivalent? Explain. 3. WRITING Which property of equality allows you to check a solution of an equation? Explain. 4. WHICH ONE DOESN’T BELONG? Which expression does not belong with the other three? Explain

your reasoning. x 8=— 2

3=x÷4

x 3

—=9

x−6=5

Monitoring Progress and Modeling with Mathematics In Exercises 5–14, solve the equation. Justify each step. Check your solution. (See Example 1.)

USING TOOLS The sum of the angle measures of a

5. x + 5 = 8

6. m + 9 = 2

quadrilateral is 360°. In Exercises 17–20, write and solve an equation to find the value of x. Use a protractor to check the reasonableness of your answer.

7. y − 4 = 3

8. s − 2 = 1

17.

9. w + 3 = −4

10. n − 6 = −7

11. −14 = p − 11

12. 0 = 4 + q

13. r + (−8) = 10



x° 150°

100° 77°

48°

120° 100°

14. t − (−5) = 9

15. MODELING WITH MATHEMATICS A discounted

18.

19. 76°

amusement park ticket costs $12.95 less than the original price p. Write and solve an equation to find the original price.



20. 115°

122°

85°



92° 60°

In Exercises 21–30, solve the equation. Justify each step. Check your solution. (See Example 2.)

16. MODELING WITH MATHEMATICS You and a friend

are playing a board game. Your final score x is 12 points less than your friend’s final score. Write and solve an equation to find your final score. ROUND

9

ROUND

10

FINAL SCORE

Your Friend You

8

Chapter 1

21. 5g = 20

22. 4q = 52

23. p ÷ 5 = 3

24. y ÷ 7 = 1

25. −8r = 64

26. x ÷ (−2) = 8

x 6

w −3

27. — = 8

28. — = 6

29. −54 = 9s

30. −7 = —

t 7

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In Exercises 31– 38, solve the equation. Check your solution. 31.

3 —2

+t=

1 —2

32. b −

3 — 16

=

45. REASONING Identify the property of equality that

makes Equation 1 and Equation 2 equivalent.

5 — 16

33. —37 m = 6

34. −—5 y = 4

Equation 1

1 x x−—=—+3 2 4

35. 5.2 = a − 0.4

36. f + 3π = 7π

Equation 2

4x − 2 = x + 12

37. − 108π = 6π j

38. x ÷ (−2) = 1.4

2

46. PROBLEM SOLVING Tatami mats are used as a floor ERROR ANALYSIS In Exercises 39 and 40, describe and

correct the error in solving the equation. 39.



covering in Japan. One possible layout uses four identical rectangular mats and one square mat, as shown. The area of the square mat is half the area of one of the rectangular mats.

−0.8 + r = 12.6 r = 12.6 + (−0.8) r = 11.8

40.



3

Total area = 81 ft2

m −— = −4 3 m −— = 3 (−4) 3 m = −12

⋅( )



41. ANALYZING RELATIONSHIPS A baker orders 162 eggs.

Each carton contains 18 eggs. Which equation can you use to find the number x of cartons? Explain your reasoning and solve the equation.

A 162x = 18 ○

x B — = 162 ○ 18

C 18x = 162 ○

D x + 18 = 162 ○

MODELING WITH MATHEMATICS In Exercises 42– 44, write and solve an equation to answer the question. (See Examples 3 and 4.) 42. The temperature at 5 p.m. is 20°F. The temperature

at 10 p.m. is −5°F. How many degrees did the temperature fall?

a. Write and solve an equation to find the area of one rectangular mat. b. The length of a rectangular mat is twice the width. Use Guess, Check, and Revise to find the dimensions of one rectangular mat. 47. PROBLEM SOLVING You spend $30.40 on 4 CDs.

Each CD costs the same amount and is on sale for 80% of the original price. a. Write and solve an equation to find how much you spend on each CD. b. The next day, the CDs are no longer on sale. You have $25. Will you be able to buy 3 more CDs? Explain your reasoning. 48. ANALYZING RELATIONSHIPS As c increases, does

the value of x increase, decrease, or stay the same for each equation? Assume c is positive.

43. The length of an

American flag is 1.9 times its width. What is the width of the flag?

Equation 9.5 ft

Value of x

x−c=0

44. The balance of an investment account is $308 more

cx = 1

than the balance 4 years ago. The current balance of the account is $4708. What was the balance 4 years ago?

cx = c x c

—=1

Section 1.1

Solving Simple Equations

9

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49. USING STRUCTURE Use the values −2, 5, 9, and 10

MATHEMATICAL CONNECTIONS In Exercises 53–56, find

the height h or the area of the base B of the solid.

to complete each statement about the equation ax = b − 5.

53.

54.

a. When a = ___ and b = ___, x is a positive integer.

h

b. When a = ___ and b = ___, x is a negative integer.

7 in. B = 147 cm2

B

50. HOW DO YOU SEE IT? The circle graph shows the

Volume = 84π in.3

percents of different animals sold at a local pet store in 1 year. 55. Hamster: 5%

Volume = 1323 cm3 56.

5m

h

Rabbit: 9% Bird: 7%

B = 30 ft 2

B

Dog: 48%

Volume = 15π m3

Cat: x%

Volume = 35 ft3

57. MAKING AN ARGUMENT In baseball, a player’s

batting average is calculated by dividing the number of hits by the number of at-bats. The table shows Player A’s batting average and number of at-bats for three regular seasons.

a. What percent is represented by the entire circle? b. How does the equation 7 + 9 + 5 + 48 + x = 100 relate to the circle graph? How can you use this equation to find the percent of cats sold?

Season

Batting average

At-bats

2010

.312

596

2011

.296

446

2012

.295

599

51. REASONING One-sixth of the girls and two-sevenths

of the boys in a school marching band are in the percussion section. The percussion section has 6 girls and 10 boys. How many students are in the marching band? Explain.

a. How many hits did Player A have in the 2011 regular season? Round your answer to the nearest whole number. b. Player B had 33 fewer hits in the 2011 season than Player A but had a greater batting average. Your friend concludes that Player B had more at-bats in the 2011 season than Player A. Is your friend correct? Explain.

52. THOUGHT PROVOKING Write a real-life problem

that can be modeled by an equation equivalent to the equation 5x = 30. Then solve the equation and write the answer in the context of your real-life problem.

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Use the Distributive Property to simplify the expression.

(Skills Review Handbook)

(

1

59. —56 x + —2 + 4

58. 8(y + 3)

)

60. 5(m + 3 + n)

61. 4(2p + 4q + 6)

Copy and complete the statement. Round to the nearest hundredth, if necessary. (Skills Review Handbook) 5L min

L

h

63. — ≈ —

7 gal min

qt sec

65. — ≈ —

62. — = — 64. — ≈ —

10

Chapter 1

68 mi h

mi sec

8 km min

h

mi

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1.2

Solving Multi-Step Equations Essential Question

How can you use multi-step equations to solve

real-life problems? Solving for the Angle Measures of a Polygon Work with a partner. The sum S of the angle measures of a polygon with n sides can be found using the formula S = 180(n − 2). Write and solve an equation to find each value of x. Justify the steps in your solution. Then find the angle measures of each polygon. How can you check the reasonableness of your answers? a.

b.

c.

(30 + x)°

50° (2x + 30)° (x + 10)°

9x°

JUSTIFYING CONCLUSIONS To be proficient in math, you need to be sure your answers make sense in the context of the problem. For instance, if you find the angle measures of a triangle, and they have a sum that is not equal to 180°, then you should check your work for mistakes.

30°

(2x + 20)°

(x + 20)°

50°



d.

(x − 17)°

e.

(x + 35)°

(5x + 2)° (3x + 5)°

(2x + 8)°

f.

(3x + 16)°

(8x + 8)° (5x + 10)°

(x + 42)° x°

(4x − 18)° (3x − 7)°

(4x + 15)° (2x + 25)°

Writing a Multi-Step Equation Work with a partner.

CONNECTIONS TO GEOMETRY You will learn more about angle measures of polygons in a future course.

a. Draw an irregular polygon. b. Measure the angles of the polygon. Record the measurements on a separate sheet of paper. c. Choose a value for x. Then, using this value, work backward to assign a variable expression to each angle measure, as in Exploration 1. d. Trade polygons with your partner. e. Solve an equation to find the angle measures of the polygon your partner drew. Do your answers seem reasonable? Explain.

Communicate Your Answer 3. How can you use multi-step equations to solve real-life problems? 4. In Exploration 1, you were given the formula for the sum S of the angle measures

of a polygon with n sides. Explain why this formula works. 5. The sum of the angle measures of a polygon is 1080º. How many sides does the

polygon have? Explain how you found your answer. Section 1.2

Solving Multi-Step Equations

11

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1.2 Lesson

What You Will Learn Solve multi-step linear equations using inverse operations. Use multi-step linear equations to solve real-life problems.

Core Vocabul Vocabulary larry

Use unit analysis to model real-life problems.

Previous inverse operations mean

Solving Multi-Step Linear Equations

Core Concept Solving Multi-Step Equations To solve a multi-step equation, simplify each side of the equation, if necessary. Then use inverse operations to isolate the variable.

Solving a Two-Step Equation Solve 2.5x − 13 = 2. Check your solution.

SOLUTION 2.5x − 13 = + 13

Undo the subtraction.

Write the equation.

2 + 13

2.5x =

Add 13 to each side.

2.5x 15 —= — 2.5 2.5

Undo the multiplication.

Check

Simplify.

15

2.5x − 13 = 2 ? 2.5(6) − 13 = 2

Divide each side by 2.5.

x=6

2=2

Simplify.



The solution is x = 6.

Combining Like Terms to Solve an Equation Solve −12 = 9x − 6x + 15. Check your solution.

SOLUTION

Undo the addition.

Undo the multiplication.

−12 = 9x − 6x + 15

Write the equation.

−12 = 3x + 15

Combine like terms.

− 15

Subtract 15 from each side.

− 15

−27 = 3x

Simplify.

−27 3x —=— 3 3

Divide each side by 3.

−9 = x

Check

Simplify.

− 12 = − 12

The solution is x = −9.

Monitoring Progress

− 12 = 9x − 6x + 15 ? − 12 = 9(− 9) − 6(− 9) + 15



Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solution. 1. −2n + 3 = 9

12

Chapter 1

2. −21 = —12 c − 11

3. −2x − 10x + 12 = 18

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Using Structure to Solve a Multi-Step Equation Solve 2(1 − x) + 3 = − 8. Check your solution.

SOLUTION Method 1 One way to solve the equation is by using the Distributive Property. 2(1 − x) + 3 = −8 2(1) − 2(x) + 3 = −8

REMEMBER

2 − 2x + 3 = −8

The Distributive Property states the following for real numbers a, b, and c.

−2x + 5 = −8 −5

Sum a(b + c) = ab + ac

−5

Distributive Property Multiply. Combine like terms. Subtract 5 from each side.

−2x = −13

Simplify.

−2x −2

Divide each side by −2.

−13 −2

—=—

Difference a(b − c) = ab − ac

Write the equation.

x = 6.5 The solution is x = 6.5.

Simplify.

Check 2(1 − x) + 3 = − 8 ? 2(1− 6.5) + 3 = − 8 −8 = −8



Method 2 Another way to solve the equation is by interpreting the expression 1 − x as a single quantity. 2(1 − x) + 3 = −8

LOOKING FOR STRUCTURE

−3

First solve for the expression 1 − x, and then solve for x.

Write the equation.

−3

Subtract 3 from each side.

2(1 − x) = −11

Simplify.

2(1 − x) 2

Divide each side by 2.

−11 2

—=—

1 − x = −5.5 −1

−1

Simplify. Subtract 1 from each side.

−x = −6.5

Simplify.

−x −1

Divide each side by − 1.

−6.5 −1

—=—

x = 6.5

Simplify.

The solution is x = 6.5, which is the same solution obtained in Method 1.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solution. 4. 3(x + 1) + 6 = −9

5. 15 = 5 + 4(2d − 3)

6. 13 = −2(y − 4) + 3y

7. 2x(5 − 3) − 3x = 5

8. −4(2m + 5) − 3m = 35

9. 5(3 − x) + 2(3 − x) = 14

Section 1.2

Solving Multi-Step Equations

13

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Solving Real-Life Problems Modeling with Mathematics U the table to find the number of miles x Use yyou need to bike on Friday so that the mean nnumber of miles biked per day is 5.

SOLUTION S

Day

Miles

Monday

3.5

Tuesday

5.5

Wednesday

0

Thursday

5

Friday

x

11. Understand the Problem You know how many miles you biked Monday through Thursday. You are asked to find the number of miles you need to bike on Friday so that the mean number of miles biked per day is 5. 2. 2 Make a Plan Use the definition of mean to write an equation that represents the problem. Then solve the equation. 3. 3 Solve the Problem The mean of a data set is the sum of the data divided by the number of data values. 3.5 + 5.5 + 0 + 5 + x 5

Write the equation.

14 + x 5

Combine like terms.

—— = 5 —=5

14 + x 5 —=5 5 5



14 + x = − 14



Multiply each side by 5.

25

Simplify.

− 14

Subtract 14 from each side.

x = 11

Simplify.

You need to bike 11 miles on Friday. 4. Look Back Notice that on the days that you did bike, the values are close to the mean. Because you did not bike on Wednesday, you need to bike about twice the mean on Friday. Eleven miles is about twice the mean. So, your solution is reasonable.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

1

10. The formula d = —2 n + 26 relates the nozzle pressure n (in pounds per square

inch) of a fire hose and the maximum horizontal distance the water reaches d (in feet). How much pressure is needed to reach a fire 50 feet away?

d

14

Chapter 1

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REMEMBER When you add miles to miles, you get miles. But, when you divide miles by days, you get miles per day.

Using Unit Analysis to Model Real-Life Problems When you write an equation to model a real-life problem, you should check that the units on each side of the equation balance. For instance, in Example 4, notice how the units balance. miles

miles per day

3.5 + 5.5 + 0 + 5 + x 5

mi day

—— = 5

per

mi day

—=—



days

Solving a Real-Life Problem Your school’s drama club charges $4 per person for admission to a play. The club borrowed $400 to pay for costumes and props. After paying back the loan, the club has a profit of $100. How many people attended the play?

SOLUTION 1. Understand the Problem You know how much the club charges for admission. You also know how much the club borrowed and its profit. You are asked to find how many people attended the play. 2. Make a Plan Use a verbal model to write an equation that represents the problem. Then solve the equation. 3. Solve the Problem

REMEMBER When you multiply dollars per person by people, you get dollars.

⋅ who attended

Words

Ticket price

Variable

Let x be the number of people who attended.

Equation



$4 person

Number of people



Amount = Profit of loan

⋅ x people − $400 = $100

4x − 400 = 100

$=$



Write the equation.

4x − 400 + 400 = 100 + 400

Add 400 to each side.

4x = 500

Simplify.

4x 4

Divide each side by 4.

500 4

—=—

x = 125

Simplify.

So, 125 people attended the play. 4. Look Back To check that your solution is reasonable, multiply $4 per person by 125 people. The result is $500. After paying back the $400 loan, the club has $100, which is the profit.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

11. You have 96 feet of fencing to enclose a rectangular pen for your dog. To provide

sufficient running space for your dog to exercise, the pen should be three times as long as it is wide. Find the dimensions of the pen.

Section 1.2

Solving Multi-Step Equations

15

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1.2

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE To solve the equation 2x + 3x = 20, first combine 2x and 3x because

they are _________. 2. WRITING Describe two ways to solve the equation 2(4x − 11) = 10.

Monitoring Progress and Modeling with Mathematics In Exercises 3−14, solve the equation. Check your solution. (See Examples 1 and 2.)

23. −3(3 + x) + 4(x − 6) = − 4

3. 3w + 7 = 19

4. 2g − 13 = 3

24. 5(r + 9) − 2(1 − r) = 1

5. 11 = 12 − q

6. 10 = 7 − m

USING TOOLS In Exercises 25−28, find the value of the

z −4

a 3

7. 5 = — − 3

8. — + 4 = 6

h+6 5

d−8 −2

9. — = 2

10. — = 12

11. 8y + 3y = 44

variable. Then find the angle measures of the polygon. Use a protractor to check the reasonableness of your answer. 25.

26. a°

2k°

12. 36 = 13n − 4n

45°

2a° 2a°



Sum of angle measures: 180°

13. 12v + 10v + 14 = 80

a° Sum of angle measures: 360°

14. 6c − 8 − 2c = −16 15. MODELING WITH MATHEMATICS The altitude a

(in feet) of a plane t minutes after liftoff is given by a = 3400t + 600. How many minutes after liftoff is the plane at an altitude of 21,000 feet?

16. MODELING WITH MATHEMATICS A repair bill for

your car is $553. The parts cost $265. The labor cost is $48 per hour. Write and solve an equation to find the number of hours of labor spent repairing the car. In Exercises 17−24, solve the equation. Check your solution. (See Example 3.) 17. 4(z + 5) = 32

18. − 2(4g − 3) = 30

19. 6 + 5(m + 1) = 26

20. 5h + 2(11 − h) = − 5

27.

b° 3 b° 2

(b + 45)°

(2b − 90)° 90° Sum of angle measures: 540°

28. 120° x°

100°

120° 120°

(x + 10)° Sum of angle measures: 720°

In Exercises 29−34, write and solve an equation to find the number. 29. The sum of twice a number and 13 is 75. 30. The difference of three times a number and 4 is −19. 31. Eight plus the quotient of a number and 3 is −2. 32. The sum of twice a number and half the number is 10.

21. 27 = 3c − 3(6 − 2c)

33. Six times the sum of a number and 15 is − 42.

22. −3 = 12y − 5(2y − 7) 34. Four times the difference of a number and 7 is 12.

16

Chapter 1

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USING EQUATIONS In Exercises 35−37, write and solve

ERROR ANALYSIS In Exercises 40 and 41, describe and

an equation to answer the question. Check that the units on each side of the equation balance. (See Examples 4 and 5.)

correct the error in solving the equation. 40.

35. During the summer, you work 30 hours per week at

a gas station and earn $8.75 per hour. You also work as a landscaper for $11 per hour and can work as many hours as you want. You want to earn a total of $400 per week. How many hours must you work as a landscaper?

deep end

shallow end

d

9 ft

−2(7 − y) + 4 = −4 −14 − 2y + 4 = −4 −10 − 2y = −4 −2y = 6 y = −3

36. The area of the surface of the swimming pool is

210 square feet. What is the length d of the deep end (in feet)?



41.



1 4

—(x − 2) + 4 = 12

1 4

—(x − 2) = 8

x−2=2 x=4

10 ft

MATHEMATICAL CONNECTIONS In Exercises 42−44,

37. You order two tacos and a salad. The salad costs

$2.50. You pay 8% sales tax and leave a $3 tip. You pay a total of $13.80. How much does one taco cost?

write and solve an equation to answer the question. 42. The perimeter of the tennis court is 228 feet. What are

the dimensions of the court?

JUSTIFYING STEPS In Exercises 38 and 39, justify each step of the solution.

1 2

38. −—(5x − 8) − 1 = 6

w

Write the equation.

1 −—(5x − 8) = 7 2

2w + 6 43. The perimeter of the Norwegian flag is 190 inches.

What are the dimensions of the flag?

5x − 8 = −14 5x = −6

y

6 x = −— 5 39.

2(x + 3) + x = −9

11 y 8

Write the equation.

2(x) + 2(3) + x = −9

44. The perimeter of the school crossing sign is

102 inches. What is the length of each side?

2x + 6 + x = −9 s+6

3x + 6 = −9

s+6

3x = −15 s

x = −5

s 2s

Section 1.2

Solving Multi-Step Equations

17

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45. COMPARING METHODS Solve the equation

2(4 − 8x) + 6 = −1 using (a) Method 1 from Example 3 and (b) Method 2 from Example 3. Which method do you prefer? Explain. 46. PROBLEM SOLVING An online ticket agency charges

the amounts shown for basketball tickets. The total cost for an order is $220.70. How many tickets are purchased?

49. REASONING An even integer can be represented by

the expression 2n, where n is any integer. Find three consecutive even integers that have a sum of 54. Explain your reasoning. 50. HOW DO YOU SEE IT? The scatter plot shows the

attendance for each meeting of a gaming club.

Charge

Amount

Ticket price

$32.50 per ticket

Convenience charge

$3.30 per ticket

Processing charge

$5.90 per order

Students

Gaming Club Attendance y 25 20 15 10 5 0

18

1

48. THOUGHT PROVOKING You teach a math class and

assign a weight to each component of the class. You determine final grades by totaling the products of the weights and the component scores. Choose values for the remaining weights and find the necessary score on the final exam for a student to earn an A (90%) in the class, if possible. Explain your reasoning. Component

Student’s Weight score

Score × Weight

92% × 0.20 = 18.4%

3

4

x

a. The mean attendance for the first four meetings is 20. Is the number of students who attended the fourth meeting greater than or less than 20? Explain. b. Estimate the number of students who attended the fourth meeting. c. Describe a way you can check your estimate in part (b).

REASONING In Exercises 51−56, the letters a, b, and c

represent nonzero constants. Solve the equation for x.

Class Participation

92%

Homework

95%

52. x + a = —4

Midterm Exam

88%

53. ax − b = 12.5

0.20

2

17

Meeting

47. MAKING AN ARGUMENT You have quarters and

dimes that total $2.80. Your friend says it is possible that the number of quarters is 8 more than the number of dimes. Is your friend correct? Explain.

21

51. bx = −7 3

Final Exam

54. ax + b = c

Total

1 55. 2bx − bx = −8 56. cx − 4b = 5b

Maintaining Mathematical Proficiency Simplify the expression. 57. 4m + 5 − 3m

Reviewing what you learned in previous grades and lessons

(Skills Review Handbook) 58. 9 − 8b + 6b

59. 6t + 3(1 − 2t) − 5

Determine whether (a) x = −1 or (b) x = 2 is a solution of the equation.

18

(Skills Review Handbook)

60. x − 8 = − 9

61. x + 1.5 = 3.5

62. 2x − 1 = 3

63. 3x + 4 = 1

64. x + 4 = 3x

65. − 2(x − 1) = 1 − 3x

Chapter 1

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1.3

Solving Equations with Variables on Both Sides Essential Question

How can you solve an equation that has

variables on both sides? Perimeter Work with a partner. The two polygons have the same perimeter. Use this information to write and solve an equation involving x. Explain the process you used to find the solution. Then find the perimeter of each polygon. 3 x 2

x

5

5

3 5

2

2

x

CONNECTIONS TO GEOMETRY

4

Perimeter and Area Work with a partner.

You will learn about finding perimeters and areas of polygons in the coordinate plane in Chapter 8.



Each figure has the unusual property that the value of its perimeter (in feet) is equal to the value of its area (in square feet). Use this information to write an equation for each figure.

• •

Solve each equation for x. Explain the process you used to find the solution.

a.

Find the perimeter and area of each figure.

5

3

b.

5

4

LOOKING FOR STRUCTURE To be proficient in math, you need to visualize complex things, such as composite figures, as being made up of simpler, more manageable parts.

c.

1

2

6

2

x

x x

Communicate Your Answer 3. How can you solve an equation that has variables on both sides? 4. Write three equations that have the variable x on both sides. The equations should

be different from those you wrote in Explorations 1 and 2. Have your partner solve the equations. Section 1.3

Solving Equations with Variables on Both Sides

19

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1.3 Lesson

What You Will Learn Solve linear equations that have variables on both sides.

Core Vocabul Vocabulary larry

Identify special solutions of linear equations. Use linear equations to solve real-life problems.

identity, p. 21 Previous inverse operations

Solving Equations with Variables on Both Sides

Core Concept Solving Equations with Variables on Both Sides To solve an equation with variables on both sides, simplify one or both sides of the equation, if necessary. Then use inverse operations to collect the variable terms on one side, collect the constant terms on the other side, and isolate the variable.

Solving an Equation with Variables on Both Sides Solve 10 − 4x = −9x. Check your solution.

SOLUTION 10 − 4x = −9x + 4x

Write the equation.

+ 4x

Add 4x to each side.

10 = − 5x

Simplify.

10 −5x —=— −5 −5

Divide each side by −5.

−2 = x

Simplify.

Check 10 − 4x = −9x ? 10 − 4(−2) = −9(−2) 18 = 18



The solution is x = −2.

Solving an Equation with Grouping Symbols 1 Solve 3(3x − 4) = —(32x + 56). 4

SOLUTION 3(3x − 4) = 9x − 12 =

— (32x + 56)

1 4

Write the equation.

8x + 14

Distributive Property

+ 12

Add 12 to each side.

+ 12 9x = − 8x

8x + 26

Simplify.

− 8x

Subtract 8x from each side.

x = 26

Simplify.

The solution is x = 26.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solution. 1. −2x = 3x + 10

20

Chapter 1

2. —12 (6h − 4) = −5h + 1

3

3. −—4 (8n + 12) = 3(n − 3)

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Identifying Special Solutions of Linear Equations

Core Concept Special Solutions of Linear Equations Equations do not always have one solution. An equation that is true for all values of the variable is an identity and has infinitely many solutions. An equation that is not true for any value of the variable has no solution.

Identifying the Number of Solutions

REASONING The equation 15x + 6 = 15x is not true because the number 15x cannot be equal to 6 more than itself.

Solve each equation. a. 3(5x + 2) = 15x

b. −2(4y + 1) = −8y − 2

SOLUTION a. 3(5x + 2) =

15x

Write the equation.

15x + 6 =

15x

Distributive Property

− 15x

− 15x 6=0

Subtract 15x from each side.



Simplify.

The statement 6 = 0 is never true. So, the equation has no solution.

b. −2(4y + 1) = −8y − 2

READING

Write the equation.

−8y − 2 = −8y − 2

Distributive Property

+ 8y

Add 8y to each side.

All real numbers are solutions of an identity.

+ 8y −2 = −2

Simplify.

The statement −2 = −2 is always true. So, the equation is an identity and has infinitely many solutions.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. 5

4. 4(1 − p) = −4p + 4

5. 6m − m = —6 (6m − 10)

6. 10k + 7 = −3 − 10k

7. 3(2a − 2) = 2(3a − 3)

Concept Summary Steps for Solving Linear Equations Here are several steps you can use to solve a linear equation. Depending on the equation, you may not need to use some steps. Step 1 Use the Distributive Property to remove any grouping symbols.

STUDY TIP To check an identity, you can choose several different values of the variable.

Step 2 Simplify the expression on each side of the equation. Step 3 Collect the variable terms on one side of the equation and the constant

terms on the other side. Step 4 Isolate the variable. Step 5 Check your solution.

Section 1.3

Solving Equations with Variables on Both Sides

21

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Solving Real-Life Problems Modeling with Mathematics A boat leaves New Orleans and travels upstream on the Mississippi River for 4 hours. The return trip takes only 2.8 hours because the boat travels 3 miles per hour faster T ddownstream due to the current. How far does the boat travel upstream?

SOLUTION 1. Understand the Problem You are given the amounts of time the boat travels and the difference in speeds for each direction. You are asked to find the distance the boat travels upstream. 22. Make a Plan Use the Distance Formula to write expressions that represent the problem. Because the distance the boat travels in both directions is the same, you can use the expressions to write an equation. 33. Solve the Problem Use the formula (distance) = (rate)(time). Words

Distance upstream = Distance downstream

Variable

Let x be the speed (in miles per hour) of the boat traveling upstream.

Equation

x mi 1h



(x + 3) mi

⋅4 h = — ⋅ 2.8 h 1h

(mi = mi)



4x = 2.8(x + 3)

Write the equation.

4x = 2.8x + 8.4

Distributive Property

− 2.8x − 2.8x

Subtract 2.8x from each side.

1.2x = 8.4

Simplify.

1.2x 1.2

Divide each side by 1.2.

8.4 1.2

—=—

x=7

Simplify.

So, the boat travels 7 miles per hour upstream. To determine how far the boat travels upstream, multiply 7 miles per hour by 4 hours to obtain 28 miles. 4. Look Back To check that your solution is reasonable, use the formula for distance. Because the speed upstream is 7 miles per hour, the speed downstream is 7 + 3 = 10 miles per hour. When you substitute each speed into the Distance Formula, you get the same distance for upstream and downstream. Upstream



7 mi Distance = — 4 h = 28 mi 1h Downstream



10 mi Distance = — 2.8 h = 28 mi 1h

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

8. A boat travels upstream on the Mississippi River for 3.5 hours. The return trip

only takes 2.5 hours because the boat travels 2 miles per hour faster downstream due to the current. How far does the boat travel upstream? 22

Chapter 1

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1.3

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY Is the equation − 2(4 − x) = 2x + 8 an identity? Explain your reasoning. 2. WRITING Describe the steps in solving the linear equation 3(3x − 8) = 4x + 6.

Monitoring Progress and Modeling with Mathematics In Exercises 3–16, solve the equation. Check your solution. (See Examples 1 and 2.) 3. 15 − 2x = 3x

4. 26 − 4s = 9s

5. 5p − 9 = 2p + 12

6. 8g + 10 = 35 + 3g

7. 5t + 16 = 6 − 5t

In Exercises 19–24, solve the equation. Determine whether the equation has one solution, no solution, or infinitely many solutions. (See Example 3.) 19. 3t + 4 = 12 + 3t

20. 6d + 8 = 14 + 3d

21. 2(h + 1) = 5h − 7 22. 12y + 6 = 6(2y + 1)

8. −3r + 10 = 15r − 8

23. 3(4g + 6) = 2(6g + 9)

9. 7 + 3x − 12x = 3x + 1

1

24. 5(1 + 2m) = —2 (8 + 20m)

10. w − 2 + 2w = 6 + 5w 11. 10(g + 5) = 2(g + 9)

ERROR ANALYSIS In Exercises 25 and 26, describe and correct the error in solving the equation.

12. −9(t − 2) = 4(t − 15)

25.

13. —23 (3x + 9) = −2(2x + 6)



3

14. 2(2t + 4) = —4 (24 − 8t)

2c − 6 = 4 2c = 10 c=5

15. 10(2y + 2) − y = 2(8y − 8) 16. 2(4x + 2) = 4x − 12(x − 1)

26.

17. MODELING WITH MATHEMATICS You and your

friend drive toward each other. The equation 50h = 190 − 45h represents the number h of hours until you and your friend meet. When will you meet? 18. MODELING WITH MATHEMATICS The equation

1.5r + 15 = 2.25r represents the number r of movies you must rent to spend the same amount at each movie store. How many movies must you rent to spend the same amount at each movie store?

VIDEO CITY MEM M MBE ERSH HIP

Membership Fee: $15

5c − 6 = 4 − 3c



6(2y + 6) = 4(9 + 3y) 12y + 36 = 36 + 12y 12y = 12y

0=0 The equation has no solution. 27. MODELING WITH MATHEMATICS Write and solve an

equation to find the month when you would pay the same total amount for each Internet service. Installation fee

Price per month

Company A

$60.00

$42.95

Company B

$25.00

$49.95

Membership Fee: Free

Section 1.3

Solving Equations with Variables on Both Sides

23

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28. PROBLEM SOLVING One serving of granola provides

37. REASONING Two times the greater of two

4% of the protein you need daily. You must get the remaining 48 grams of protein from other sources. How many grams of protein do you need daily?

consecutive integers is 9 less than three times the lesser integer. What are the integers? 38. HOW DO YOU SEE IT? The table and the graph show

information about students enrolled in Spanish and French classes at a high school.

USING STRUCTURE In Exercises 29 and 30, find the value

of r. 29. 8(x + 6) − 10 + r = 3(x + 12) + 5x

Students enrolled this year

30. 4(x − 3) − r + 2x = 5(3x − 7) − 9x MATHEMATICAL CONNECTIONS In Exercises 31 and 32,

the value of the surface area of the cylinder is equal to the value of the volume of the cylinder. Find the value of x. Then find the surface area and volume of the cylinder. 32.

2.5 cm

355

French

229

1 5

7 ft

x ft

x cm

y 400 350 300 250 200 150 0

33. MODELING WITH MATHEMATICS A cheetah that

speed for only about 20 seconds. If an antelope is too far away for a cheetah to catch it in 20 seconds, the antelope is probably safe. Your friend claims the antelope in Exercise 33 will not be safe if the cheetah starts running 650 feet behind it. Is your friend correct? Explain.

1 2 3 4 5 6 7 8 9 10 x

39. WRITING EQUATIONS Give an example of a linear

equation that has (a) no solution and (b) infinitely many solutions. Justify your answers.

REASONING In Exercises 35 and 36, for what value of a is the equation an identity? Explain your reasoning.

40. THOUGHT PROVOKING Draw

35. a(2x + 3) = 9x + 15 + x

a different figure that has the same perimeter as the triangle shown. Explain why your figure has the same perimeter.

36. 8x − 8 + 3ax = 5ax − 2a

Maintaining Mathematical Proficiency

x+3

2x + 1

3x

Reviewing what you learned in previous grades and lessons

(Skills Review Handbook)

41. 9, ∣ −4 ∣, −4, 5, ∣ 2 ∣

42.

43. −18, ∣ −24 ∣, −19, ∣ −18 ∣, ∣ 22 ∣

44. −∣ −3 ∣, ∣ 0 ∣, −1, ∣ 2 ∣, −2

Chapter 1

Spanish

a. Use the graph to determine after how many years there will be equal enrollment in Spanish and French classes. b. How does the equation 355 − 9x = 229 + 12x relate to the table and the graph? How can you use this equation to determine whether your answer in part (a) is reasonable?

34. MAKING AN ARGUMENT A cheetah can run at top

24

French

Years from now

is running 90 feet per second is 120 feet behind an antelope that is running 60 feet per second. How long will it take the cheetah to catch up to the antelope? (See Example 4.)

Order the values from least to greatest.

9 fewer students each year 12 more students each year

Predicted Language Class Enrollment Students enrolled

31.

Spanish

Average rate of change

∣ −32 ∣, 22, −16, −∣ 21 ∣, ∣ −10 ∣

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1.1–1.3

What Did You Learn?

Core Vocabulary conjecture, p. 3 rule, p. 3 theorem, p. 3 equation, p. 4 linear equation in one variable, p. 4

solution, p. 4 inverse operations, p. 4 equivalent equations, p. 4 identity, p. 21

Core Concepts Section 1.1 Addition Property of Equality, p. 4 Subtraction Property of Equality, p. 4 Substitution Property of Equality, p. 4 Multiplication Property of Equality, p. 5

Division Property of Equality, p. 5 Four-Step Approach to Problem Solving, p. 6 Common Problem-Solving Strategies, p. 7

Section 1.2 Solving Multi-Step Equations, p. 12

Unit Analysis, p. 15

Section 1.3 Solving Equations with Variables on Both Sides, p. 20

Special Solutions of Linear Equations, p. 21

Mathematical Practices 1.

How did you make sense of the relationships between the quantities in Exercise 46 on page 9?

2.

What is the limitation of the tool you used in Exercises 25–28 on page 16?

3.

What definition did you use in your reasoning in Exercises 35 and 36 on page 24?

Using the Features of Yourr Textbook d Tests ests to Prepare for Quizzes and • Read and understand the core vocabulary and the contents of the Core Concept boxes. • Review the Examples and the Monitoring Progress questions. Use the tutorials at BigIdeasMath.com for additional help. • Review previously completed homework assignments.

25 5 Copyright © Big Ideas Learning, LLC. All rights reserved.

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1.1–1.3

Quiz

Solve the equation. Justify each step. Check your solution. (Section 1.1) 1. x + 9 = 7

2. 8.6 = z − 3.8

3. 60 = −12r

4. —34 p = 18

Solve the equation. Check your solution. (Section 1.2) 5. 2m − 3 = 13

6. 5 = 10 − v

7. 5 = 7w + 8w + 2

8. −21a + 28a − 6 = −10.2 1

9. 2k − 3(2k − 3) = 45

10. 68 = —5 (20x + 50) + 2

Solve the equation. (Section 1.3) 11. 3c + 1 = c + 1

12. −8 − 5n = 64 + 3n

13. 2(8q − 5) = 4q

14. 9(y − 4) − 7y = 5(3y − 2)

15. 4(g + 8) = 7 + 4g

16. −4(−5h − 4) = 2(10h + 8)

17. To estimate how many miles you are from a thunderstorm, count the seconds between

when you see lightning and when you hear thunder. Then divide by 5. Write and solve an equation to determine how many seconds you would count for a thunderstorm that is 2 miles away. (Section 1.1) 18. You want to hang three equally-sized travel posters on a wall so that the posters on the ends

are each 3 feet from the end of the wall. You want the spacing between posters to be equal. Write and solve an equation to determine how much space you should leave between the posters. (Section 1.2)

3 ft

2 ft

2 ft

2 ft

3 ft

15 ft

19. You want to paint a piece of pottery at an art studio. The total cost is the cost of the piece

plus an hourly studio fee. There are two studios to choose from. (Section 1.3)

a. After how many hours of painting are the total costs the same at both studios? Justify your answer. b. Studio B increases the hourly studio fee by $2. How does this affect your answer in part (a)? Explain.

26

Chapter 1

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1.4

Solving Absolute Value Equations Essential Question

How can you solve an absolute value equation?

Solving an Absolute Value Equation Algebraically Work with a partner. Consider the absolute value equation

∣ x + 2 ∣ = 3.

MAKING SENSE OF PROBLEMS To be proficient in math, you need to explain to yourself the meaning of a problem and look for entry points to its solution.

a. Describe the values of x + 2 that make the equation true. Use your description to write two linear equations that represent the solutions of the absolute value equation. b. Use the linear equations you wrote in part (a) to find the solutions of the absolute value equation. c. How can you use linear equations to solve an absolute value equation?

Solving an Absolute Value Equation Graphically Work with a partner. Consider the absolute value equation

∣ x + 2 ∣ = 3. a. On a real number line, locate the point for which x + 2 = 0. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1

0

1

2

3

4

5

6

7

8

9 10

b. Locate the points that are 3 units from the point you found in part (a). What do you notice about these points? c. How can you use a number line to solve an absolute value equation?

Solving an Absolute Value Equation Numerically Work with a partner. Consider the absolute value equation

∣ x + 2 ∣ = 3. a. Use a spreadsheet, as shown, to solve the absolute value equation. b. Compare the solutions you found using the spreadsheet with those you found in Explorations 1 and 2. What do you notice? c. How can you use a spreadsheet to solve an absolute value equation?

Communicate Your Answer

1 2 3 4 5 6 7 8 9 10 11

A x -6 -5 -4 -3 -2 -1 0 1 2

B |x + 2| 4

abs(A2 + 2)

4. How can you solve an absolute value equation? 5. What do you like or dislike about the algebraic, graphical, and numerical methods

for solving an absolute value equation? Give reasons for your answers.

Section 1.4

Solving Absolute Value Equations

27

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1.4 Lesson

What You Will Learn Solve absolute value equations. Solve equations involving two absolute values.

Core Vocabul Vocabulary larry

Identify special solutions of absolute value equations.

absolute value equation, p. 28 extraneous solution, p. 31 Previous absolute value opposite

Solving Absolute Value Equations An absolute value equation is an equation that contains an absolute value expression. You can solve these types of equations by solving two related linear equations.

Core Concept Properties of Absolute Value Let a and b be real numbers. Then the following properties are true. 1. ∣ a ∣ ≥ 0 2. ∣ −a ∣ = ∣ a ∣ 3.

∣ ∣

∣a∣ a 4. — = —, b ≠ 0 b ∣b∣

∣ ab ∣ = ∣ a ∣ ∣ b ∣

Solving Absolute Value Equations To solve ∣ ax + b ∣ = c when c ≥ 0, solve the related linear equations ax + b = c

or

ax + b = − c.

When c < 0, the absolute value equation ∣ ax + b ∣ = c has no solution because absolute value always indicates a number that is not negative.

Solving Absolute Value Equations Solve each equation. Graph the solutions, if possible. a. ∣ x − 4 ∣ = 6

b. ∣ 3x + 1 ∣ = −5

SOLUTION a. Write the two related linear equations for ∣ x − 4 ∣ = 6. Then solve. x−4=6

or

x − 4 = −6

x = 10

Write related linear equations.

x = −2

Add 4 to each side.

The solutions are x = 10 and x = −2. −4

−2

0

2

4

6

6

8

10

12

6

Each solution is 6 units from 4. Property of Absolute Value

b. The absolute value of an expression must be greater than or equal to 0. The expression ∣ 3x + 1 ∣ cannot equal −5. So, the equation has no solution.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Graph the solutions, if possible. 1. ∣ x ∣ = 10

28

Chapter 1

2. ∣ x − 1 ∣ = 4

3. ∣ 3 + x ∣ = −3

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Solving an Absolute Value Equation Solve ∣ 3x + 9 ∣ − 10 = −4.

SOLUTION First isolate the absolute value expression on one side of the equation.

ANOTHER WAY

∣ 3x + 9 ∣ − 10 = −4

Using the product property of absolute value, |ab| = |a| |b|, you could rewrite the equation as

∣ 3x + 9 ∣ = 6

Write the equation. Add 10 to each side.

Now write two related linear equations for ∣ 3x + 9 ∣ = 6. Then solve. 3x + 9 =

3|x + 3| − 10 = −4 and then solve.

6

3x + 9 = −6

or

3x = −3

3x = −15

x = −1

x = −5

Write related linear equations. Subtract 9 from each side. Divide each side by 3.

The solutions are x = −1 and x = −5.

Writing an Absolute Value Equation

REASONING Mathematically proficient students have the ability to decontextualize problem situations.

In a cheerleading competition, the minimum length of a routine is 4 minutes. The maximum length of a routine is 5 minutes. Write an absolute value equation that represents the minimum and maximum lengths.

SOLUTION 1. Understand the Problem You know the minimum and maximum lengths. You are asked to write an absolute value equation that represents these lengths. 2. Make a Plan Consider the minimum and maximum lengths as solutions to an absolute value equation. Use a number line to find the halfway point between the solutions. Then use the halfway point and the distance to each solution to write an absolute value equation. 33. Solve the Problem 4.0

4.1

4.2

4.3

4.4

4.5

4.6

0.5

4.7

4.8

4.9

5.0

0.5

halfway point

distance from halfway point

∣ x − 4.5 ∣ = 0.5 The equation is ∣ x − 4.5 ∣ = 0.5. 4. 4 Look Back To check that your equation is reasonable, substitute the minimum and maximum lengths into the equation and simplify. Minimum

∣ 4 − 4.5 ∣ = 0.5

Maximum



∣ 5 − 4.5 ∣ = 0.5

Monitoring Progress M



Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solutions. S 4. ∣ x − 2 ∣ + 5 = 9

5. 4∣ 2x + 7 ∣ = 16

6. −2∣ 5x − 1 ∣ − 3 = −11

7. For a poetry contest, the minimum length of a poem is 16 lines. The maximum

length is 32 lines. Write an absolute value equation that represents the minimum and maximum lengths. Section 1.4

Solving Absolute Value Equations

29

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Solving Equations with Two Absolute Values If the absolute values of two algebraic expressions are equal, then they must either be equal to each other or be opposites of each other.

Core Concept Solving Equations with Two Absolute Values To solve ∣ ax + b ∣ = ∣ cx + d ∣, solve the related linear equations ax + b = cx + d

or

ax + b = −(cx + d).

Solving Equations with Two Absolute Values Solve (a) ∣ 3x − 4 ∣ = ∣ x ∣ and (b) ∣ 4x − 10 ∣ = 2∣ 3x + 1 ∣.

SOLUTION a. Write the two related linear equations for ∣ 3x − 4 ∣ = ∣ x ∣. Then solve.

Check

3x − 4 =

∣ 3x − 4 ∣ = ∣ x ∣

−x

? ∣ 3(2) − 4 ∣ = ∣2∣



+4

∣ 3x − 4 ∣ = ∣ x ∣

? ∣ 3(1) − 4 ∣ = ∣1∣ ? ∣ −1 ∣ = ∣1∣ 1=1



3x − 4 = −x

or

−x

2x − 4 =

? ∣2∣ = ∣2∣ 2=2

x

+x

+x

4x − 4 =

0 +4

+4

0 +4

2x = 4

4x = 4

—=—

2x 2

4 2

—=—

x=2

x=1

4x 4

4 4

The solutions are x = 2 and x = 1. b. Write the two related linear equations for ∣ 4x − 10 ∣ = 2∣ 3x + 1 ∣. Then solve. 4x − 10 =

2(3x + 1)

4x − 10 =

6x + 2

− 6x

4x − 10 = 2[−(3x + 1)]

or

4x − 10 = 2(−3x − 1)

− 6x

− 2x − 10 = + 10

2

+ 6x

+ 10

−2x = −2x −2

4x − 10 = −6x − 2

+ 6x

10x − 10 = −2 + 10

12 12 −2

—=—

+ 10

10x = 8 10x 10

8 10

—=—

x = −6

x = 0.8 The solutions are x = −6 and x = 0.8.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solutions. 8. ∣ x + 8 ∣ = ∣ 2x + 1 ∣

30

Chapter 1

9. 3∣ x − 4 ∣ = ∣ 2x + 5 ∣

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Identifying Special Solutions When you solve an absolute value equation, it is possible for a solution to be extraneous. An extraneous solution is an apparent solution that must be rejected because it does not satisfy the original equation.

Identifying Extraneous Solutions Solve ∣ 2x + 12 ∣ = 4x. Check your solutions.

SOLUTION Write the two related linear equations for ∣ 2x + 12 ∣ = 4x. Then solve.

Check

∣ 2x + 12 ∣ = 4x

2x + 12 = 4x

? ∣ 2(6) + 12 ∣ = 4(6) ? ∣ 24 ∣ = 24 24 = 24

12 = 2x

∣ 2x + 12 ∣ = 4x ? ∣ 2(−2) + 12 ∣ = 4(−2)

8 ≠ −8



2x + 12 = −4x 12 = −6x

6=x



? ∣8∣ = −8

or

−2 = x

Write related linear equations. Subtract 2x from each side. Solve for x.

Check the apparent solutions to see if either is extraneous. The solution is x = 6. Reject x = −2 because it is extraneous. When solving equations of the form ∣ ax + b ∣ = ∣ cx + d ∣, it is possible that one of the related linear equations will not have a solution.

Solving an Equation with Two Absolute Values Solve ∣ x + 5 ∣ = ∣ x + 11 ∣.

SOLUTION By equating the expression x + 5 and the opposite of x + 11, you obtain x + 5 = −(x + 11)

Write related linear equation.

x + 5 = −x − 11

Distributive Property

2x + 5 = −11

Add x to each side.

2x = −16

Subtract 5 from each side.

x = −8.

Divide each side by 2.

However, by equating the expressions x + 5 and x + 11, you obtain x + 5 = x + 11

REMEMBER Always check your solutions in the original equation to make sure they are not extraneous.

x=x+6 0=6

Write related linear equation. Subtract 5 from each side.



Subtract x from each side.

which is a false statement. So, the original equation has only one solution. The solution is x = −8.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the equation. Check your solutions. 10. ∣ x + 6 ∣ = 2x

11.

∣ 3x − 2 ∣ = x

12. ∣ 2 + x ∣ = ∣ x − 8 ∣

13.

∣ 5x − 2 ∣ = ∣ 5x + 4 ∣

Section 1.4

Solving Absolute Value Equations

31

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1.4

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check 1. VOCABULARY What is an extraneous solution? 2. WRITING Without calculating, how do you know that the equation ∣ 4x − 7 ∣ = −1 has no solution?

Monitoring Progress and Modeling with Mathematics In Exercises 3−10, simplify the expression. 3.

∣ −9 ∣

4. −∣ 15 ∣

5.

∣ 14 ∣ − ∣ −14 ∣

6.

∣ −3 ∣ + ∣ 3 ∣



8.

∣ −0.8 ⋅ 10 ∣

7. −∣ −5 (−7) ∣ 9.

∣ −327 ∣ —

10.

26. WRITING EQUATIONS The shoulder heights of the

shortest and tallest miniature poodles are shown.

∣ −−124 ∣ —

15 in.

10 in.

In Exercises 11−24, solve the equation. Graph the solution(s), if possible. (See Examples 1 and 2.) 11.

∣w∣ = 6

12.

∣ r ∣ = −2

13.

∣ y ∣ = −18

14.

∣ x ∣ = 13

15.

∣m + 3∣ = 7

16.

∣ q − 8 ∣ = 14

17.

∣ −3d ∣ = 15

18.

∣ ∣

19.

∣ 4b − 5 ∣ = 19

20.

t — =6 2

∣x − 1∣ + 5 = 2

a. Represent these two heights on a number line. b. Write an absolute value equation that represents these heights. USING STRUCTURE In Exercises 27−30, match the

absolute value equation with its graph without solving the equation. 27.

∣x + 2∣ = 4

28.

∣x − 4∣ = 2

29.

∣x − 2∣ = 4

30.

∣x + 4∣ = 2

21. −4∣ 8 − 5n ∣ = 13



2 3



A.

22. −3 1 − — v = −9

∣ 14

−10



−4

−8

−6

−4

−2

4

25. WRITING EQUATIONS The minimum distance from

0

2

0

2

4

6

8

8

10

4

C. −4

Earth to the Sun is 91.4 million miles. The maximum distance is 94.5 million miles. (See Example 3.) a. Represent these two distances on a number line. b. Write an absolute value equation that represents the minimum and maximum distances.

−2

2

B.

24. 9∣ 4p + 2 ∣ + 8 = 35

Chapter 1

−6

2

23. 3 = −2 — s − 5 + 3

32

−8

−2

0

2

4

4

4

D. −2

0

2

4

2

6

2

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In Exercises 31−34, write an absolute value equation that has the given solutions. 31. x = 8 and x = 18

32. x = −6 and x = 10

33. x = 2 and x = 9

34. x = −10 and x = −5

48. MODELING WITH MATHEMATICS The recommended

weight of a soccer ball is 430 grams. The actual weight is allowed to vary by up to 20 grams. a. Write and solve an absolute value equation to find the minimum and maximum acceptable soccer ball weights. b. A soccer ball weighs 423 grams. Due to wear and tear, the weight of the ball decreases by 16 grams. Is the weight acceptable? Explain.

In Exercises 35−44, solve the equation. Check your solutions. (See Examples 4, 5, and 6.) 35.

∣ 4n − 15 ∣ = ∣ n ∣

36.

∣ 2c + 8 ∣ = ∣ 10c ∣

37.

∣ 2b − 9 ∣ = ∣ b − 6 ∣

38.

∣ 3k − 2 ∣ = 2∣ k + 2 ∣

39. 4∣ p − 3 ∣ = ∣ 2p + 8 ∣

40. 2∣ 4w − 1 ∣ = 3∣ 4w + 2 ∣

41.

∣ 3h + 1 ∣ = 7h

42.

∣ 6a − 5 ∣ = 4a

43.

∣f − 6∣ = ∣f + 8∣

44.

∣ 3x − 4 ∣ = ∣ 3x − 5 ∣

ERROR ANALYSIS In Exercises 49 and 50, describe and correct the error in solving the equation.



49.

45. MODELING WITH MATHEMATICS Starting from

2x − 1 = −(−9) 2x = 10 x=5

The solutions are x = −4 and x = 5.



∣ 5x + 8 ∣ = x 5x + 8 = x

or

5x + 8 = −x

4x + 8 = 0

47. MODELING WITH MATHEMATICS You randomly

6x + 8 = 0

4x = −8

survey students about year-round school. The results are shown in the graph.

6x = −8

4 x = −— 3 4 The solutions are x = −2 and x = −—. 3 x = −2

Year-Round School

68%

Oppose

51. ANALYZING EQUATIONS Without solving completely,

32% Error: ±5%

Favor 0%

or

x = −4

50.

absolute value equation ∣ 3x + 8 ∣ − 9 = −5 has no solution because the constant on the right side of the equation is negative. Is your friend correct? Explain.

2x − 1 = −9 2x = −8

300 feet away, a car drives toward you. It then passes by you at a speed of 48 feet per second. The distance d (in feet) of the car from you after t seconds is given by the equation d = ∣ 300 − 48t ∣. At what times is the car 60 feet from you? 46. MAKING AN ARGUMENT Your friend says that the

∣ 2x − 1 ∣ = −9

20%

40%

60%

place each equation into one of the three categories.

80%

The error given in the graph means that the actual percent could be 5% more or 5% less than the percent reported by the survey. a. Write and solve an absolute value equation to find the least and greatest percents of students who could be in favor of year-round school. b. A classmate claims that —13 of the student body is actually in favor of year-round school. Does this conflict with the survey data? Explain.

Section 1.4

No solution

One solution

Two solutions

∣x − 2∣ + 6 = 0

∣x + 3∣ − 1 = 0

∣x + 8∣ + 2 = 7

∣x − 1∣ + 4 = 4

∣ x − 6 ∣ − 5 = −9

∣ x + 5 ∣ − 8 = −8

Solving Absolute Value Equations

33

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52. USING STRUCTURE Fill in the equation

∣x −

60. HOW DO YOU SEE IT? The circle graph shows the

∣=

results of a survey of registered voters the day of an election.

with a, b, c, or d so that the equation is graphed correctly. a

b

d

Which Party’s Candidate Will Get Your Vote?

c

d

Other: 4% Green: 2% Libertarian: 5%

ABSTRACT REASONING In Exercises 53−56, complete

the statement with always, sometimes, or never. Explain your reasoning. 53. If x 2 = a 2, then ∣ x ∣ is ________ equal to ∣ a ∣.

Democratic: 47% Republican: 42%

54. If a and b are real numbers, then ∣ a − b ∣ is

_________ equal to ∣ b − a ∣.

Error: ±2%

55. For any real number p, the equation ∣ x − 4 ∣ = p will

The error given in the graph means that the actual percent could be 2% more or 2% less than the percent reported by the survey.

________ have two solutions. 56. For any real number p, the equation ∣ x − p ∣ = 4 will

________ have two solutions.

a. What are the minimum and maximum percents of voters who could vote Republican? Green? b. How can you use absolute value equations to represent your answers in part (a)? c. One candidate receives 44% of the vote. Which party does the candidate belong to? Explain.

57. WRITING Explain why absolute value equations can

have no solution, one solution, or two solutions. Give an example of each case. 58. THOUGHT PROVOKING Describe a real-life situation

that can be modeled by an absolute value equation with the solutions x = 62 and x = 72.

61. ABSTRACT REASONING How many solutions does

the equation a∣ x + b ∣ + c = d have when a > 0 and c = d? when a < 0 and c > d ? Explain your reasoning.

59. CRITICAL THINKING Solve the equation shown.

Explain how you found your solution(s). 8∣ x + 2 ∣ − 6 = 5∣ x + 2 ∣ + 3

Maintaining Mathematical Proficiency

Reviewing what you learned in previous grades and lessons

Identify the property of equality that makes Equation 1 and Equation 2 equivalent. 62.

Equation 1

3x + 8 = x − 1

Equation 2

3x + 9 = x

Use a geometric formula to solve the problem.

63.

Equation 1

4y = 28

Equation 2

y=7

(Section 1.1)

(Skills Review Handbook)

64. A square has an area of 81 square meters. Find the side length. 65. A circle has an area of 36π square inches. Find the radius. 66. A triangle has a height of 8 feet and an area of 48 square feet. Find the base. 67. A rectangle has a width of 4 centimeters and a perimeter of 26 centimeters. Find the length.

34

Chapter 1

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1.5

Rewriting Equations and Formulas Essential Question

How can you use a formula for one measurement to write a formula for a different measurement? Using an Area Formula Work with a partner. a. Write a formula for the area A of a parallelogram.

REASONING QUANTITATIVELY To be proficient in math, you need to consider the given units. For instance, in Exploration 1, the area A is given in square inches and the height h is given in inches. A unit analysis shows that the units for the base b are also inches, which makes sense.

A = 30 in.2 h = 5 in.

b. Substitute the given values into the formula. Then solve the equation for b. Justify each step.

b

c. Solve the formula in part (a) for b without first substituting values into the formula. Justify each step. d. Compare how you solved the equations in parts (b) and (c). How are the processes similar? How are they different?

Using Area, Circumference, and Volume Formulas Work with a partner. Write the indicated formula for each figure. Then write a new formula by solving for the variable whose value is not given. Use the new formula to find the value of the variable. a. Area A of a trapezoid

b. Circumference C of a circle

b1 = 8 cm

h

C = 24π ft r

A = 63 cm2

b2 = 10 cm

c. Volume V of a rectangular prism

d. Volume V of a cone

V = 75 yd3

V = 24π m3

h

B = 12π m2

h B = 15 yd2

Communicate Your Answer 3. How can you use a formula for one measurement to write a formula for a

different measurement? Give an example that is different from those given in Explorations 1 and 2. Section 1.5

Rewriting Equations and Formulas

35

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1.5 Lesson

What You Will Learn Rewrite literal equations.

Core Vocabul Vocabulary larry

Rewrite and use formulas for area. Rewrite and use other common formulas.

literal equation, p. 36 formula, p. 37

Rewriting Literal Equations

Previous surface area

An equation that has two or more variables is called a literal equation. To rewrite a literal equation, solve for one variable in terms of the other variable(s).

Rewriting a Literal Equation Solve the literal equation 3y + 4x = 9 for y.

SOLUTION 3y + 4x = 9

Write the equation.

3y + 4x − 4x = 9 − 4x

Subtract 4x from each side.

3y = 9 − 4x 3y 3

Simplify.

9 − 4x 3

—=—

Divide each side by 3.

4 y = 3 − —x 3

Simplify.

4 The rewritten literal equation is y = 3 − — x. 3

Rewriting a Literal Equation Solve the literal equation y = 3x + 5xz for x.

SOLUTION y = 3x + 5xz

Write the equation.

y = x(3 + 5z)

Distributive Property

x(3 + 5z) y 3 + 5z 3 + 5z y —=x 3 + 5z

—=—

REMEMBER Division by 0 is undefined.

Divide each side by 3 + 5z. Simplify.

y The rewritten literal equation is x = —. 3 + 5z 3

In Example 2, you must assume that z ≠ −—5 in order to divide by 3 + 5z. In general, if you have to divide by a variable or variable expression when solving a literal equation, you should assume that the variable or variable expression does not equal 0.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the literal equation for y. 1. 3y − x = 9

2. 2x − 2y = 5

3. 20 = 8x + 4y

Solve the literal equation for x. 4. y = 5x − 4x

36

Chapter 1

5. 2x + kx = m

6. 3 + 5x − kx = y

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Rewriting and Using Formulas for Area A formula shows how one variable is related to one or more other variables. A formula is a type of literal equation.

Rewriting a Formula for Surface Area The formula for the surface area S of a rectangular prism is S = 2ℓw + 2ℓh + 2wh. Solve the formula for the lengthℓ.

SOLUTION S = 2ℓw + 2ℓh + 2wh

h w

Write the equation.

S − 2wh = 2ℓw + 2ℓh + 2wh − 2wh

Subtract 2wh from each side.

S − 2wh = 2ℓw + 2ℓh

Simplify.

S − 2wh =ℓ(2w + 2h)

Distributive Property

ℓ(2w + 2h) 2w + 2h

S − 2wh 2w + 2h

— = ——

Divide each side by 2w + 2h.

S − 2wh 2w + 2h

Simplify.

— =ℓ

S − 2wh When you solve the formula forℓ, you obtainℓ= —. 2w + 2h

Using a Formula for Area You own a rectangular lot that is 500 feet deep. It has an area of 100,000 square feet. To pay for a new water system, you are assessed $5.50 per foot of lot frontage. a. Find the frontage of your lot. b. How much are you assessed for the new water system?

SOLUTION a. In the formula for the area of a rectangle, let the width w represent the lot frontage.

w

frontage

500 ft

A =ℓw

Write the formula for area of a rectangle.

A ℓ

Divide each side byℓ to solve for w.

—=w

100,000 500

—=w

Substitute 100,000 for A and 500 forℓ.

200 = w

Simplify.

The frontage of your lot is 200 feet.



$5.50 b. Each foot of frontage costs $5.50, and — 200 ft = $1100. 1 ft So, your total assessment is $1100.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the formula for the indicated variable. 1

7. Area of a triangle: A = —2 bh; Solve for h. 8. Surface area of a cone: S = πr 2 + π rℓ; Solve for ℓ.

Section 1.5

Rewriting Equations and Formulas

37

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Rewriting and Using Other Common Formulas

Core Concept Common Formulas F = degrees Fahrenheit, C = degrees Celsius

Temperature

5 C = — (F − 32) 9 I = interest, P = principal, r = annual interest rate (decimal form), t = time (years)

Simple Interest

I = Prt d = distance traveled, r = rate, t = time d = rt

Distance

Rewriting the Formula for Temperature Solve the temperature formula for F.

SOLUTION 5 C = —(F − 32) 9 9 5

— C = F − 32

Write the temperature formula. 9 Multiply each side by —. 5

— C + 32 = F − 32 + 32

9 5

Add 32 to each side.

9 5

Simplify.

— C + 32 = F

The rewritten formula is F = —95 C + 32.

Using the Formula for Temperature Which has the greater surface temperature: Mercury or Venus?

SOLUTION Convert the Celsius temperature of Mercury to degrees Fahrenheit. 9 F = — C + 32 5

Mercury 427°C

Venus 864°F

Write the rewritten formula from Example 5.

9 = —(427) + 32 5

Substitute 427 for C.

= 800.6

Simplify.

Because 864°F is greater than 800.6°F, Venus has the greater surface temperature.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

9. A fever is generally considered to be a body temperature greater than 100°F. Your

friend has a temperature of 37°C. Does your friend have a fever? 38

Chapter 1

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Using the Formula for Simple Interest You deposit $5000 in an account that earns simple interest. After 6 months, the account earns $162.50 in interest. What is the annual interest rate?

COMMON ERROR The unit of t is years. Be sure to convert months to years. 1 yr 12 mo





SOLUTION To find the annual interest rate, solve the simple interest formula for r. I = Prt I Pt

—=r

6 mo = 0.5 yr

Write the simple interest formula. Divide each side by Pt to solve for r.

162.50 (5000)(0.5)

—=r

Substitute 162.50 for I, 5000 for P, and 0.5 for t.

0.065 = r

Simplify.

The annual interest rate is 0.065, or 6.5%.

Solving a Real-Life Problem A truck driver averages 60 miles per hour while delivering freight to a customer. On the return trip, the driver averages 50 miles per hour due to construction. The total driving time is 6.6 hours. How long does each trip take?

SOLUTION Step 1

Rewrite the Distance Formula to write expressions that represent the two trip d d times. Solving the formula d = rt for t, you obtain t = —. So, — represents r 60 d the delivery time, and — represents the return trip time. 50

Step 2

Use these expressions and the total driving time to write and solve an equation to find the distance one way. d 60

d 50

— + — = 6.6

The sum of the two trip times is 6.6 hours.

— = 6.6

11d 300

Add the left side using the LCD.

11d = 1980

Multiply each side by 300 and simplify.

d = 180

Divide each side by 11 and simplify.

The distance one way is 180 miles. Step 3

Use the expressions from Step 1 to find the two trip times.

60 mi So, the delivery takes 180 mi ÷ — = 3 hours, and the return trip takes 1h 50 mi 180 mi ÷ — = 3.6 hours. 1h

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

10. How much money must you deposit in a simple interest account to earn $500 in

interest in 5 years at 4% annual interest? 11. A truck driver averages 60 miles per hour while delivering freight and 45 miles

per hour on the return trip. The total driving time is 7 hours. How long does each trip take? Section 1.5

Rewriting Equations and Formulas

39

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1.5

Exercises

Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check π 5

1. VOCABULARY Is 9r + 16 = — a literal equation? Explain. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

Solve 3x + 6y = 24 for x.

Solve 24 − 3x = 6y for x.

Solve 6y = 24 − 3x for y in terms of x.

Solve 24 − 6y = 3x for x in terms of y.

Monitoring Progress and Modeling with Mathematics In Exercises 3–12, solve the literal equation for y. (See Example 1.) 3. y − 3x = 13

4. 2x + y = 7

5. 2y − 18x = −26

6. 20x + 5y = 15

7. 9x − y = 45

8. 6 − 3y = −6

9. 4x − 5 = 7 + 4y 1

11. 2 + —6 y = 3x + 4

24. MODELING WITH MATHEMATICS The penny

size of a nail indicates the length of the nail. The penny size d is given by the literal equation d = 4n − 2, where n is the length (in inches) of the nail.

n

a. Solve the equation for n. b. Use the equation from part (a) to find the lengths of nails with the following penny sizes: 3, 6, and 10.

10. 16x + 9 = 9y − 2x 1

12. 11 − —2 y = 3 + 6x ERROR ANALYSIS In Exercises 25 and 26, describe and

In Exercises 13–22, solve the literal equation for x. (See Example 2.) 13. y = 4x + 8x

14. m = 10x − x

15. a = 2x + 6xz

16. y = 3bx − 7x

17. y = 4x + rx + 6

18. z = 8 + 6x − px

19. sx + tx = r

20. a = bx + cx + d

21. 12 − 5x − 4kx = y

22. x − 9 + 2wx = y

correct the error in solving the equation for x. 25.



−2x = −2(y − x) − 12 x = (y − x) + 6

26.



10 = ax − 3b 10 = x(a − 3b) 10 a − 3b

—=x

23. MODELING WITH MATHEMATICS The total cost

C (in dollars) to participate in a ski club is given by the literal equation C = 85x + 60, where x is the number of ski trips you take.

12 − 2x = −2(y − x)

In Exercises 27–30, solve the formula for the indicated variable. (See Examples 3 and 5.)

a. Solve the equation for x.

27. Profit: P = R − C; Solve for C.

b. How many ski trips do you take if you spend a total of $315? $485?

28. Surface area of a cylinder: S = 2π r 2 + 2π rh;

Solve for h. 1

29. Area of a trapezoid: A = —2 h(b1 + b2); Solve for b2.

v −v t

1 0; 30. Average acceleration of an object: a = —

40

Chapter 1

Solving Linear Equations

Solve for v1.

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31. REWRITING A FORMULA A common statistic used in

35. PROBLEM SOLVING You deposit $2000 in an account

professional football is the quarterback rating. This rating is made up of four major factors. One factor is the completion rating given by the formula

that earns simple interest at an annual rate of 4%. How long must you leave the money in the account to earn $500 in interest? (See Example 7.)

(

C R = 5 — − 0.3 A

)

36. PROBLEM SOLVING A flight averages 460 miles per

hour. The return flight averages 500 miles per hour due to a tailwind. The total flying time is 4.8 hours. How long is each flight? Explain. (See Example 8.)

where C is the number of completed passes and A is the number of attempted passes. Solve the formula for C. 32. REWRITING A FORMULA Newton’s law of gravitation

is given by the formula m1m2 F=G — d2

( ) 37. USING STRUCTURE An athletic facility is building an

where F is the force between two objects of masses m1 and m2, G is the gravitational constant, and d is the distance between the two objects. Solve the formula for m1.

indoor track. The track is composed of a rectangle and two semicircles, as shown. x

33. MODELING WITH MATHEMATICS The sale price

S (in dollars) of an item is given by the formula S = L − rL, where L is the list price (in dollars) and r is the discount rate (in decimal form). (See Examples 4 and 6.)

r

r

a. Solve the formula for r. a. Write a formula for the perimeter of the indoor track.

b. The list price of the shirt is $30. What is the discount rate?

b. Solve the formula for x.

Sale price:

$18

34. MODELING WITH MATHEMATICS The density d of a

c. The perimeter of the track is 660 feet, and r is 50 feet. Find x. Round your answer to the nearest foot. 38. MODELING WITH MATHEMATICS The distance

m substance is given by the formula d = —, where m is V its mass and V is its volume.

d (in miles) you travel in a car is given by the two equations shown, where t is the time (in hours) and g is the number of gallons of gasoline the car uses.

Pyrite Density: 5.01g/cm3

Volume: 1.2 cm3 d = 55t d = 20g

a. Write an equation that relates g and t. b. Solve the equation for g. c. You travel for 6 hours. How many gallons of gasoline does the car use? How far do you travel? Explain.

a. Solve the formula for m. b. Find the mass of the pyrite sample.

Section 1.5

Rewriting Equations and Formulas

41

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39. MODELING WITH MATHEMATICS One type of stone

41. MAKING AN ARGUMENT Your friend claims that

formation found in Carlsbad Caverns in New Mexico is called a column. This cylindrical stone formation connects to the ceiling and the floor of a cave.

Thermometer A displays a greater temperature than Thermometer B. Is your friend correct? Explain yyour reasoning. g 100 90 80 70 60 50 40 30 20 10 0 −10

column

°F

Thermometer A Thermometer B

stalagmite

42. THOUGHT PROVOKING Give a possible value for h.

a. Rewrite the formula for the circumference of a circle, so that you can easily calculate the radius of a column given its circumference.

Justify your answer. Draw and label the figure using your chosen value of h.

b. What is the radius (to the nearest tenth of a foot) of a column that has a circumference of 7 feet? 8 feet? 9 feet?

A = 40 cm2

h

c. Explain how you can find the area of a cross section of a column when you know its circumference.

8 cm

40. HOW DO YOU SEE IT? The rectangular prism shown

MATHEMATICAL CONNECTIONS In Exercises 43 and 44,

has bases with equal side lengths.

write a formula for the area of the regular polygon. Solve the formula for the height h. 43.

44. center

center b

h

b

h

b

b

a. Use the figure to write a formula for the surface area S of the rectangular prism.

REASONING In Exercises 45 and 46, solve the literal

equation for a.

b. Your teacher asks you to rewrite the formula by solving for one of the side lengths, b orℓ. Which side length would you choose? Explain your reasoning.

a+b+c ab

45. x = —

( a ab− b )

46. y = x —

Maintaining Mathematical Proficiency Evaluate the expression.

(Skills Review Handbook)



47. 15 − 5 + 52

48. 18 2 − 42 ÷ 8



49. 33 + 12 ÷ 3 5

Solve the equation. Graph the solutions, if possible. 51.

42

∣x − 3∣ + 4 = 9 Chapter 1

Reviewing what you learned in previous grades and lessons

52.

∣ 3y − 12 ∣ − 7 = 2

50. 25(5 − 6) + 9 ÷ 3

(Section 1.4)

53. 2∣ 2r + 4 ∣ = −16

54. −4∣ s + 9 ∣ = −24

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1.4–1.5

What Did You Learn?

Core Vocabulary absolute value equation, p. 28 extraneous solution, p. 31

literal equation, p. 36 formula, p. 37

Core Concepts Section 1.4 Properties of Absolute Value, p. 28 Solving Absolute Value Equations, p. 28 Solving Equations with Two Absolute Values, p. 30 Special Solutions of Absolute Value Equations, p. 31

Section 1.5 Rewriting Literal Equations, p. 36 Common Formulas, p. 38

Mathematical Practices 1.

How did you decide whether your friend’s argument in Exercise 46 on page 33 made sense?

2.

How did you use the structure of the equation in Exercise 59 on page 34 to rewrite the equation?

3.

What entry points did you use to answer Exercises 43 and 44 on page 42?

Performance Task:

Dead Reckoning Have you ever wondered how sailors navigated the oceans before the Global Positioning System (GPS)? One method sailors used is called dead reckoning. How does dead reckoning use mathematics to track locations? Could you use this method today? To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

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1

Chapter Review 1.1

Dynamic Solutions available at BigIdeasMath.com

Solving Simple Equations (pp. 3–10)

a. Solve x − 5 = −9. Justify each step. x − 5 = −9 Addition Property of Equality

+5

Write the equation.

+5

Add 5 to each side.

x = −4

Simplify.

The solution is x = −4. b. Solve 4x = 12. Justify each step. 4x = 12

Write the equation.

4x 4

Divide each side by 4.

12 4

—=—

Division Property of Equality

x=3

Simplify.

The solution is x = 3. Solve the equation. Justify each step. Check your solution. 1. z + 3 = − 6

3.2 3 .2 2 1.2

n 5

3. −— = −2

2. 2.6 = −0.2t

Solving Multi-Step Equations (pp. 11–18)

Solve −6x + 23 + 2x = 15. −6x + 23 + 2x = 15

Write the equation.

−4x + 23 = 15

Combine like terms.

−4x = −8

Subtract 23 from each side.

x=2

Divide each side by −4.

The solution is x = 2. Solve the equation. Check your solution. 4. 3y + 11 = −16 7. −4(2z + 6) − 12 = 4

5. 8.

6=1−b 3 —2 (x

6. n + 5n + 7 = 43 1

− 2) − 5 = 19

7

9. 6 = —5 w + —5 w − 4

Find the value of x. Then find the angle measures of the polygon. 10.

110° 5x° 2x° Sum of angle measures: 180°

44

Chapter 1

(x − 30)°

11. (x − 30)°





(x − 30)°

Sum of angle measures: 540°

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1.3

Solving Equations with Variables on Both Sides (pp. 19–24)

Solve 2( y − 4) = −4( y + 8). 2( y − 4) = −4( y + 8)

Write the equation.

2y − 8 = −4y − 32

Distributive Property

6y − 8 = −32

Add 4y to each side.

6y = −24

Add 8 to each side.

y = −4

Divide each side by 6.

The solution is y = −4. Solve the equation. 12. 3n − 3 = 4n + 1

1.4

1

13. 5(1 + x) = 5x + 5

14. 3(n + 4) = —2 (6n + 4)

Solving Absolute Value Equations (pp. 27–34)

a. Solve ∣ x − 5 ∣ = 3. x−5= +5

3

x − 5 = −3

or

+5

+5

x=8

+5

x=2

Write related linear equations. Add 5 to each side. Simplify.

The solutions are x = 8 and x = 2. Check

b. Solve ∣ 2x + 6 ∣ = 4x. Check your solutions. 2x + 6 = 4x −2x

−2x

or

2x + 6 = −4x −2x

−2x

Write related linear equations. Subtract 2x from each side.

6 = 2x

6 = −6x

6 2x —=— 2 2

6 −6x —=— −6 −6

Solve for x.

3=x

−1 = x

Simplify.

Simplify.

∣ 2x + 6 ∣ = 4x

? ∣ 2(3) + 6 ∣ = 4(3) ? ∣ 12 ∣ = 12 12 = 12



∣ 2x + 6 ∣ = 4x

? ∣ 2(−1) + 6 ∣ = 4(−1) ? ∣ 4 ∣ = −4

Check the apparent solutions to see if either is extraneous. The solution is x = 3. Reject x = −1 because it is extraneous.

∕ −4 4=



Solve the equation. Check your solutions. 15.

∣ y + 3 ∣ = 17

16. −2∣ 5w − 7 ∣ + 9 = − 7

17. ∣ x − 2 ∣ = ∣ 4 + x ∣

18. The minimum sustained wind speed of a Category 1 hurricane is 74 miles per hour. The maximum

sustained wind speed is 95 miles per hour. Write an absolute value equation that represents the minimum and maximum speeds.

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Rewriting Equations and Formulas (pp. 35–42)

a. The slope-intercept form of a linear equation is y = mx + b. Solve the equation for m. y = mx + b

Write the equation.

y − b = mx + b − b

Subtract b from each side.

y − b = mx

Simplify.

y − b mx x x y−b —=m x

—=—

Divide each side by x. Simplify.

y−b When you solve the equation for m, you obtain m = —. x

b. The formula for the surface area S of a cylinder is S = 2𝛑 r 2 + 2𝛑 rh. Solve the formula for the height h. 2π r 2 + 2πrh

S= − 2πr 2

− 2πr 2

Write the equation. Subtract 2πr 2 from each side.

S − 2πr 2 = 2πrh

Simplify.

—=—

S − 2πr 2 2πr

Divide each side by 2πr.

S − 2πr 2 2πr

Simplify.

2πrh 2πr

—=h

S − 2π r 2 When you solve the formula for h, you obtain h = —. 2πr Solve the literal equation for y. 19. 2x − 4y = 20

20. 8x − 3 = 5 + 4y

21. a = 9y + 3yx 1

22. The volume V of a pyramid is given by the formula V = —3 Bh, where B is the area of the

base and h is the height. a. Solve the formula for h. b. Find the height h of the pyramid.

V = 216 cm3

B = 36 cm2 9

23. The formula F = —5 (K − 273.15) + 32 converts a temperature from kelvin K to degrees

Fahrenheit F. a. Solve the formula for K. b. Convert 180°F to kelvin K. Round your answer to the nearest hundredth.

46

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Chapter Test

Solve the equation. Justify each step. Check your solution. 2. —23 x + 5 = 3

1. x − 7 = 15

3. 11x + 1 = −1 + x

Solve the equation. 4. 2∣ x − 3 ∣ − 5 = 7

5.

∣ 2x − 19 ∣ = 4x + 1

6. −2 + 5x − 7 = 3x − 9 + 2x

7. 3(x + 4) − 1 = −7

8.

∣ 20 + 2x ∣ = ∣ 4x + 4 ∣

9. —13 (6x + 12) − 2(x − 7) = 19

Describe the values of c for which the equation has no solution. Explain your reasoning. 10. 3x − 5 = 3x − c

11.

∣x − 7∣ = c

12. A safety regulation states that the minimum height of a handrail is 30 inches. The

maximum height is 38 inches. Write an absolute value equation that represents the minimum and maximum heights. 13. The perimeter P (in yards) of a soccer field is represented by the formula P = 2ℓ+ 2w,

whereℓ is the length (in yards) and w is the width (in yards). a. Solve the formula for w. b. Find the width of the field. c. About what percent of the field is inside the circle?

P = 330 yd

10 yd = 100 yd

14. Your car needs new brakes. You call a dealership and a local

mechanic for prices. Cost of parts

Labor cost per hour

Dealership

$24

$99

Local Mechanic

$45

$89

a. After how many hours are the total costs the same at both places? Justify your answer. b. When do the repairs cost less at the dealership? at the local mechanic? Explain. 15. Consider the equation ∣ 4x + 20 ∣ = 6x. Without calculating, how do you know that x = −2 is an

extraneous solution? 16. Your friend was solving the equation shown and was confused by the result

“−8 = −8.” Explain what this result means. 4(y − 2) − 2y = 6y − 8 − 4y 4y − 8 − 2y = 6y − 8 − 4y 2y − 8 = 2y − 8 −8 = −8 Chapter 1

Chapter Test

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1

Cumulative Assessment

1. A mountain biking park has 48 trails, 37.5% of which are beginner trails. The rest are

divided evenly between intermediate and expert trails. How many of each kind of trail are there? A 12 beginner, 18 intermediate, 18 expert ○ B 18 beginner, 15 intermediate, 15 expert ○ C 18 beginner, 12 intermediate, 18 expert ○ D 30 beginner, 9 intermediate, 9 expert ○ 2. Which of the equations are equivalent to cx − a = b?

cx − a + b = 2b

0 = cx − a + b

b 2cx − 2a = — 2

b x−a=— c

a+b x=— c

b + a = cx

3. Let N represent the number of solutions of the equation 3(x − a) = 3x − 6. Complete

each statement with the symbol , or =. a. When a = 3, N ____ 1. b. When a = −3, N ____ 1. c. When a = 2, N ____ 1. d. When a = −2, N ____ 1. e. When a = x, N ____ 1. f. When a = −x, N ____ 1.

4. You are painting your dining room white and your living room blue. You spend

$132 on 5 cans of paint. The white paint costs $24 per can, and the blue paint costs $28 per can. a. Use the numbers and symbols to write an equation that represents how many cans of each color you bought. x

132

5

24

28

=

(

)

+



×

÷

b. How much would you have saved by switching the colors of the dining room and living room? Explain.

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5. Which of the equations are equivalent?

6x + 6 = −14

8x + 6 = −2x − 14

5x + 3 = −7

7x + 3 = 2x − 13

6. The perimeter of the triangle is 13 inches. What is the length of the shortest

side? (x − 5) in.

A 2 in. ○ B 3 in. ○

x 2

in.

C 4 in. ○ 6 in.

D 8 in. ○

7. You pay $45 per month for cable TV. Your friend buys a satellite TV receiver for $99 and

pays $36 per month for satellite TV. Your friend claims that the expenses for a year of satellite TV are less than the expenses for a year of cable. a. Write and solve an equation to determine when you and your friend will have paid the same amount for TV services. b. Is your friend correct? Explain.

8. Place each equation into one of the four categories. No solution

One solution

Two solutions

Infinitely many solutions

∣ 8x + 3 ∣ = 0

−6 = 5x − 9

3x − 12 = 3(x − 4) + 1

−2x + 4 = 2x + 4

0 = ∣ x + 13 ∣ + 2 9 = 3∣ 2x − 11 ∣

−4(x + 4) = −4x − 16

12x − 2x = 10x − 8

7 − 2x = 3 − 2(x − 2)

9. A car travels 1000 feet in 12.5 seconds. Which of the expressions do not represent the

average speed of the car? second 80 — feet

feet 80 — second

80 feet second





second 80 feet

Chapter 1

Cumulative Assessment

49

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Integrated Mathematics I Teaching Edition Chapter 1 Solving Linear Equations Copyright © Big Ideas Learning, LLC. All rights reserved.

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1

Chapter 1 Pacing Guide Chapter Opener/ Mathematical Practices

1 Day

Section 1

1 Day

Section 2

2 Days

Section 3

1 Day

Quiz

1 Day

Section 4

2 Days

Section 5

2 Days

Chapter Review/ Chapter Tests

2 Days

Total Chapter 1

12 Days

Year-to-Date

12 Days

1.1 1.2 1.3 1.4 1.5

Solving Linear Equations Solving Simple Equations Solving Multi-Step Equations Solving Equations with Variables on Both Sides Solving Absolute Value Equations Rewriting Equations and Formulas

Density of Pyrite (p. 41)

SEE the Big Idea Cheerleading Competition (p. 29)

Boat (p. 22)

Biking (p. 14)

Average Speed (p. 6)

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Laurie’s Notes

Dynamic Teaching Tools Dynamic Assessment & Progress Monitoring Tool Lesson Planning Tool

Chapter Summary Welcome to a new school year, and for many students, a new school. There is always great excitement, and students are anxious to start anew. As teachers, we need to capitalize on the opportunity, establishing norms and routines for student discourse and classroom climate. • In this book, students are expected to work together on explorations, to make conjectures, to construct viable arguments, and to critique the reasoning of others. Take time in this first chapter to make explicit what classroom productive dialogue sounds like. Listen for students explaining their thinking, not just their process. • Chapter 1 presents the foundational skills related to solving linear equations and the connected skills of solving absolute value equations and rewriting equations and formulas. • Most students will have prior experience with the Properties of Equality and techniques presented in the first three sections. It will sound familiar that whatever operation is performed on one side of the equation, the same operation must be performed on the other side of the equation to keep equality, or balance. • The fourth section of the chapter applies the techniques of equation solving to the context of absolute value equations. Understanding absolute value as a function concept and not simply two vertical lines can be challenging for students. • Solving literal equations in the last section requires students to see the structure of equations and perform operations on variable terms (i.e., 4x) as they would perform operations on constants (i.e., 4). • Essential to success in this chapter is accuracy in computation. Feedback to students should distinguish between an error in computation and a process error.

Interactive Whiteboard Lesson Library Dynamic Classroom with Dynamic Investigations Real-Life STEM Videos

Scaffolding in the Classroom Graphic Organizers: Word Magnet A Word Magnet can be used to organize information associated with a vocabulary word or term. Students write the word or term inside the magnet. Students write associated information on the blank lines that “radiate” from the magnet. Associated information can include, but is not limited to: other vocabulary words or terms, definitions, formulas, procedures, examples, and visuals. This type of organizer serves as a good summary tool because any information related to a topic can be included.

What Your Students Have Learned Middle School • Add, subtract, multiply, and divide rational numbers. • Find the absolute values of numbers and use absolute value to compare numbers in real-life situations. • Use variables to represent quantities in real-life problems. • Write simple equations to solve real-life problems. • Solve linear equations using the Distributive Property and combining like terms.

What Your Students Will Learn 9/23/14 7:37 AM

Math I • Solve multi-step linear equations with variables on one or both sides. • Solve absolute value equations involving one or two absolute value expressions. • Identify equations with no solution or infinitely many solutions. • Identify extraneous solutions. • Use unit analysis to model and solve real-life problems. • Rewrite and use literal equations and common formulas.

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Questioning in the Classroom But why? Ask questions that require critical thinking so that follow-up questions may be asked. Avoid questions having yes or no answers. If this cannot be avoided, always follow with why?

Laurie’s Notes Maintaining Mathematical Proficiency Adding and Subtracting Integers • Remind students how to add integers with the same sign. They should add the absolute values of the integers and then use the common sign. • Remind students how to add integers with different signs. They should subtract the lesser absolute value from the greater absolute value and use the sign of the integer with the greater absolute value. COMMON ERROR Students may ignore the signs and just add the integers.

Multiplying and Dividing Integers • Remind students that the product or quotient of two integers with the same sign is always positive. • Remind students that the product or quotient of two integers with different signs is always negative. COMMON ERROR When both factors are negative, students may write the product as negative.

Mathematical Practices (continued on page 2) • The Mathematical Practices page focuses attention on how mathematics is learned—process versus content. Page 2 demonstrates that focusing on unit analysis leads to labeling answers with correct units of measures, providing an entry point into solving problems. • Use the Mathematical Practices page to help students develop mathematical habits of mind —how mathematics can be explored and how mathematics is thought about.

If students need help...

If students got it...

Student Journal • Maintaining Mathematical Proficiency

Game Closet at BigIdeasMath.com

Lesson Tutorials

Start the next Section

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Skills Review Handbook

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What Your Students Have Learned

Maintaining Mathematical Proficiency Adding and Subtracting Integers Example 1

(Grade 7)

• Add integers with the same sign or with different signs using absolute value.

Evaluate 4 + (−12). ∣ −12 ∣ > ∣ 4 ∣. So, subtract ∣ 4 ∣ from ∣ −12 ∣.

• Subtract integers by adding the opposite.

4 + (−12) = −8 Use the sign of −12.

Example 2

• Multiply integers with the same sign or with different signs.

Evaluate −7 − (−16). −7 − (−16) = −7 + 16

• Divide integers with the same sign or with different signs and recognize the quotient as a rational number.

Add the opposite of −16.

=9

Add.

Add or subtract. 1. −5 + (−2)

2. 0 + (−13)

3. −6 + 14

4. 19 − (−13)

5. −1 − 6

6. −5 − (−7)

ANSWERS

7. 17 + 5

8. 8 + (−3)

9. 11 − 15

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Multiplying and Dividing Integers

(Grade 7)



Example 3 Evaluate −3 (−5). The integers have the same sign.



−3 (−5) = 15 The product is positive.

Example 4 Evaluate 15 ÷ (−3). The integers have different signs.

15 ÷ (−3) = −5 The quotient is negative.

Multiply or divide.





10. −3 (8)

11. −7 (−9)

12. 4 (−7)

13. −24 ÷ (−6)

−16 14. — 2

15. 12 ÷ (−3)

17. 36 ÷ 6

18. −3(−4)



16. 6 8

19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers,

(c) multiplying integers, and (d) dividing integers. Give an example of each.

Dynamic Solutions available at BigIdeasMath.com

1

−7 −13 8 32 −7 2 22 5 −4 −24 63 −28 4 −8 −4 48 6 12 a. If the signs are the same, add the absolute values and attach the sign. If the signs are different, subtract the absolute values and attach the sign of the number with the greatest absolute value; Sample answer: −6 + 2 = −4

19b–d. See Additional Answers. Int_Math1_PE_01OP.indd 1

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Vocabulary Review Have students make Information Frames for the following words. • Integer • Opposite • Absolute value

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Mathematical Practices

MONITORING PROGRESS ANSWERS 1. about 3 million per year 2. 30 mi/gal 3. about 17 min

Mathematically proficient students carefully specify units of measure.

Specifying Units of Measure

Core Concept Operations and Unit Analysis Addition and Subtraction

When you add or subtract quantities, they must have the same units of measure. The sum or difference will have the same unit of measure. Example

Perimeter of rectangle = (3 ft) + (5 ft) + (3 ft) + (5 ft)

3 ft

= 16 feet

5 ft

When you add feet, you get feet.

Multiplication and Division

When you multiply or divide quantities, the product or quotient will have a different unit of measure. Example

Area of rectangle = (3 ft) × (5 ft) = 15 square feet

When you multiply feet, you get feet squared, or square feet.

Specifying Units of Measure You work 8 hours and earn $72. What is your hourly wage?

SOLUTION dollars per hour

dollars per hour The units on each side of the equation balance. Both are specified in dollars per hour.

Hourly wage = $72 ÷ 8 h ($ per h) = $9 per hour Your hourly wage is $9 per hour.

Monitoring Progress Solve the problem and specify the units of measure. 1. The population of the United States was about 280 million in 2000 and about

310 million in 2010. What was the annual rate of change in population from 2000 to 2010? 2. You drive 240 miles and use 8 gallons of gasoline. What was your car’s gas mileage

(in miles per gallon)? 3. A bathtub is in the shape of a rectangular prism. Its dimensions are 5 feet by 3 feet by

18 inches. The bathtub is three-fourths full of water and drains at a rate of 1 cubic foot per minute. About how long does it take for all the water to drain?

2

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(continued from page T-1)

• Allow students time to read through the Core Concept and example. Ask probing questions to assess students’ understanding of the units of measure being the same on each side of the equation. • Students could work with partners or in groups on Monitoring Progress. Allow private think time before dialogue begins.

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1.1–1.3

What Did You Learn?

Dynamic Teaching Tools Dynamic Assessment & Progress Monitoring Tool

Core Vocabulary conjecture, p. 3 rule, p. 3 theorem, p. 3 equation, p. 4 linear equation in one variable, p. 4

Interactive Whiteboard Lesson Library solution, p. 4 inverse operations, p. 4 equivalent equations, p. 4 identity, p. 21

Dynamic Classroom with Dynamic Investigations

ANSWERS 1. Sample answer: Let A represent the area of a single rectangle. Because each rectangle has the same area, and the area of the square is half the area of a rectangle, use the expression 4A + —12A to represent the total area. 2. Sample answer: A limitation of a protractor is centering it exactly on each vertex point. 3. the definition of “identity”

Core Concepts Section 1.1 Addition Property of Equality, p. 4 Subtraction Property of Equality, p. 4 Substitution Property of Equality, p. 4 Multiplication Property of Equality, p. 5

Division Property of Equality, p. 5 Four-Step Approach to Problem Solving, p. 6 Common Problem-Solving Strategies, p. 7

Section 1.2 Solving Multi-Step Equations, p. 12

Unit Analysis, p. 15

Section 1.3 Solving Equations with Variables on Both Sides, p. 20

Special Solutions of Linear Equations, p. 21

Mathematical Practices 1.

How did you make sense of the relationships between the quantities in Exercise 46 on page 9?

2.

What is the limitation of the tool you used in Exercises 25–28 on page 16?

3.

What definition did you use in your reasoning in Exercises 35 and 36 on page 24?

Using the Features of Yourr Textbook d Tests ests to Prepare for Quizzes and • Read and understand the core vocabulary and the contents of the Core Concept boxes. • Review the Examples and the Monitoring Progress questions. Use the tutorials at BigIdeasMath.com for additional help. • Review previously completed homework assignments.

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ANSWERS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

x = −2; Subtract 9 from each side. z = 12.4; Add 3.8 to each side. r = −5; Divide each side by −12. p = 24; Multiply each side by —43 . m=8 v=5 w = —51 a = −0.6 k = −9 x = 14 c=0 n = −9 q = —56 y = −2 no solution infinitely many solutions s 17. 2 = —; 10 sec 5 18. 12 + 2x = 15; 1—12 ft 19. a. 3 h; The cost at studio A is 10 + 8h and the cost at Studio B is 16 + 6h. To find when the costs are the same, set these two expressions equal and solve for the time. b. The costs will never be the same; Sample answer: The cost for studio B changes to 16 + 8h, and the new equation has no solution.

1.1–1.3

Quiz

Solve the equation. Justify each step. Check your solution. (Section 1.1) 1. x + 9 = 7

2. 8.6 = z − 3.8

3. 60 = −12r

4. —34 p = 18

Solve the equation. Check your solution. (Section 1.2) 5. 2m − 3 = 13

6. 5 = 10 − v

7. 5 = 7w + 8w + 2

8. −21a + 28a − 6 = −10.2 1

9. 2k − 3(2k − 3) = 45

10. 68 = —5 (20x + 50) + 2

Solve the equation. (Section 1.3) 11. 3c + 1 = c + 1

12. −8 − 5n = 64 + 3n

13. 2(8q − 5) = 4q

14. 9(y − 4) − 7y = 5(3y − 2)

15. 4(g + 8) = 7 + 4g

16. −4(−5h − 4) = 2(10h + 8)

17. To estimate how many miles you are from a thunderstorm, count the seconds between

when you see lightning and when you hear thunder. Then divide by 5. Write and solve an equation to determine how many seconds you would count for a thunderstorm that is 2 miles away. (Section 1.1) 18. You want to hang three equally-sized travel posters on a wall so that the posters on the ends

are each 3 feet from the end of the wall. You want the spacing between posters to be equal. Write and solve an equation to determine how much space you should leave between the posters. (Section 1.2)

3 ft

2 ft

2 ft

2 ft

3 ft

15 ft

19. You want to paint a piece of pottery at an art studio. The total cost is the cost of the piece

plus an hourly studio fee. There are two studios to choose from. (Section 1.3)

a. After how many hours of painting are the total costs the same at both studios? Justify your answer. b. Studio B increases the hourly studio fee by $2. How does this affect your answer in part (a)? Explain.

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Solving Linear Equations

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Laurie’s Notes

Dynamic Teaching Tools Dynamic Assessment & Progress Monitoring Tool Lesson Planning Tool

Overview of Section 1.5

Interactive Whiteboard Lesson Library Dynamic Classroom with Dynamic Investigations

Introduction • In the study of high school mathematics, students will work with equations involving more than one variable. In Math I, this occurs when students start to represent linear equations in two variables, x and y. This chapter presents equation-solving techniques, often referred to as symbolic manipulation, that apply to equations involving more than one variable. • This lesson uses formulas that should be familiar to students, in order to practice the skill of rewriting equations. In addition, students will rewrite an equation in two variables, x and y, to solve for y. Rewriting an equation in function form is an important skill that you want students to be secure with now. Any weaknesses in rational number operations will surface, so it is important to differentiate where students are having a problem in this lesson. Is the challenge with using inverse operations correctly, or is it problems with rational numbers, or both?

Common Misconceptions • When an equation involves more than one variable, students can be uncertain how to treat the variable(s) that is not being solved for. It is helpful to review vocabulary such as coefficient, variable term, and constant term before you begin this lesson.

Formative Assessment Tips • Whiteboarding: Whiteboards can be used to provide individual responses, or used with small groups to encourage student collaboration and consensus on a problem or solution method. Whiteboards can be used at the beginning of class for warm-ups or throughout the lesson to elicit student responses. Unlike writing on scrap paper (individual response) or chart paper (group response), responses can be erased and modified easily. As understanding progresses, responses can reflect this growth. • Use boards for more than quick responses. Sizable boards can be used to communicate thinking, providing evidence of how a problem was solved. When students display their whiteboards in the front of the room, classmates can critique their reasoning or method of solution.

Pacing Suggestion • The formal lesson is quite long. You might have students complete only Exploration 1 and then move to Examples 3, 4, 7, and 8 in the formal lesson. That would leave Examples 1 and 2 (literal equations) and Examples 5 and 6 (temperature formulas) for the second day.

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Section 1.5

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What Your Students Will Learn • Rewrite literal equations using the Properties of Equality. • Rewrite formulas by performing operations on variable terms.

Laurie’s Notes Exploration Motivate Distribute whiteboards and ask students to answer the first three questions. • “I drove 2.1 hours at 65 miles per hour. How far did I go?” 136.5 miles • “It took me 32 minutes to walk 4 kilometers. How fast was I walking on average?” 1 —8 kilometer per minute or 7.5 kilometers per hour

• “At 8 miles per hour, how long will it take me to bike 6 miles?” —34 hour “How are these three equations alike?” They all involve distance, rate, and time. • Write the three equations relating distance, rate, and time. d d d = rt, r = —, t = — t r • Explain that today they will be rewriting familiar formulas and solving them for different variables, just as the familiar formula d = rt can be solved for r or t.

Exploration 1 Draw a parallelogram without a horizontal side and ask, “How do you find the area of a parallelogram?” A = bh “Are the two adjacent sides the base and height?” No, not unless the parallelogram is a rectangle. • Circulate as students begin the exploration. Note student work in part (c). Assessing Question: “How did you solve for b (base) when you knew the area and height?” Divide each side by 5. “Can this same process be used when you do not know the area or height? Explain.” Yes. Divide each side of the literal equation by h. • Have partners share their answers to part (d). You might also ask which was easier, part (b) or part (c). The steps are the same; however, students often view working with constants as easier than working with variables.

Exploration 2 • Have students read through the directions. Check to see that students understand that they are solving the equation for a variable and then making a substitution, just as they did in part (c) of Exploration 1. • Differentiate: You might assign one of the four problems to different groups or pairs of students in the class. Work could be recorded on a whiteboard, enabling presentation of the work to the whole class. • Teaching Tip: The formula for the volume of a cone involves the fraction —13 . Suggest to students that they first work with the fraction before they consider the variables involved. 1 V = — Bh 3 3V = Bh

Int_Math1_PE_01

Communicate Your Answer • Students should refer to solving an equation for a particular variable. • Students should be familiar with many formulas, and not always from mathematics!

Connecting to Next Step • If you feel students are comfortable with solving a formula for a variable, assign Exercises 28 and 29 on page 40 and Exercise 34 on page 41.

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1.5

Rewriting Equations and Formulas

Dynamic Teaching Tools

Essential Question

Dynamic Assessment & Progress Monitoring Tool

How can you use a formula for one measurement to write a formula for a different measurement?

Lesson Planning Tool Interactive Whiteboard Lesson Library

Using an Area Formula

Dynamic Classroom with Dynamic Investigations

Work with a partner. a. Write a formula for the area A of a parallelogram.

REASONING QUANTITATIVELY To be proficient in math, you need to consider the given units. For instance, in Exploration 1, the area A is given in square inches and the height h is given in inches. A unit analysis shows that the units for the base b are also inches, which makes sense.

A = 30 in.2

ANSWERS

h = 5 in.

b. Substitute the given values into the formula. Then solve the equation for b. Justify each step.

b

c. Solve the formula in part (a) for b without first substituting values into the formula. Justify each step. d. Compare how you solved the equations in parts (b) and (c). How are the processes similar? How are they different?

Technology for the Teacher Using Area, Circumference, and

Volume Formulas Computer

Work with a partner. Write the indicated formula for each figure. Then write a new Calculator formula by solving for the variable whose value is not given. Use the new formula to find the value of the variable.

Online Ancillaries

a. Area A of a trapezoid

b. Circumference C of a circle

Smartboard

b1 = 8 cm

C = 24π ft

Video h

r

A = 63 cm2

b2 = 10 cm

c. Volume V of a rectangular prism

d. Volume V of a cone

V = 75 yd3

1. a. A = bh b. 30 = b ⋅ 5; Write the equation. 6 = b; Divide each side by 5. c. A = bh; Write the formula. A — = b; Divide each side by h. h d. applied the same steps to solve; used variables instead of constants 1 2A 2. a. A = — h (b1 + b2); h = —; 2 b1 + b2 h = 7 cm C b. C = 2πr; r = —; r = 12 ft 2π V c. V = Bh; h = —; h = 5 yd B 1 3V d. V = —Bh; h = —; h = 6 m 3 B 3. Solve the formula for a different variable; Sample answer: (volume of V a cylinder) V = πr2h; h = —2 πr

V = 24π m3 h

B = 12π m2

h B = 15 yd2

Communicate Your Answer 3. How can you use a formula for one measurement to write a formula for a

different measurement? Give an example that is different from those given in Explorations 1 and 2. Section 1.5

Rewriting Equations and Formulas

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English Language Learners Vocabulary Write the word literal on the board. Explain that in everyday speech, the literal meaning of a phrase is based on the strict definition of each word. Compare the literal and intended meanings of the expression It’s raining cats and dogs. Explain that, in mathematics, a literal equation is an equation with two or more variables. When you solve for one variable, the solution includes the other variables.

1.5 Lesson

What You Will Learn Rewrite literal equations.

Core Vocabul Vocabulary larry

Rewrite and use formulas for area. Rewrite and use other common formulas.

literal equation, p. 36 formula, p. 37

Rewriting Literal Equations

Previous surface area

An equation that has two or more variables is called a literal equation. To rewrite a literal equation, solve for one variable in terms of the other variable(s).

Rewriting a Literal Equation Solve the literal equation 3y + 4x = 9 for y.

SOLUTION 3y + 4x = 9

Write the equation.

3y + 4x − 4x = 9 − 4x

Extra Example 1

Subtract 4x from each side.

3y = 9 − 4x

Simplify.

3y 9 − 4x —=— 3 3

Divide each side by 3.

4 y = 3 − —x 3

Solve the literal equation 4x − 7y = 12 4x − 12 for y. y = — 7

Simplify.

4 The rewritten literal equation is y = 3 − — x. 3

Rewriting a Literal Equation

Extra Example 2

Solve the literal equation y = 3x + 5xz for x.

Solve the literal equation 3w + 4wp = a a for w. w = — 3 + 4p

SOLUTION

MONITORING PROGRESS ANSWERS 2. y = x −

Write the equation.

y = x(3 + 5z)

Distributive Property

x(3 + 5z) y —=— 3 + 5z 3 + 5z y —=x 3 + 5z

1. y = 3 + —13x 5 —2

y = 3x + 5xz

REMEMBER Division by 0 is undefined.

3. y = 5 − 2x 4. x = y

Divide each side by 3 + 5z. Simplify.

y The rewritten literal equation is x = —. 3 + 5z 3

In Example 2, you must assume that z ≠ −—5 in order to divide by 3 + 5z. In general, if you have to divide by a variable or variable expression when solving a literal equation, you should assume that the variable or variable expression does not equal 0.

m 2+k y−3 6. x = — 5−k

5. x = —

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

Solve the literal equation for y. 1. 3y − x = 9

2. 2x − 2y = 5

3. 20 = 8x + 4y

Solve the literal equation for x. 4. y = 5x − 4x

36

Chapter 1

5. 2x + kx = m

6. 3 + 5x − kx = y

Solving Linear Equations

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• Look For and Make Use of Structure: When solving literal equations, you want students to verbalize the operations represented. For example, when solving 3y + 4x = 9, students should understand there are two variable terms. To isolate the 3y-term, subtract 4x. In other words, approach solving 3y + 4x = 9 for y in the same way as you would solve 3y + 4 = 9 for y. • Write the definition of a literal equation. “In Example 1, can 9 and 4x be combined? Explain.” No. They are not like terms. • In Example 2, use color to highlight each x: y = 3x + 5xz. Using the Distributive Property is not obvious to most students because they do not think of the property in a factoring context. Turn and Talk: “Why can’t z = −—35?” The denominator would be 0 and, therefore, undefined.

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Rewriting and Using Formulas for Area A formula shows how one variable is related to one or more other variables. A formula is a type of literal equation.

Differentiated Instruction Inclusion

Rewriting a Formula for Surface Area The formula for the surface area S of a rectangular prism is S = 2ℓw + 2ℓh + 2wh. Solve the formula for the lengthℓ.

SOLUTION h w

S = 2ℓw + 2ℓh + 2wh

Write the equation.

S − 2wh = 2ℓw + 2ℓh + 2wh − 2wh

Subtract 2wh from each side.

S − 2wh = 2ℓw + 2ℓh

Simplify.

S − 2wh =ℓ(2w + 2h)

Distributive Property

S − 2wh ℓ(2w + 2h) — = —— 2w + 2h 2w + 2h

Divide each side by 2w + 2h.

S − 2wh — =ℓ 2w + 2h

Simplify.

Extra Example 3 The formula for the area of a circle is A = π r2 . Solve the formula for r.

S − 2wh When you solve the formula forℓ, you obtainℓ= —. 2w + 2h





A r= — π

Using a Formula for Area You own a rectangular lot that is 500 feet deep. It has an area of 100,000 square feet. To pay for a new water system, you are assessed $5.50 per foot of lot frontage. a. Find the frontage of your lot. b. How much are you assessed for the new water system?

SOLUTION a. In the formula for the area of a rectangle, let the width w represent the lot frontage. frontage

500 ft w

A =ℓw

Write the formula for area of a rectangle.

A —=w ℓ

Divide each side byℓ to solve for w.

100,000 500

—=w

Substitute 100,000 for A and 500 forℓ.

200 = w

Some students may have difficulty keeping track of the variable. Have students circle the variable for which they are solving. Remind them that they have found the answer when that variable is alone on one side of the literal equation.

Extra Example 4 A patio is in the shape of a parallelogram. Its base, which is up against the side of the house, is 13 feet. The area of the patio is 156 square feet. The height of the parallelogram represents the distance from the house to the edge of the patio. The yard is 24 yards deep from the house to the back fence. a. Find the distance from the house to the edge of the patio. 12 feet

Simplify.

b. How far is it from that edge of the patio to the back fence? 60 feet, or 20 yards

The frontage of your lot is 200 feet.



$5.50 b. Each foot of frontage costs $5.50, and — 200 ft = $1100. 1 ft So, your total assessment is $1100.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

2A b S − πr2 8. ℓ = — πr 7. h = —

Solve the formula for the indicated variable. 7. Area of a triangle: A =

1 —2 bh;

Solve for h.

8. Surface area of a cone: S = πr 2 + π rℓ; Solve for ℓ.

Section 1.5

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Rewriting Equations and Formulas

Laurie’s Notes Teacher Actions

MONITORING PROGRESS ANSWERS

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• Use color to highlight eachℓ: S = 2ℓw + 2ℓh + 2wh • Look For and Make Use of Structure: Students must see the 2wh as a constant term that can be subtracted from each side of the equation. • Pose the problem in Example 4. Give time for partners to work the example on whiteboards. As you circulate, observe solution methods. Determine whether there are unique and/or particularly clear solutions that should be seen by the class. Either ask for those solutions to be shared, or make it appear random by placing particular Popsicle sticks in the center of the can.

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Rewriting and Using Other Common Formulas

Extra Example 5

Core Concept

Solve the distance formula d = rt for r. d r=— t

Common Formulas F = degrees Fahrenheit, C = degrees Celsius

Temperature

5 C = — (F − 32) 9 I = interest, P = principal, r = annual interest rate (decimal form), t = time (years)

Extra Example 6 Simple Interest

On a July day, the temperature at the peak of Mt. Everest is −16°C. Is that temperature warmer or colder than 0°F? warmer

I = Prt d = distance traveled, r = rate, t = time d = rt

Distance

MONITORING PROGRESS ANSWER

Rewriting the Formula for Temperature

9. no

Solve the temperature formula for F.

SOLUTION 5 C = —(F − 32) 9 9 5

— C = F − 32

Write the temperature formula. 9 Multiply each side by —. 5

— C + 32 = F − 32 + 32

9 5

Add 32 to each side.

9 5

Simplify.

— C + 32 = F

The rewritten formula is F = —95 C + 32.

Using the Formula for Temperature Which has the greater surface temperature: Mercury or Venus?

SOLUTION Convert the Celsius temperature of Mercury to degrees Fahrenheit. 9 F = — C + 32 5

Mercury 427°C

Venus 864°F

Write the rewritten formula from Example 5.

9 = —(427) + 32 5

Substitute 427 for C.

= 800.6

Simplify.

Because 864°F is greater than 800.6°F, Venus has the greater surface temperature.

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

9. A fever is generally considered to be a body temperature greater than 100°F. Your

friend has a temperature of 37°C. Does your friend have a fever? 38

Chapter 1

Solving Linear Equations

Laurie’s Notes Teacher Actions

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• Attend to Precision: In the simple interest formula, students should note that both rate and time are in terms of years. • Students may know the formula for converting Celsius to Fahrenheit. If so, that should be the common formula stated and change Example 5 to rewriting the formula to solve for Celsius. • Extension: Students have likely heard about the Kelvin temperature scale. Have students research a formula for converting Kelvin to Fahrenheit or Celsius. Rewrite the formula.

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Using the Formula for Simple Interest

Extra Example 7 You deposit $3500 in an account that earns 4% simple interest. How long will you have to leave the money in the account to earn $350 interest? 2.5 years

You deposit $5000 in an account that earns simple interest. After 6 months, the account earns $162.50 in interest. What is the annual interest rate?

COMMON ERROR The unit of t is years. Be sure to convert months to years. 1 yr 12 mo



⋅ 6 mo = 0.5 yr

SOLUTION To find the annual interest rate, solve the simple interest formula for r. I = Prt I —=r Pt

Write the simple interest formula.

Extra Example 8

Divide each side by Pt to solve for r.

162.50 (5000)(0.5)

—=r

Dan drove 330 miles at 60 miles per hour. Ryan drove 275 miles at 55 miles per hour. How much more time did Dan drive than Ryan? Dan drove 0.5 hour longer than Ryan.

Substitute 162.50 for I, 5000 for P, and 0.5 for t.

0.065 = r

Simplify.

The annual interest rate is 0.065, or 6.5%.

Solving a Real-Life Problem

MONITORING PROGRESS ANSWERS

A truck driver averages 60 miles per hour while delivering freight to a customer. On the return trip, the driver averages 50 miles per hour due to construction. The total driving time is 6.6 hours. How long does each trip take?

10. $2500 11. delivery: 3h, return: 4h

SOLUTION Step 1

Rewrite the Distance Formula to write expressions that represent the two trip d d times. Solving the formula d = rt for t, you obtain t = —. So, — represents r 60 d the delivery time, and — represents the return trip time. 50

Step 2

Use these expressions and the total driving time to write and solve an equation to find the distance one way. d 60

d 50

— + — = 6.6

The sum of the two trip times is 6.6 hours.

— = 6.6

11d 300

Add the left side using the LCD.

11d = 1980

Multiply each side by 300 and simplify.

d = 180

Divide each side by 11 and simplify.

The distance one way is 180 miles. Step 3

Use the expressions from Step 1 to find the two trip times.

60 mi So, the delivery takes 180 mi ÷ — = 3 hours, and the return trip takes 1h 50 mi 180 mi ÷ — = 3.6 hours. 1h

Monitoring Progress

Help in English and Spanish at BigIdeasMath.com

10. How much money must you deposit in a simple interest account to earn $500 in

interest in 5 years at 4% annual interest? 11. A truck driver averages 60 miles per hour while delivering freight and 45 miles

per hour on the return trip. The total driving time is 7 hours. How long does each trip take? Section 1.5

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Rewriting Equations and Formulas

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Laurie’s Notes Teacher Actions COMMON ERROR In solving the simple interest formula for r in

Example 7, students often divide by P and then divide by t. By doing so, they end up with a compound fraction on the left side of the equation. Turn and Talk: “Why can you divide by P and t in one step?” Sample answer: P and t are multiplied together and can be considered the coefficient of r.

• Whiteboarding: Pose the question in Example 8 and give partners time to work the example. Circulate and ask advancing questions to assist students in making progress. • Construct Viable Arguments and Critique the Reasoning of Others: Have a pair of students share their solution.

Closure

11 − 2x • Exit Ticket: Solve 2x + 4y = 11 for y. y = — 4

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1.5

Assignment Guide and Homework Check

Exercises

Dynamic Solutions available at BigIdeasMath.com

ASSIGNMENT

Vocabulary and Core Concept Check

Basic: 1–2, 3–33 odd, 40–41, 47–54

1. VOCABULARY Is 9r + 16 = — a literal equation? Explain.

Average: 1–2, 4–22 even, 23–34, 39–41, 47–54

2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

π 5

Advanced: 1–2, 10–22 even, 26–34 even, 35–37, 39–54 HOMEWORK CHECK

Average: 12, 22, 30, 32, 34

ANSWERS

Solve 24 − 6y = 3x for x in terms of y.

3. y − 3x = 13

4. 2x + y = 7

5. 2y − 18x = −26

6. 20x + 5y = 15

7. 9x − y = 45

1. no; It only has one variable. 2. Solve 6y = 24 − 3x for y in terms of x; y = 4 − —12 x; x = 8 − 2y 3. y = 13 + 3x 4. y = 7 − 2x 5. y = −13 + 9x 6. y = 3 − 4x 7. y = 9x − 45 8. y = 4 9. y = x − 3 10. y = 2x + 1 11. y = 18x + 12 12. y = 16 − 12x 1 13. x = — y 12

9. 4x − 5 = 7 + 4y 1

11. 2 + —6 y = 3x + 4

24. MODELING WITH MATHEMATICS The penny

size of a nail indicates the length of the nail. The penny size d is given by the literal equation d = 4n − 2, where n is the length (in inches) of the nail.

10. 16x + 9 = 9y − 2x 1

12. 11 − —2 y = 3 + 6x

13. y = 4x + 8x

14. m = 10x − x

15. a = 2x + 6xz

16. y = 3bx − 7x

17. y = 4x + rx + 6

18. z = 8 + 6x − px

19. sx + tx = r

20. a = bx + cx + d

21. 12 − 5x − 4kx = y

22. x − 9 + 2wx = y

ERROR ANALYSIS In Exercises 25 and 26, describe and correct the error in solving the equation for x. 25.



C (in dollars) to participate in a ski club is given by the literal equation C = 85x + 60, where x is the number of ski trips you take.

12 − 2x = −2(y − x) −2x = −2(y − x) − 12 x = (y − x) + 6

26.



10 = ax − 3b 10 = x(a − 3b) 10 —=x a − 3b

23. MODELING WITH MATHEMATICS The total cost

1 —9 m

n

a. Solve the equation for n. b. Use the equation from part (a) to find the lengths of nails with the following penny sizes: 3, 6, and 10.

8. 6 − 3y = −6

In Exercises 13–22, solve the literal equation for x. (See Example 2.)

In Exercises 27–30, solve the formula for the indicated variable. (See Examples 3 and 5.)

a. Solve the equation for x.

27. Profit: P = R − C; Solve for C.

b. How many ski trips do you take if you spend a total of $315? $485?

28. Surface area of a cylinder: S = 2π r 2 + 2π rh;

Solve for h. 1

29. Area of a trapezoid: A = —2 h(b1 + b2); Solve for b2.

v −v t

1 0; 30. Average acceleration of an object: a = —

40

Chapter 1

Solving Linear Equations

Solve for v1.

40 25.Int_Math1_PE_0105.indd The equation is not solved for x because

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there is still a term with x on both sides; y+6 x = y − x + 6; 2x = y + 6; x = — 2 26. There is no x in 3b to factor out;

27. 28. 29. 30.

40

Solve 6y = 24 − 3x for y in terms of x.

In Exercises 3–12, solve the literal equation for y. (See Example 1.)

Advanced: 12, 22, 35, 37, 40

a 15. x = — 2 + 6z y 16. x = — 3b − 7 y−6 17. x = — 4+r z−8 18. x = — 6−p r 19. x = — s+t a−d 20. x = — b+c y − 12 21. x = — −5 − 4k y+9 22. x = — 1 + 2w C − 60 23. a. x = — 85 b. 3 trips; 5 trips d+2 24. a. n = — 4 1 b. 1— in.; 2 in.; 3 in. 4

Solve 24 − 3x = 6y for x.

Monitoring Progress and Modeling with Mathematics

Basic: 7, 17, 23, 27, 31

14. x =

Solve 3x + 6y = 24 for x.

10 = ax − 3b; 10 + 3b = ax; 10 + 3b x=— a C=R−P S − 2πr2 h=— 2π r 2A b2 = — − b1 h v1 = at + v0

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31. REWRITING A FORMULA A common statistic used in

35. PROBLEM SOLVING You deposit $2000 in an account

professional football is the quarterback rating. This rating is made up of four major factors. One factor is the completion rating given by the formula

that earns simple interest at an annual rate of 4%. How long must you leave the money in the account to earn $500 in interest? (See Example 7.)

(

C R = 5 — − 0.3 A

36. PROBLEM SOLVING A flight averages 460 miles per

)

hour. The return flight averages 500 miles per hour due to a tailwind. The total flying time is 4.8 hours. How long is each flight? Explain. (See Example 8.)

where C is the number of completed passes and A is the number of attempted passes. Solve the formula for C.

m1m2 F=G — d2

( ) 37. USING STRUCTURE An athletic facility is building an

indoor track. The track is composed of a rectangle and two semicircles, as shown. x

33. MODELING WITH MATHEMATICS The sale price r

S (in dollars) of an item is given by the formula S = L − rL, where L is the list price (in dollars) and r is the discount rate (in decimal form). (See Examples 4 and 6.)

r

a. Solve the formula for r. a. Write a formula for the perimeter of the indoor track. b. Solve the formula for x.

Sale price:

$18

34. MODELING WITH MATHEMATICS The density d of a

c. The perimeter of the track is 660 feet, and r is 50 feet. Find x. Round your answer to the nearest foot.

Dynamic Classroom with Dynamic Investigations

)

R 31. C = A — + 0.3 5 Fd2 32. m1 = — m2G L−S 33. a. r = — L b. 0.4 34. a. m = dV b. 6.012 g 35. 6.25 yr d 36. 2.5 h; 2.3 h; — represents the 460 d original trip time, and — 500 represents the return trip time. Add these expressions and solve for the one-way distance. Substituting the distance into each of the expressions gives the time for each flight. 37. a. P = 2x + 2πr b. x = —12 P − πr

38. MODELING WITH MATHEMATICS The distance

m substance is given by the formula d = —, where m is V its mass and V is its volume.

c. 173 ft 38. a. 55t = 20g 11t b. g = — 4 c. 16.5 gal; 330 mi; The amount of gasoline used can be found using the formula from part (b). Either of the original formulas can be used to find the distance.

d (in miles) you travel in a car is given by the two equations shown, where t is the time (in hours) and g is the number of gallons of gasoline the car uses.

Pyrite Density: 5.01g/cm3

Interactive Whiteboard Lesson Library

(

is given by the formula

b. The list price of the shirt is $30. What is the discount rate?

Dynamic Assessment & Progress Monitoring Tool

ANSWERS

32. REWRITING A FORMULA Newton’s law of gravitation

where F is the force between two objects of masses m1 and m2, G is the gravitational constant, and d is the distance between the two objects. Solve the formula for m1.

Dynamic Teaching Tools

Volume: 1.2 cm3 d = 55t d = 20g

a. Write an equation that relates g and t. b. Solve the equation for g. c. You travel for 6 hours. How many gallons of gasoline does the car use? How far do you travel? Explain.

a. Solve the formula for m. b. Find the mass of the pyrite sample.

Section 1.5

Rewriting Equations and Formulas

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41

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39. MODELING WITH MATHEMATICS One type of stone

ANSWERS 39.

40.

41. 42.

41. MAKING AN ARGUMENT Your friend claims that

formation found in Carlsbad Caverns in New Mexico is called a column. This cylindrical stone formation connects to the ceiling and the floor of a cave.

C a. r = — 2π b. 1.1 ft; 1.3 ft; 1.4 ft c. First find the radius using the formula from part (a), then substitute this into the formula for the area of a circle. a. S = 4ℓb + 2b2 b. Sample answer: ℓ; The formula contains terms with both b and b2, but only one term withℓ. no; 70°F is about 21.1°C, which is greater than 20°C. Sample answer: h = 8; missing base = 2; A = —12 h (b1 + b2), 40 = —12 8 (2 + 8)

Thermometer A displays a greater temperature than Thermometer B. Is your friend correct? Explain your y reasoning. g 100 90 80 70 60 50 40 30 20 10 0 −10

column

°F

Thermometer A Thermometer B

stalagmite

42. THOUGHT PROVOKING Give a possible value for h.

a. Rewrite the formula for the circumference of a circle, so that you can easily calculate the radius of a column given its circumference.

Justify your answer. Draw and label the figure using your chosen value of h.

b. What is the radius (to the nearest tenth of a foot) of a column that has a circumference of 7 feet? 8 feet? 9 feet?

⋅ ⋅

h

A = 40 cm2

c. Explain how you can find the area of a cross section of a column when you know its circumference.

2 cm

8 cm

40. HOW DO YOU SEE IT? The rectangular prism shown

MATHEMATICAL CONNECTIONS In Exercises 43 and 44,

has bases with equal side lengths.

8 cm

write a formula for the area of the regular polygon. Solve the formula for the height h.

A = 40 cm

2

43.

44.

8 cm center

center

2A 5 43. A = — bh; h = — 2 5b A 44. A = 4bh; h = — 4b b+c 45. a = — bx − 1 yb 46. a = — y − xb 47–54. See Additional Answers.

b

h

b

h

b

b

a. Use the figure to write a formula for the surface area S of the rectangular prism.

REASONING In Exercises 45 and 46, solve the literal

equation for a.

b. Your teacher asks you to rewrite the formula by solving for one of the side lengths, b orℓ. Which side length would you choose? Explain your reasoning.

a+b+c ab

45. x = —

( a ab− b )

46. y = x —

Mini-Assessment

Maintaining Mathematical Proficiency

1. Solve 9 − c = 17d for c. c = 9 − 17d 2. Solve nt + 6 − xt = s for t. s−6 t=— n−x 3. Solve the formula for the volume of a rectangular prism V =ℓhw for w. V w=— ℓh 4. A circular trampoline has a padded rope that goes around the outside of the frame. The rope is sold for $2.75 per foot and costs $104.50 before tax for this trampoline. What is the diameter of the trampoline, to the nearest foot? 12 feet 5. Kayla drives 144 miles in 3 hours. Kendra drives 224 miles in 4 hours. How much faster does Kendra drive on average? Kendra drives an average of 8 miles per hour faster than Kayla.

42

(Skills Review Handbook)

Evaluate the expression. 47. 15 − 5 + 52



48. 18 2 − 42 ÷ 8



49. 33 + 12 ÷ 3 5

Solve the equation. Graph the solutions, if possible. 51.

42

∣x − 3∣ + 4 = 9 Chapter 1

Reviewing what you learned in previous grades and lessons

52.

∣ 3y − 12 ∣ − 7 = 2

50. 25(5 − 6) + 9 ÷ 3

(Section 1.4)

53. 2∣ 2r + 4 ∣ = −16

54. −4∣ s + 9 ∣ = −24

Solving Linear Equations

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If students got it...

Resources by Chapter • Practice A and Practice B • Puzzle Time

Resources by Chapter • Enrichment and Extension • Cumulative Review

Student Journal • Practice

Start the next Section

Differentiating the Lesson Skills Review Handbook

Chapter 1

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1.4–1.5

What Did You Learn?

Dynamic Teaching Tools Dynamic Assessment & Progress Monitoring Tool

Core Vocabulary absolute value equation, p. 28 extraneous solution, p. 31

Interactive Whiteboard Lesson Library literal equation, p. 36 formula, p. 37

Dynamic Classroom with Dynamic Investigations

ANSWERS 1. Sample answer: An absolute value equation must be set equal to a positive value to have a solution. Isolate the absolute value on one side of the equation to determine whether it has a solution. 2. Sample answer: The absolute values are the same, so treat them as a single variable. 3. Sample answer: the center and two adjacent vertices to form congruent triangles

Core Concepts Section 1.4 Properties of Absolute Value, p. 28 Solving Absolute Value Equations, p. 28 Solving Equations with Two Absolute Values, p. 30 Special Solutions of Absolute Value Equations, p. 31

Section 1.5 Rewriting Literal Equations, p. 36 Common Formulas, p. 38

Mathematical Practices 1.

How did you decide whether your friend’s argument in Exercise 46 on page 33 made sense?

2.

How did you use the structure of the equation in Exercise 59 on page 34 to rewrite the equation?

3.

What entry points did you use to answer Exercises 43 and 44 on page 42?

Performance Task:

Dead Reckoning Have you ever wondered how sailors navigated the oceans before the Global Positioning System (GPS)? One method sailors used is called dead reckoning. How does dead reckoning use mathematics to track locations? Could you use this method today? To explore the answers to these questions and more, check out the Performance Task and Real-Life STEM video at BigIdeasMath.com.

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1

ANSWERS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

z = −9; Subtract 3 from each side. t = −13; Divide each side by −0.2. n = 10; Multiply each side by −5. y = −9 b = −5 n=6 z = −5 x = 18 25 w=— 4 x = 10; 110°, 50°, 20° x = 126; 126°, 96°, 126°, 96°, 96°

Chapter Review 1.1

Dynamic Solutions available at BigIdeasMath.com

Solving Simple Equations (pp. 3–10)

a. Solve x − 5 = −9. Justify each step. x − 5 = −9 Addition Property of Equality

+5

Write the equation.

+5

Add 5 to each side.

x = −4

Simplify.

The solution is x = −4. b. Solve 4x = 12. Justify each step.

Division Property of Equality

4x = 12

Write the equation.

4x 12 —=— 4 4

Divide each side by 4.

x=3

Simplify.

The solution is x = 3. Solve the equation. Justify each step. Check your solution. 1. z + 3 = − 6

3.2 3 .2 2 1.2

n 5

3. −— = −2

2. 2.6 = −0.2t

Solving Multi-Step Equations (pp. 11–18)

Solve −6x + 23 + 2x = 15. −6x + 23 + 2x = 15

Write the equation.

−4x + 23 = 15

Combine like terms.

−4x = −8

Subtract 23 from each side.

x=2

Divide each side by −4.

The solution is x = 2. Solve the equation. Check your solution. 4. 3y + 11 = −16

5.

7. −4(2z + 6) − 12 = 4

8. —32 (x − 2) − 5 = 19

6=1−b

6. n + 5n + 7 = 43 1

7

9. 6 = —5 w + —5 w − 4

Find the value of x. Then find the angle measures of the polygon. 10.

11.

110° 5x° 2x° Sum of angle measures: 180°

44

Chapter 1

(x − 30)° (x − 30)°





(x − 30)°

Sum of angle measures: 540°

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ANSWERS 1.3

12. 13. 14. 15. 16. 17. 18.

Solving Equations with Variables on Both Sides (pp. 19–24)

Solve 2( y − 4) = −4( y + 8). 2( y − 4) = −4( y + 8)

Write the equation.

2y − 8 = −4y − 32

Distributive Property

6y − 8 = −32

Add 4y to each side.

6y = −24

n = −4 infinitely many solutions no solution y = 14, y = −20 1 w = 3, w = −—5 x = −1 ∣ v − 84.5 ∣ = 10.5

Add 8 to each side.

y = −4

Divide each side by 6.

The solution is y = −4. Solve the equation. 12. 3n − 3 = 4n + 1

1.4

1

13. 5(1 + x) = 5x + 5

14. 3(n + 4) = —2 (6n + 4)

Solving Absolute Value Equations (pp. 27–34)

a. Solve ∣ x − 5 ∣ = 3. x−5= +5

3

x − 5 = −3

or

+5

+5

x=8

+5

x=2

Write related linear equations. Add 5 to each side. Simplify.

The solutions are x = 8 and x = 2. Check

b. Solve ∣ 2x + 6 ∣ = 4x. Check your solutions. 2x + 6 = 4x −2x

−2x

or

2x + 6 = −4x −2x

−2x

Write related linear equations. Subtract 2x from each side.

6 = 2x

6 = −6x

—=—

2x 2

—=—

Solve for x.

3=x

−1 = x

Simplify.

6 2

6 −6

−6x −6

Simplify.

∣ 2x + 6 ∣ = 4x ? ∣ 2(3) + 6 ∣ = 4(3) ∣ 12 ∣ =? 12 12 = 12



∣ 2x + 6 ∣ = 4x ? ∣ 2(−1) + 6 ∣ = 4(−1) ∣ 4 ∣ =? −4

Check the apparent solutions to see if either is extraneous. The solution is x = 3. Reject x = −1 because it is extraneous.

∕ −4 4=



Solve the equation. Check your solutions. 15.

∣ y + 3 ∣ = 17

16. −2∣ 5w − 7 ∣ + 9 = − 7

17. ∣ x − 2 ∣ = ∣ 4 + x ∣

18. The minimum sustained wind speed of a Category 1 hurricane is 74 miles per hour. The maximum

sustained wind speed is 95 miles per hour. Write an absolute value equation that represents the minimum and maximum speeds.

Chapter 1

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ANSWERS

19. y = —12 x − 5 20. y = 2x − 2 a 21. y = — 9 + 3x 3V 22. a. h = — B b. 18 cm 23. a. K = —59 (F − 32) + 273.15 b. about 355.37 K

1.5

Rewriting Equations and Formulas (pp. 35–42)

a. The slope-intercept form of a linear equation is y = mx + b. Solve the equation for m. y = mx + b

Write the equation.

y − b = mx + b − b

Subtract b from each side.

y − b = mx

Simplify.

y − b mx —=— x x y−b —=m x

Divide each side by x. Simplify.

y−b When you solve the equation for m, you obtain m = —. x

b. The formula for the surface area S of a cylinder is S = 2𝛑 r 2 + 2𝛑 rh. Solve the formula for the height h. 2π r 2 + 2πrh

S= −

2πr 2

S−

2πr 2

− 2πr 2 = 2πrh

S − 2πr 2 2πrh —=— 2πr 2πr 2πr 2

S− 2πr

—=h

Write the equation. Subtract 2πr 2 from each side. Simplify. Divide each side by 2πr. Simplify.

S − 2π r 2 When you solve the formula for h, you obtain h = —. 2πr Solve the literal equation for y. 19. 2x − 4y = 20

20. 8x − 3 = 5 + 4y

21. a = 9y + 3yx 1

22. The volume V of a pyramid is given by the formula V = —3 Bh, where B is the area of the

base and h is the height. a. Solve the formula for h. b. Find the height h of the pyramid.

V = 216 cm3

B = 36 cm2 9

23. The formula F = —5 (K − 273.15) + 32 converts a temperature from kelvin K to degrees

Fahrenheit F. a. Solve the formula for K. b. Convert 180°F to kelvin K. Round your answer to the nearest hundredth.

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1

Chapter Test

ANSWERS 1. x = 22; Add 7 to each side. 2. x = −3; Subtract 5 from each side; Multiply each side by —32 . 1 3. x = −—5 ; Subtract x and 1 from each side; Divide each side by 10. 4. x = −3, x = 9 5. x = 3 6. infinitely many solutions 7. x = −6 8. x = −4, x = 8 9. no solution 10. c ≠ 5; If c is 5, then the equation is an identity. For all other values of c, subtracting 3x from each side will give a statement that is always false. 11. c < 0; An absolute value cannot be negative. 12. ∣ h − 34 ∣ = 4 P − 2ℓ 13. a. w = — 2 b. 65 yd c. about 4.8% 14. a. 2.1 h; The cost at the dealership is 24 + 99t and the cost at the local mechanic is 45 + 89t. Set these two expressions equal and solve for the time. b. time is less than 2.1 h; time is greater than 2.1 h; Because the expressions are equal for 2.1 hours, that is the cutoff point from the dealership being less expensive to the local mechanic being less expensive. 15. It will give a negative value on the right and absolute value cannot be negative. 16. It is an identity, meaning that all real numbers are solutions of the equation.

Solve the equation. Justify each step. Check your solution. 2. —23 x + 5 = 3

1. x − 7 = 15

3. 11x + 1 = −1 + x

Solve the equation. 4. 2∣ x − 3 ∣ − 5 = 7

5.

∣ 2x − 19 ∣ = 4x + 1

6. −2 + 5x − 7 = 3x − 9 + 2x

7. 3(x + 4) − 1 = −7

8.

∣ 20 + 2x ∣ = ∣ 4x + 4 ∣

9. —13 (6x + 12) − 2(x − 7) = 19

Describe the values of c for which the equation has no solution. Explain your reasoning. 10. 3x − 5 = 3x − c

11.

∣x − 7∣ = c

12. A safety regulation states that the minimum height of a handrail is 30 inches. The

maximum height is 38 inches. Write an absolute value equation that represents the minimum and maximum heights. 13. The perimeter P (in yards) of a soccer field is represented by the formula P = 2ℓ+ 2w,

whereℓ is the length (in yards) and w is the width (in yards). a. Solve the formula for w. b. Find the width of the field. c. About what percent of the field is inside the circle?

P = 330 yd

10 yd = 100 yd

14. Your car needs new brakes. You call a dealership and a local

mechanic for prices. Cost of parts

Labor cost per hour

Dealership

$24

$99

Local Mechanic

$45

$89

a. After how many hours are the total costs the same at both places? Justify your answer. b. When do the repairs cost less at the dealership? at the local mechanic? Explain. 15. Consider the equation ∣ 4x + 20 ∣ = 6x. Without calculating, how do you know that x = −2 is an

extraneous solution? 16. Your friend was solving the equation shown and was confused by the result

“−8 = −8.” Explain what this result means. 4(y − 2) − 2y = 6y − 8 − 4y 4y − 8 − 2y = 6y − 8 − 4y 2y − 8 = 2y − 8 −8 = −8 Chapter 1

Chapter Test

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Lesson Tutorials

Resources by Chapter • Enrichment and Extension • Cumulative Review

Skills Review Handbook

Performance Task

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1

ANSWERS 1. B a+b 2. cx − a + b = 2b, x = —, c b + a = cx 3. a. < b. < c. > d. < e. = f. = 4. a. 24x + 28(5 − x) = 132 or 24(5 − x) + 28x = 132 b. $4; Sample answer: Switching gives a total cost of $128, which is $4 less than $132.

Cumulative Assessment

1. A mountain biking park has 48 trails, 37.5% of which are beginner trails. The rest are

divided evenly between intermediate and expert trails. How many of each kind of trail are there? A 12 beginner, 18 intermediate, 18 expert ○ B 18 beginner, 15 intermediate, 15 expert ○ C 18 beginner, 12 intermediate, 18 expert ○ D 30 beginner, 9 intermediate, 9 expert ○ 2. Which of the equations are equivalent to cx − a = b?

cx − a + b = 2b

0 = cx − a + b

b 2cx − 2a = — 2

b x−a=— c

a+b x=— c

b + a = cx

3. Let N represent the number of solutions of the equation 3(x − a) = 3x − 6. Complete

each statement with the symbol , or =. a. When a = 3, N ____ 1. b. When a = −3, N ____ 1. c. When a = 2, N ____ 1. d. When a = −2, N ____ 1. e. When a = x, N ____ 1. f. When a = −x, N ____ 1.

4. You are painting your dining room white and your living room blue. You spend

$132 on 5 cans of paint. The white paint costs $24 per can, and the blue paint costs $28 per can. a. Use the numbers and symbols to write an equation that represents how many cans of each color you bought. x

132

5

24

28

=

(

)

+



×

÷

b. How much would you have saved by switching the colors of the dining room and living room? Explain.

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ANSWERS 5. 8x + 6 = −2x − 14 and 5x + 3 = −7 6. B 7. a. 45m = 99 + 36m; month 11 b. yes; Because the expenses are equal for month 11, any time after that satellite TV will be less expensive. 8. no solution: 12x − 2x = 10x − 8, 0 = ∣ x + 13 ∣ + 2, 3x − 12 = 3(x − 4) + 1 one solution: ∣ 8x + 3 ∣ = 0, −2x + 4 = 2x + 4, −6 = 5x − 9 two solutions: 9 = 3∣ 2x − 11 ∣ infinitely many solutions: −4(x + 4) = −4x − 16, 7 − 2x = 3 − 2(x − 2) sec sec 9. 80 —, — ft 80 ft

5. Which of the equations are equivalent?

6x + 6 = −14

8x + 6 = −2x − 14

5x + 3 = −7

7x + 3 = 2x − 13

6. The perimeter of the triangle is 13 inches. What is the length of the shortest

side? (x − 5) in.

A 2 in. ○ B 3 in. ○

x 2

C 4 in. ○

in.

6 in.

D 8 in. ○

7. You pay $45 per month for cable TV. Your friend buys a satellite TV receiver for $99 and

pays $36 per month for satellite TV. Your friend claims that the expenses for a year of satellite TV are less than the expenses for a year of cable. a. Write and solve an equation to determine when you and your friend will have paid the same amount for TV services. b. Is your friend correct? Explain.

8. Place each equation into one of the four categories. No solution

One solution

Two solutions

Infinitely many solutions

∣ 8x + 3 ∣ = 0

−6 = 5x − 9

3x − 12 = 3(x − 4) + 1

−2x + 4 = 2x + 4

0 = ∣ x + 13 ∣ + 2

−4(x + 4) = −4x − 16

12x − 2x = 10x − 8

9 = 3∣ 2x − 11 ∣

7 − 2x = 3 − 2(x − 2)

9. A car travels 1000 feet in 12.5 seconds. Which of the expressions do not represent the

average speed of the car? second 80 — feet

feet 80 — second

80 feet second





second 80 feet

Chapter 1

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Integrated Mathematics I Ancillaries Assessment Book .....................................................A2 Resources by Chapter ..............................................A5 Student Journal .......................................................A8 Differentiating the Lesson ....................................A10

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Assessment Book The Assessment Book contains formative and summative assessment options, providing teachers with the ability to assess students on the same content in a variety of ways. It is available in print and online in an editable format. Name _________________________________________________________ Date _________

Skills Test

Prerequisite Skills Test

Add or subtract.

u Prerequisite Skills Test with Item Analysis

Answers

1. − 9 + ( −15)

2. 2 + ( − 3)

3. 6 − 9

1.

4. − 6 − 11

5. 13 + 8

6. −12 − ( −10)

2.

8. − 8 • 2

9. 9 ÷ 3

4.

12. −1( − 7)

5.

3.

Multiply or divide. 7. 2( − 7)

The Prerequisite Skills Test checks students’ understanding of previously learned mathematical skills they will need to be successful in their math class. You can use the Item Analysis to diagnose topics your students need to review to prepare them for the school year.

10. 25 ÷ ( − 5)

11. − 30 ÷ ( − 6)

6.

Solve the problem and specify the units of measure. 13. The length of a rectangle is 6 feet and the width is 3 feet. Find the perimeter

of the rectangle.

8. 9.

14. One side of a square measures 9 centimeters. Find the area of the square.

10.

Graph the number. 15. 4

16.

−5 −4 −3 −2 −1

7.

0

1

2

3

4

−3

11.

−5 −4 −3 −2 −1

5

0

1

2

3

4

5

12. 13.

17. − 6 + 5

18. 1 − − 3

−5 −4 −3 −2 −1

0

1

2

3

4

14.

−5 −4 −3 −2 −1

5

0

1

2

3

4

5

Complete the statement with , or = . 19. 3____7

20. −1____4

21. − 4____ −10

22.

− 6 ____ − 3

15.

See left.

16.

See left.

17.

See left.

18.

See left.

19.

Evaluate the expression for the given value of x. 23. 2 x − 6; x = 9

24. − 7 + 9 x; x = 3

20.

25. 12 x + 13; x = 5

21. 22. 23. 24. 25.

Name _________________________________________________________ Date _________

Post Course

u Pre-Course Test with Item Analysis/Post-Course Test with Item Analysis

Post Course Test

Solve the equation. Check your solution.

Answers 2. 3x − 2 = 4 − x

1. x + 14 = − 5

1.

Solve the equation.

2.

4. − 23 = − 3( − 4b − 3) − 8(1 + b)

3. − 3 + 4 8n + 10 = 53

Describe the values of c for which the equation has no solution.

The Pre-Course Test and PostCourse Test cover key concepts that students will learn in their math class. You can gauge how much your students learned throughout the year by comparing their PreTest score against their Post-Test score. These tests also can be given as practice for state assessments.

4.

5. − 3 x + c = − 3 x + 8 6. − 3 + x = c Name _________________________________________________________ Date _________ 7. The quotient of n and 3 is less than 5. Pre-Course Test

7.

8. 10 more than y is greater than or equal to 17. Solve the equation. Check your solution. Solve the inequality. 2 x + 7the = solution. −5 + x 1. x − 12 = 9 2. Graph 9. −3 − 3 ≤ − 3(1 − x)

Solve the equation.

8.

Answers

9. 1.

10. − 3 + 2 x + 1 ≤ −2

See left.

2.

(− 61 x

3. 8 2 − 9 p − 2 = 14

−5 −4 −3 −24.−1 − 30

+2 63) + 6( 4 x − 3) −6 = −−478−2

0

3.

2

4

6

8

Write and graph a compound inequality that represents the numbers that are 4. Describe the values of c for which the equation has no solution. nott solutions of the inequality represented by the graph shown. 5. 5 x + 4 = 5 x − c 6. 2 x + 6 = c 5. 11. −2 −1

0

1

2

Write the sentence as an inequality. 12. 7. A number m increased by 12−2 is less0 than248.

3

4

5

y 2

3

4

5

6

7

8

−3

−4

−5

−2 −1

12.

−2

0

1

0

2

3

−6

0

5

2

4

6

8

10.

Determine whether the relation is a function. If the relation is a function, determine whether the function is linear or nonlinear. 13.

x

1

2

3

4

5

y

1

4

9

16

25

15. 16.

See left.

6

16. through: ( − 3, − 4), perpendicular to y = − 3 x + 5 4 4 6 8

2

See left. 13.

14.

−7

−2

15. through: ( − 3, 1), slope = 2 4

See left. 12.

7.

8

See left. Write an equation in slope-intercept form of the line Write and graph a compound inequality that represents the numbers that arewith the given characteristics. not solutions of the inequality represented by the graph shown. 11.

11.

See left. 11.

6

6

−4

9 10

10.

6. 4

Determine whether the relation is a function, 8. The product of x and 10 is greater than or equal to 23.is a function. If the relation 8. determine whether the function is linearr or nonlinear. r 9. Solve the inequality. Graph the solution. 2 13. 14. y = − ( x + 1) 010. 32 2 −6x + 94 ≤ 16 x −3 9. − 6 x − ( − 7 x − 1) ≤ 6 See left. 1

5. 6.

Write the sentence as an inequality.

PreCourse

3.

12. See left. 13.

14. y = 3 x + 1

Write an equation in slope-intercept form of the line with the given characteristics. 15. through: (3, − 3) and ( − 3, 4)

14.

15. 16.

16. through: ( − 3, 5), parallel to y = 2 x + 2

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Integrated Math I PE_TE sampler.indd A2

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Name _________________________________________________________ Date _________

Chapter

1

Quiz For use after Section 1.3

Solve the equation.

Answers

1. x + 7 = 3

2. 5.3 = z − 2.7

3. 72 = − 6r

4.

2.

5. 4m − 6 = 14

6. 8 = 11 − v

7. 9 = − 7 w + 12 w − 6

8. 40 = −

3.

11. − 3( g − 4) = 2 − 3 g

The Quiz provides ongoing assessment of student understanding. You can use this as a gradable quiz or as practice.

4.

1 (9 x + 30) + 2 3

5. 6.

Solve the equation. Determine whether the equation has one solution, no solution, or infinitely many solutions. 9. 6c + 3 = c − 12

u Quiz

1.

1 p = 15 5

7.

10. 4( 2q − 3) = 8q − 12

8.

12. 8( y − 1) − 3 y = 6( 2 y − 6)

9.

13. To estimate your average monthly salary, divide your yearly salary by the

number of months in a year. Write and solve an equation to determine your yearly salary when your average monthly salary is $4560.

10.

14. You are a contractor and charge $45 for a site visit plus an additional 11.

$24 per hour for each hour you spend working at the site. Write and solve an equation to determine how many total hours you have to work to earn $810 working at two separate work sights.

12.

15. You and a friend are both traveling from the Seattle area. You start 38 miles

east of Seattle and travel east on Interstate 90 at 62 miles per hour. Your friend starts 20 miles south of Seattle and travels south on Interstate 5 at 65 miles per hour.

13.

a. Who would be farther from Seattle in two hours and by how much? 14.

b. How many hours will it take for you and your friend to be the same

distance from Seattle? 15. a.

b.

Name_________________________________________________________

Chapter

1

Solve the equation. Justify each step. 1. x +

Date __________

Test A Answers

1 3 = 2 4

2.

z = 12 4

Solve the equation. Determine whether the equation has one solution, no solution, or infinitely many solutions.

2.

3. 5n = − 20

4. g + 5 = 17

5. 13 + 3 p + 10 = 23 + 3 p

6. 7 + 4 y = 39

3.

7. 3 = t + 11.5 − t

8. 4 x + 8 + 6 x − 5 = 33

4.

9. 5( 2c + 7) − 3c = 7(c + 5)

10.

5.

3 1 b + 6 + b = 15 + 2b 2 2

6.

Describe the values of c for which the equation has no solution. 11. 2 x − 6 = 2 x − c

12.

7.

x +8 = c

8.

Find the value of the variable. Then find the angle measures of the polygon. 13. 5x°

14.

u Chapter Tests

1.

5b°

4b°

9. 10.

The Chapter Tests provide assessment of student understanding of key concepts taught in the chapter. There are two tests for each chapter. You can use these as gradable tests or as practice for your students to master an upcoming test.

11.



12.

4b°

Sum of angle measures: 180°

Sum of angle measures: 360°

Solve the equation. 15. 7 n + 3 = 2n + 23

17. 19.

5b°

3 1 (d + 12) = (2d − 6) 2 2

16.

1 (6 x + 4) = 5(2 x − 8) 2

18.

b − 12 = 15

20.

2k + 6 = k

13.

14.

15. 16.

2 r + 5 = 3r

17. 18. 19.

Name_________________________________________________________

Chapter

1

Date __________

20.

Alternative Assessment

1. Which of the following equations have only one solution? Which have two

solutions? Which have no solution? Which have infinitely many solutions? a.

3 x −8 =1 5

b. 5 x − 2( x − 2) = 7 x + 4 − 4 x c.

u Alternative Assessment with Scoring Rubric

3x + 2 − 2 = 6

d. 3 x − 2 + 4 = 2

1 e. (6 x − 3) − x = 2 x + 1 2 f.

8 x − 5 = 3x + 5

2. A doormat is in the shape of a trapezoid. The area A of the doormat is represented

by the formula A =

1 h(b1 + b2 ). 2

a. Solve the formula for h. b. Show that solving the formula for b1 by first multiplying both sides by 2 and

then dividing both sides by h leads to b1 =

2A − b2 . h

c. Show that solving the formula for b1 by first multiplying both sides by 2 and

then using the Distributive Property to distribute the h leads to b1 =

2 A − b2h . h

d. Show that the final formula in part (b) is equivalent to the final formula in part

(c) by showing the steps to transform one formula to the other one. e. Explain how you could solve the original formula for b2.

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Integrated Math I PE_TE sampler.indd A3

Each Alternative Assessment includes at least one multi-step problem that combines a variety of concepts from the chapter. Students are asked to explain their solutions, write about the mathematics, or compare and analyze different situations. You can use this as an alternative to traditional tests. It can be graded, assigned as a project, or given as practice.

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Name_________________________________________________________

Chapter

Performance Task (continued)

1

u Performance Task The Performance Task presents an assessment in a real-world situation. Every chapter has a task that allows students to work with multiple standards and apply their knowledge to realistic scenarios. You can use this task as an in-class project, a take-home assignment or as a graded assessment.

Date __________

Magic of Mathematics Have you ever watched a magician perform a number trick? You can use algebra to explain how these types of tricks work. Work through problems 1 and 2 with a partner. Take turns having one of you read the steps aloud as the other calculates and records the results. Then repeat the steps with a variable in place of a specific number. Work together using algebra to explain why the magic trick works. 1. Step 1: Think of any number.

Step 2: Double the number. Step 3: Add 5. Step 4: Double your new number. Step 5: Subtract 6. Step 6: Divide by 4. Step 7: Subtract your original number. Did you get 1? Use algebra to show why this works. 2. Step 1: Select a two-digit number less than 100.

Step 2: Add 3 to your number. Step 3: Double the result. Step 4: Subtract 9. Step 5: Multiply by 5. Step 6: Add 15. Step 7: Divide by 10. Did you get your original number? Use algebra to show why this works. How is this problem different from the previous problem? 3. Create your own magic number trick and use algebra to explain why it works.

u Quarterly Standards Based Test The Quarterly Standards Based Test provides students practice answering questions in standardized test format. The assessments are cumulative and cover material from multiple chapters of the textbook. The questions represent problem types and reasoning patterns frequently found on standardized tests. You can give this test to your students as a cumulative assessment for the quarter or as practice for state assessment.

Name_________________________________________________________

Chapters

1–3

Date __________

Quarterly Standards Based Test (continued)

8. Translate the equation 4 x − 5 = 3( x + 2) into a verbal sentence. A. Four times the difference of x and five equals three times the sum of x and two. B. Four times x less than five is the same as three times x plus two. C. The sum of four times x and five is three times the sum of x and two. D. The difference of four times x and five is three times the sum of x and two. 9. A boat travels upstream on the Allegheny River for 2 hours. The return trip only

takes 1.7 hours because the boat travels 2.5 miles per hour faster downstream due to the current. How far does the boat travel one way? A. 14 1 mi 6

B. 14 mi

C. 28 1 mi 3

D. 28 mi

10. Solve 1 (6 x + 30) = 4 x + 2( x − 7). 3 A. x = 6

B. x = 11

C. x = 4 1 4

D. x = −1

11. Solve 3n + 6 = 21. A. n = − 5 and n = 5

B. n = − 9 and n = 9

C. n = − 9 and n = 5

D. no solution

12. Translate 5 x + 7 ≥ 2 into a verbal sentence. A. Five times the sum of x and seven is greater than two. B. Five times x, increased by seven, is less than or equal to two. C. The product of five and x, increased by seven, is greater than or equal to two. D. Seven more than the product of five and x is greater than two.

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Integrated Math I PE_TE sampler.indd A4

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Resources by Chapter The Resources by Chapter ancillary includes a number of supplemental resources for every chapter in the program. It is available in print and online in an editable format.

Nombre _______________________________________________________

Capítulo

1

Fecha _________

Resolver ecuaciones lineales

Estimada familia:

Resolver ecuaciones es una destreza importante en la clase de matemáticas, pero ¿qué pasa en la vida cotidiana? ¿Alguna vez han considerado cómo pueden usar esta destreza en la vida real?

u Family Communication Letters (English and Spanish) The Family Communication Letters provide a way to quickly communicate to family members how they can help their student with the material of the chapter. You can send this home with your students to help make the mathematics less intimidating and provide suggestions for families to help their students see mathematical concepts in common activities.

Consideren la siguiente situación: •

Usted y su familia quieren comprar un nuevo sistema de videojuegos. El sistema cuesta $400. Ya han ahorrado $250 para el sistema. ¿Cuánto dinero aún necesitan ahorrar para comprar el sistema?

Podrían simplemente restar $250 de $400 para obtener la respuesta. En cambio, consideren escribir una ecuación. ¿Cómo sería? En este caso, la incógnita, la cantidad de dinero que falta ahorrar, puede representarse con una variable. Un ejemplo de la ecuación sería y + 250 = 400, donde

y es el valor desconocido. Comenten cómo hallarían el valor desconocido. Ahora, en familia, propongan maneras en que podrían ganar el dinero. ¿Ganarán el dinero trabajando juntos? ¿O dividirán la Date restante cantidad Name _________________________________________________________ _________ que deben ahorrar entre los integrantes de su familia y trabajarán por Chapter separado?

1

Solving Linear Equations

Dear Family,

Luego, escriban una ecuación para cada situación. ¿En qué se parecen las ecuaciones? ¿En qué se diferencian? ¿El valor de la variable es igual para ambas situaciones?

A medida que su hijo avanza en el capítulo 1, aprenderá cómo resolver Solving equations is an important skill in the math classroom, but how about in tipos de ecuaciones semejantes. Compartan otras maneras en que su familia everyday life? Have you ever considered how you may use this skill in real life? usa ecuaciones en la vida cotidiana, quizás sin ni siquiera darse cuenta. Consider the following scenario:



You and your family want to purchase a new video game system. The system costs $400. You already have $250 saved to put toward the system. How much money do you still need to save to buy the system?

You could simply subtract $250 from $400 to get your answer. Consider writing an equation instead. What would that look like? The unknown, in this case, the amount of money left to save, can be represented by a variable. One example of an equation is y + 250 = 400, where y is the unknown value. Discuss how you would find the unknown value. Now, as a family, brainstorm ways you could earn the money. Will you earn the money working together? Or will you divide the remaining amount needed to be earned among the members of your family and work independently? Copyrightfor © Big Ideasscenario. Learning, LLC Next, write an equation each How are the equations the same? All rights reserved. How are they different? Is the value of the variable the same for both scenarios?

Algebra 1 Resources by Chapter

3

As your child works through Chapter 1, he or she will learn how to solve similar types of equations. Share together other ways you as family use equations in everyday life, maybe without even realizing it.

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Integrated Math I PE_TE sampler.indd A5

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u Start Thinking/Warm Up/ Cumulative Review Warm Up Each Start Thinking/Warm Up/ Cumulative Review Warm Up includes two options for getting the class started. The Start Thinking questions provide students with an opportunity to discuss thought-provoking questions and analyze real-world situations. You can use this to begin a class discussion. The Warm Up questions review prerequisite skills needed for the lesson. You can give this to students at the beginning of class to prepare them for the upcoming lesson. The Cumulative Review Warm Up questions review material from earlier lessons or courses. You can give these to students to continue their mastery of concepts they have been taught.

1.1

Start Thinking

The angle measures of any four-sided figure sum to 360 degrees. How can you find the fourth angle measure when you know the sum of the other three angle measures?

1.1

Warm Up

Simplify the expression. 1. 5 + ( −15)

2. 6 − 7

3. 10 • ( −1)

− 30 4. 2

5. −1 × 0

6. 4 − ( − 5)

1.1

Cumulative Review Warm Up

Tell which property the statement illustrates. 1. 2 + 4 = 4 + 2

2. (3 • 7) 4 = 3(7 • 4)

3. 8 + 0 = 8

4. 7 • ¨ ¸ = 1

5. 4 • 0 = 0

6. 12(8 + 3) = 12 • 8 + 12 • 3

§1· ©7¹

Name_________________________________________________________

u Practice The Practice exercises provide additional practice on the key concepts taught in the lesson. There are two levels of practice provided for each lesson: A (basic) and B (average). You can assign these exercises for students that need the extra work or as a gradable assignment.

1.1

Date __________

Practice A

In Exercises 1–6, solve the equation. Justify each step. Check your solution. 1. x + 2 = 5

2. g − 4 = 3

4. d + 4 = − 2

5.

3. m − 1 = 8

p + 7 = 5

6. k − 6 = − 4

The sum of the angle measures of a quadrilateral is 360°. In Exercises 7 and 8, write and solve an equation to find the value of x. Use a protractor to check the reasonableness of your answer.

7.

8. x°

125° 132°

110°

72°

x° 90°

80°

In Exercises 9–14, solve the equation. Justify each step. Check your solutions. 9. 3t = 24 12.

j ÷5 = 2

10. 7 p = 28

11. s ÷ 4 = 3

13. − 6q = 54

14. c ÷ ( − 9) = 2

In Exercises 15–20, solve the equation. Check your solution. 15. h + 1 = 5 3 3

16. w − 7 = 2 9 9

17. 3 f = 9 5

18. u + 2.7 = 1.5

19. 32π t = 64π

20. m ÷ ( − 7) = 2.1

In Exercises 21–23, write and solve an equation to answer the question. 21. The width of a laptop is 11.25 inches. The width is 0.75 times the length. What is

the length of the laptop? 22. The temperature at 10 A.M. is 12 degrees Fahrenheit. The temperature at 6:00 A.M.

was − 7 degrees Fahrenheit. How many degrees did the temperature rise? 23. The population of a city is 645 people less than it was 5 years ago. The current

population is 13,500. What was the population 5 years ago? 24. Identify the property of equality that makes Equation 1 and Equation 2 equivalent.

A6

Integrated Math I PE_TE sampler.indd A6

Equation 1

4.2 x − 1.5 = 1.7 x + 8.3

Equation 2

42 x − 15 = 17 x + 83

Copyright © Big Ideas Learning, LLC. All rights reserved.

10/20/14 1:42 PM

Name_________________________________________________________

1.1

Date __________

Enrichment and Extension

u Enrichment and Extension

Solving Simple Equations Solve the equation. Justify each step. Check your solution. 1. x +

4.

4 5 = 5 2

2.

m 1 =1 −7 4

5.

7. 3.2t = 6.4

9 3 = t 16 4

π 2

t =

5π 6

8. 150 = 7.5x

3. w −

1 2 = 2 2 3

6. x ÷

4 7 = − 5 8

Each Enrichment and Extension extends the lesson and provides a challenging application of the key concepts. These rigorous exercises can be assigned to challenge your students to use higher order thinking skills or to build on a concept via an extension of the lesson.

9. w ÷ 2.4 = 6.52

The sum of the angle measures of a polygon follows the general rule of ( n − 2) • 180°, where the variable n represents the number of sides. In Exercises 10–15, write and solve an equation to find the value of x. Use a protractor to check the reasonableness of your answer. 10.

11. 96°

72° x°

35°



114° 48°

12.



131°

13.



47° x°

x° x°





14.



15. 90°



2x + 20° x°

2x − 10°



16. It takes a plane 4 hours and 15 minutes to fly from Orlando, Florida, to Boston,

ance between th iti iis 1114 miles. il Massachusetts. The distance the ttwo cities a. What is the average sspeed of the plane in miles per hour?

_________________________________________________________ Date _________ b. If every mile is approximately o 1.6Name kilometers, what is the speed of the airplane

in kilometers per hour? u

1.1 Puzzle Time

u Puzzle Time

Did You Hear About The Tree's Birthday? Circle the letter of each correct answer in the boxes below. The circled letters will spell out the answer to the riddle.

Each Puzzle Time provides additional practice in a fun format in which students use their mathematical knowledge to solve a riddle. This format allows students to self-check their work. Your students can learn the lesson concepts by finding the answers to silly jokes and riddles.

Find the value of the variable of the equation. 2. x + 11 = 4

1. m + 7 = 9 3. n −

3 2 = 5 5

4. −18 = r − 12

5. s − ( −10) = 2

6. 6.3 = b − 1.5 8. y ÷ 9 = − 3

7. 1.4h = 5.6 9. − 7c = − 63

10.

6 11. − a = 18 7

x = −3 8

12. −144π = −12π k

Solve an equation to answer the question. 13. The students on a decorating committee create a banner. The length of the

banner is 2.5 times its width. The length of the banner is 20 feet. What is the width (in feet) of the banner? 14. The student council consists of 32 members. There are 27 members decorating

for the dance. How many members are not decorating?

B

L

I

B

M

15

−18

4

2.1

13

A

Q

G

U

S

T

T

−16 −21 A

C

N

W

D

A

O

E

S

P

4.6

17

9

−13

5

9.1

8.2

7.8

11.6

P

R

I

P

Y

O

J

N

E

−7

8

−20

12

−6

H

−8 18 1 −14 −27 20 −17 16 2 −24 14 Name_________________________________________________________ Date __________

Chapter

1

Cumulative Review

In Exercises 1–8, add or subtract. 1. 4 − 6

2. 0 + ( −17)

3. − 5 + 7

4. 10 + ( − 2)

5. 4 − ( −13)

6. −11 − ( − 23)

7. 17 + 8

8. −19 + 21

In Exercises 9 –24, multiply or divide. 10. − 5 • ( − 6)

11. 3 • ( −1)

13. 19 • 2

9. − 4(5)

14. − 7 ( − 6)

15. 9 • ( − 8)

12. 7( − 2) 16. 4( 2)

17. − 20 ÷ 5

18. 48 ÷ ( − 3)

19. − 38 ÷ ( − 2)

20. 8 ÷ ( − 2)

21. −18 ÷ ( − 3)

22. 32 ÷ 8

23. − 48 ÷ 6

24. 63 ÷ ( − 9)

In Exercises 25–28, solve the problem and specify the units of measure. 25. You mow your neighbor’s lawn in 5 hours and earn $45. What is your hourly wage? 26. How many packages of diapers can you buy with $36 when one package costs $9? 27. At a gas station you buy 15 gallons of gas. The total cost is $60. What is the cost per

gallon? 28. On Saturday, you run 4 miles more than your friend. Your friend ran 3 miles. How

many miles did you run? 29. A flower bed is in the shape of a rectangular prism. Its dimensions are 3 feet wide by 16

u Cumulative Review The Cumulative Review includes exercises covering concepts and skills from the current chapter and previous chapters. Students can work on their mastery of previously learned material.

feet long by 6 inches deep. a. How many cubic feet of topsoil do you need to fill the flower bed? b. You can spread topsoil at a rate of 4 cubic feet per minute. How long will it take you

to spread all the topsoil? In Exercises 30 –37, solve the equation, justify each step, and check your answer. 30. x + 5 = 7

31.

32. x + ( − 3) = 5

33. 10 − m = − 3

y −4 = 2

34. 4 g = 36

35. 3b = 39

36. c ÷ 7 = 14

37.

y = 15 2

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Integrated Math I PE_TE sampler.indd A7

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Student Journal The Student Journal serves as a valuable component where students may work extra problems, take notes about new concepts and classroom lessons, and internalize new concepts by expressing their findings in their own words. Available in English and Spanish Name_________________________________________________________

Chapter

Date __________

Maintaining Mathematical Proficiency

1

Add or subtract. 1. −1 + ( − 3)

u Maintaining Mathematical Proficiency The Maintaining Mathematical Proficiency corresponds to the Pupil Edition Chapter Opener. Your students have the opportunity to practice prior skills necessary to move forward.

2. 0 + ( −12)

3. 5 − ( − 2)

4. − 4 − 7

5. Find two pairs of integers whose sum is − 6.

6. In a city, the record monthly high temperature for March is 56°F. The record

monthly low temperature for March is −4°F. What is the range of temperatures for the month of March?

Multiply or divide. 7. − 2(13)

8. − 8 • ( − 5)

10. − 30 ÷ ( − 3)

9. −14 ÷ 2

11. Find two pairs of integers whose product is − 20.

12. A football team loses 3 yards in 3 consecutive plays. What is the total yardage

gained?

u Exploration Journal The Exploration pages correspond to the Explorations and accompanying exercises in the Pupil Edition. Your students can use the room given on these pages to show their work and record their answers.

Name _________________________________________________________ Date _________

Solving Simple Equations

1.1

For use with Exploration 1.1

Essential Question How can you use simple equations to solve real-life problems? 1

EXPLORATION: Measuring Angles Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use a protractor to measure the angles of each quadrilateral. Complete the table to organize your results. (The notation m∠ A denotes the measure of angle A.) How precise are your measurements? a.

A

B

b.

c.

A

D

B A

B

C D

C

D C

Quadrilateral

m∠ A

(degrees)

m∠B

(degrees)

m∠ C (degrees)

m∠D

(degrees)

m∠ A + m∠B + m∠C + m∠D

a. b. c.

2

EXPLORATION: Making a Conjecture Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use the completed table in Exploration 1 to write a conjecture about the sum of the angle measures of a quadrilateral. Draw three quadrilaterals that are different from those in Exploration 1 and use them to justify your conjecture.

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Integrated Math I PE_TE sampler.indd A8

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Name _________________________________________________________ Date _________

Notetaking with Vocabulary

1.1

For use after Lesson 1.1

u Notetaking with Vocabulary

In your own words, write the meaning of each vocabulary term.

conjecture

This student-friendly notetaking component is designed to be a reference for key vocabulary, properties, and core concepts from the lesson. Students can write the definitions of the vocabulary terms in their own words and take notes about the core concepts.

rule

theorem

equation

linear equation in one variable

solution

inverse operations

equivalent equations

Core Concepts Addition Property of Equality Words Adding the same number to each side of an equation produces an equivalent

equation. Algebra If a = b, then a + c = b + c.

Notes:

Name _________________________________________________________ Date _________

1.1

Notetaking with Vocabulary (continued)

u Extra Practice

Common Problem-Solving Strategies Use a verbal model.

Guess, check, and revise.

Draw a diagram.

Sketch a graph or number line.

Write an equation.

Make a table.

Look for a pattern.

Make a list.

Work backward.

Break the problem into parts.

Each section of the Pupil Edition has an Extra Practice in the Student Journal with room for students to show their work.

Notes:

Extra Practice In Exercises 1–9, solve the equation. Justify each step. Check your solution. 1. w + 4 = 16

2. x + 7 = −12

3. −15 + w = 6

4. z − 5 = 8

5. − 2 = y − 9

6. 7 q = 35

7. 4b = − 52

8. 3 =

q 11

9.

n = −15 −2

10. A coupon subtracts $17.95 from the price p of a pair of headphones. You pay $71.80 for the

headphones after using the coupon. Write and solve an equation to find the original price of the headphones.

11. After a party, you have 2 of the brownies you made left over. There are 16 brownies left. 5

How many brownies did you make for the party?

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Integrated Math I PE_TE sampler.indd A9

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Differentiating the Lesson The Differentiating the Lesson online ancillary provides complete teaching notes and worksheets that address the needs of diverse learners. Lessons engage students in activities that often incorporate visual and kinesthetic learning. Some lessons present an alternative approach to teaching the content, while other lessons extend the concepts of the text in a challenging way for advanced students. Each chapter also begins with an overview of the differentiated lessons in the chapter and describes the students who would most benefit from the approach used in each lesson.

Lesson

1A

Solving Simple Linear Equations Using the Addition and Subtraction Properties For use with Section 1.1

Lesson Preparation Materials: math tiles*, Resource Sheet 1A: Math Tile Mat for Solving Equations, Resource Sheet 1B: Tape Diagram Mat (optional) *If math tiles are not available, laminate copies of Resource Sheet 1A and have students draw the math tiles with dry erase markers as they complete the lesson. Beforehand: Pre-assess students’ understanding of integer operations using two-color counters. Students will need to understand the concept of a zero pair. Classroom Management: Use the pre-assessment information to group students into pairs.

Lesson Procedure Distribute math tiles to each student, along with Resource Sheet 1A. Explain that a yellow square (or a square with + ) represents +1, and a red square (or a square with − ) represents −1. Make connections to two-color counters used in previous years. Introduce that a green rod (or a rod with + ) represents 1x, and a red rod (or a rod with − ) represents −1x. Remind students that placing one yellow square and one red square together equals 0, which is called a zero pair. Ask students what is being represented when placing one green rod and one red rod together. Be sure to have students recognize that this is also a zero pair. = +1 = −1

= +1x

= −1x

Present students with the equation x + 4 = 7. Explain that an equation must always be balanced, which is why students will represent each side of the equation on the balance provided in Resource Sheet 1A. Have students represent the equation on their resource sheets using the math tiles. Reference the example below.

=

A10

Integrated Math I PE_TE sampler.indd A10

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Credits Cover © Abidal | Dreamstime.com, Jason Winter/Shutterstock.com

Front Matter vi Goodluz/Shutterstock.com

Chapter 1 0 top left Doug Meek/Shutterstock.com; top right aceshot1/ Shutterstock.com; center left Stephanie Lupoli/Shutterstock.com; bottom right Stefan Schurr/Shutterstock.com; bottom left Ververidis Vasilis/Shutterstock.com; 6 Ververidis Vasilis/Shutterstock.com; 9 cowardlion/Shutterstock.com; 14 Stefan Schurr/Shutterstock.com; 15 Eric Isselee/Shutterstock.com; 22 Stephanie Lupoli/ Shutterstock.com; 25 Nattika/Shutterstock.com; 26 Invisible Studio/ Shutterstock.com, avNY/Shutterstock.com, barragan/ Shutterstock.com; 29 aceshot1/Shutterstock.com; 32 Jagodka/ Shutterstock.com; 38 Petr84/Shutterstock.com; 40 Maxim Tupikov/ Shutterstock.com; 41 bottom left Vladimir Dokovski/Shutterstock.com; top right Potapov Alexander/Shutterstock.com; 42 John Blanton/ Shutterstock.com; 43 tuulijumala/Shutterstock.com, Kokhanchikov/ Shutterstock.com

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Integrated Math I PE_TE sampler.indd A11

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Integrated Math I PE_TE sampler.indd A12

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