Springboard Unit 1 - Equations, Inequalities and Functions

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Unit 1

Planning the Unit

Vocabulary Development

The key terms for this unit can be found on the Unit

Opener page. These terms are divided into Academic

Vocabulary and Math Terms. Academic Vocabulary

includes terms that have additional meaning outside of

math. These terms are listed separately to help students

transition from their current understanding of a term

to its meaning as a mathematics term. To help students

learn new vocabulary:

Have students discuss meaning and use graphic

organizers to record their understanding of new words.

Remind students to place their graphic organizers in

their math notebooks and revisit their notes as their

understanding of vocabulary grows.

As needed, pronounce new words and place

pronunciation guides and definitions on the class

Word Wall.

Embedded Assessments

Embedded Assessments allow students to do the

following:

Demonstrate their understanding of new concepts.

Integrate previous and new knowledge by solving

real-world problems presented in new settings.

They also provide formative information to help you

adjust instruction to meet your students’ learning needs.

Prior to beginning instruction, have students unpack the

first Embedded Assessment in the unit to identify the

skills and knowledge necessary for successful completion

of that assessment. Help students create a visual display

of the unpacked assessment and post it in your class. As

students learn new knowledge and skills, remind them

that they will be expected to apply that knowledge to

the assessment. After students complete each Embedded

Assessment, turn to the next one in the unit and repeat

the process of unpacking that assessment with students.

I

n this unit, students model real-world situations by

using one- and two-variable equations. They study

inverse functions, composite functions, and

piecewise-defi ned functions, perform operations on functions, and

solve systems of equations and inequalities.

Equations, Inequalities, and Systems,

Gaming Systems

Systems of equations

Systems of inequalities

Absolute value equations

Embedded Assessment 1

Unpacking the Embedded Assessments

The following are the key skills and knowledge students

will need to know for each assessment.

Piecewise-Defi ned, Composite, and Inverse

Functions, Currency Conversion

Embedded Assessment 2

Piecewise-defined functions

Composition of functions

Inverse functions

AP / College Readiness

Unit 1 continues to prepare students for Advanced

Placement courses by:

Modeling real-world situations using one- and

two-variable equations.

Increasing student ability to work with a wide

variety of functions.

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Planning the Unit

continued

Additional Resources

Additional resources that you may find helpful for your instruction include the

following, which may be found in the Teacher Resources at SpringBoard Digital.

Unit Practice (additional problems for each activity)

Getting Ready Practice (additional lessons and practice problems for the

prerequisite skills)

Mini-Lessons (instructional support for concepts related to lesson content)

45-Minute Period

Your Comments on Pacing

Unit Overview/Getting Ready

1 Activity 1

3 Activity 2

2 Activity 3

4 Embedded Assessment 1

1 Activity 4

3 Activity 5

3 Activity 6

2 Embedded Assessment 2

1 Total 45-Minute Periods

20 Suggested Pacing

The following table provides suggestions for pacing using a 45-minute class

period. Space is left for you to write your own pacing guidelines based on

your experiences in using the materials.

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EMBEDDED ASSESSMENTS This unit has two embedded assessments, following Activities 3 and 6. They will give you an opportunity to demonstrate your understanding of equations, inequalities, and functions.

Embedded Assessment 1:

Equations, Inequalities, and

Systems p. 55

Embedded Assessment 2:

Piecewise-Defined, Composite, and Inverse Functions p. 99

Unit Overview

In this unit, you will model real-world situations by using one- and two-variable linear equations. You will extend your knowledge of linear relationships through the study of inverse functions, composite functions, piecewise-defined functions, operations on functions, and systems of linear equations and inequalities.

Key Terms

As you study this unit, add these and other terms to your math notebook. Include in your notes your prior knowledge of each word, as well as your experiences in using the word in different mathematical examples. If needed, ask for help in pronouncing new words and add information on pronunciation to your math notebook. It is important that you learn new terms and use them correctly in your class discussions and in your problem solutions.

Academic Vocabulary

• interpret • feasible

• compare • confirm

• contrast • prove

Math Terms

• absolute value equation • absolute value inequality • constraints • consistent • inconsistent • independent • dependent • ordered triple • Gaussian elimination • matrix • dimensions of a matrix • square matrix

• multiplicative identity matrix • multiplicative inverse matrix • matrix equation • coefficient matrix • variable matrix • constant matrix • piecewise-defined function • step function • parent function • composition • composite function • inverse function

1 Equations,

Inequalities,

Functions

ESSENTIAL QUESTIONS How are linear equations and systems of equations and inequalities used to model and solve real-world problems?

How are composite and inverse functions useful in problem solving?

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Developing Math Language

As this unit progresses, help students make the transition from general words they may already know (the Academic Vocabulary) to the meanings of those words in mathematics. You may want students to work in pairs or small groups to facilitate discussion and to build confidence and fluency as they internalize new language. Ask students to discuss new academic and mathematics terms as they are introduced, identifying meaning as well as pronunciation and common usage. Remind students to use their math notebooks to record their understanding of new terms and concepts.

As needed, pronounce new terms clearly and monitor students’ use of words in their discussions to ensure that they are using terms correctly. Encourage students to practice fluency with new words as they gain greater understanding of mathematical and other terms.

Unit Overview

Ask students to read the unit overview and mark the text to identify key phrases that indicate what they will learn in this unit.

Key Terms

As students encounter new terms in this unit, help them to choose an appropriate graphic organizer for their word study. As they complete a graphic organizer, have them place it in their math notebooks and revisit as needed as they gain additional knowledge about each word or concept.

Essential Questions

Read the essential questions with students and ask them to share possible answers. As students complete the unit, revisit the essential questions to help them adjust their initial answers as needed.

Unpacking Embedded

Assessments

Prior to beginning the first activity in this unit, turn to Embedded Assessment 1 and have students unpack the assessment by identifying the skills and knowledge they will need to complete the assessment successfully. Guide students through a close reading of the assessment, and use a graphic organizer or other means to capture their identification of the skills and knowledge. Repeat the process for each Embedded Assessment in the unit.

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Getting Ready

Write your answers on notebook paper. Show your work.

1. Given f(x) = x2_{− 4x + 5, find each value.}

a. f(2) b. f(−6) 2. Find the slope and y-intercept.

a. y = 3x − 4 b. 4x − 5y = 15 3. Graph each equation.

a. 2x + 3y = 12 b. x = 7 4. Write an equation for each line.

a. line with slope 3 and y-intercept −2 b. line passing through (2, 5) and (−4, 1) 5. Write the equation of the line below.

6. Using the whole number 5, define the additive inverse and the multiplicative inverse. 7. Solve 3(x + 2) + 4 = 5x + 7.

8. What is the absolute value of 2 and of −2? Explain your response.

9. Solve the equation for x. 3x y ₂ z + ₌ x y 8 10 6 4 2 –8 –10 –6 –4 –2 2 4 6 8 10 –2 –4 –6 –8 –10

10. Which point is a solution to the equation 6x − 5y = 4? Justify your choice. A. (1, 2) B. (1, −2) C. (−1, −2) D. (−1, 2)

11. Find the domain and range of each relation. a. y = 2x + 1

b.

c.

d.

12. How many lines of symmetry exist in the figure shown in Item 11c?

input ₃ ₇ ₁₁ output ₋₁ ₋₃ ₋₅ x y 4 2 –6 –4 –2 2 4 6 –2 –4 x y 4 2 –6 –8 –4 –2 2 4 6 8 –2 –4 –6 –8

2 SpringBoard®_{Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions}

UNIT 1

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Use some or all of these exercises for formative evaluation of students’ readiness for Unit 1 topics.

Prerequisite Skills

• Evaluating functions (Item 1) HSF-IF.A.2 • Finding slope and intercepts

(Item 2) HSF-IF.B.4 • Graphing linear equations

(Item 3) HSF-IF.C.7a • Writing linear equations

(Items 4–5) HSF-IF.B.4

• Finding additive and multiplicative inverses (Item 6) 7.NS.A.1b • Solving linear and literal equations

(Items 7, 9, 10) HSA-REI.B.3, HSA-REI.D.10

• Understanding absolute value (Item 8) 6.NS.C.7

• Finding domain and range (Item 11) HSF-IF.B.5 • Identifying lines of symmetry

(Item 12) 4.G.A.3

Answer Key

1. a. 1 b. 65 2. a. slope: 3, y-intercept: −4 b._{slope: 45, y-intercept: −3} 3. a–b. x 8 10 6 4 2 –8 –6 –4 –2 –10 2 4 6 8 10 –2 –4 –6 –8 –10 y x = 7 2x + 3y = 12 4. a. y = 3x − 2 b. 2x − 3y = −11 5. 3x + 4y = 24

6. Sample answer: The additive inverse

of 5 is −5 because 5 + (−5) = 0 and (−5) + 5 = 0. The multiplicative inverse of 5 is 15 because 5 15 1

( )

= _{and 15 5 1}

( )

= .

7. x=3₂

8. 2; Absolute value is the distance from

0 on a number line, so it cannot be a negative number. The absolute value of both 2 and −2 is 2.

9. x=2z y₃−

10. C. Sample explanation: When −1 is

substituted for x and −2 is substituted for y in the equation, you get 6(−1) − 5(−2) = −6 + 10 = 4. So, (−1, −2) is a solution.

11. a. domain and range: all real numbers

b. domain: 3, 7, 11; range: −1, −3, −5

c. domain: −5 ≤ x ≤ 5; range: −1 ≤ y ≤ 1

d. domain and range: all real numbers

12. There are two lines of symmetry, the

x- and y-axes.

UNIT

1 Getting Ready

Getting Ready Practice

For students who may need additional instruction on one or more of the prerequisite skills for this unit, Getting Ready practice pages are available in the Teacher Resources at SpringBoard Digital. These practice pages include worked-out examples as well as multiple opportunities for students to apply concepts learned.

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My Notes

Creating Equations

One to Two

Lesson 1-1 One-Variable Equations

Learning Targets:

•

Create an equation in one variable from a real-world context.

•

Solve an equation in one variable.

SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Create Representations, Identify a Subtask, Think-Pair-Share, Close Reading

A new water park called Sapphire Island is about to have its official grand opening. The staff is putting up signs to provide information to customers before the park opens to the general public. As you read the following scenario, mark the text to identify key information and parts of sentences that help you make meaning from the text.

The Penguin, one of the park’s tube rides, has two water slides that share a single line of riders. The table presents information about the number of riders and tubes that can use each slide.

Penguin Water Slides Slide

Number Tube Size Tube Release Time

1 2 riders every 0.75 min

2 4 riders every 1.25 min

Jaabir places a sign in the waiting line for the Penguin. When a rider reaches the sign, there will be approximately 100 people in front of him or her waiting for either slide. The sign states, “From this point, your wait time is approximately minutes.” Jaabir needs to determine the number of minutes to write on the sign. Work with a partner or with your group on Items 1–7.

1. Let the variable r represent the number of riders taking slide 1. Write an

algebraic expression for the number of tubes this many riders will need,

assuming each tube is full.

2. Next, write an expression for the time in minutes it will take r riders to go down slide 1.

3. Assuming that r riders take slide 1 and that there are 100 riders in all, write an expression for the number of riders who will take slide 2.

An algebraic expression includes at least one variable. It may also include numbers and operations, such as addition, subtraction, multiplication, and division. It does not include an equal sign.

MATH TIP r 2 r 20 75

( )

. 100 − r

Activity 1 • Creating Equations 3

ACTIVITY 1

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Common Core State Standards for Activity 1

HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions.

HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

ACTIVITY

1

Directed

Activity Standards Focus

In Activity 1, students write and solve linear equations in one variable, including multistep equations and equations with variables on both sides. They also write equations in two variables and show solutions to those equations on a coordinate plane. Finally, they write, solve, and graph absolute value equations and inequalities. Throughout this activity, emphasize the importance of performing the same operation on both sides of an equation or inequality in an effort to keep the equation or inequality balanced.

Lesson 1-1

PLAN

Pacing:1 class period

Chunking the Lesson

#1–2 #3–5 #6–7

#8 #9 #10–11

Check Your Understanding Lesson Practice

TEACH

Bell-Ringer Activity

Ask students to translate each phrase to an equation.

1. Six more than twice a number c is 24. [6 + 2c = 24] 2. One-third of a number y is 45. 1 3 y =   45

3. Seven less than the product of a number and 10 is 50. [10n − 7 = 50] Discuss with students the methods they used to translate the sentences into equations.

Developing Math Language

This lesson refers to both expressions and equations. An expression can be made up of numbers, variables, constants, arithmetic operation symbols, and grouping symbols. If an expression is algebraic, then it must contain at least one variable. Equal signs are never a part of an expression. An equation is a mathematical sentence that contains an equal sign, showing that two expressions are equivalent to each other.

1–2 Activating Prior Knowledge, Visualization, Create Representations

If students are struggling with Item 2, give them a hint to multiply the number of tubes needed for r riders by the time needed for each tube to go down the slide. This expression will combine the result from Item 1 with the information that is found in the table.

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My Notes

Lesson 1-1 One-Variable Equations

4. Using the expression you wrote in Item 3, write an expression for the number of tubes the riders taking slide 2 will need, assuming each tube is full.

5. Write an expression for the time in minutes needed for the riders taking slide 2 to go down the slide.

6. Since Jaabir wants to know how long it takes for 100 riders to complete the ride when both slides are in use, the total time for the riders taking slide 1 should equal the total time for the riders taking slide 2. Write an equation that sets your expression from Item 2 (the time for the slide 1 riders) equal to your expression from Item 5 (the time for the slide 2 riders).

7. Reason abstractly and quantitatively. Solve your equation from

Item 6. Describe each step to justify your solution. These properties of real numbers

can help you solve equations.

Addition Property of Equality

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

Subtraction Property of Equality

If a = b, then a − c = b − c.

Multiplication Property of Equality

If a = b, then ca = cb.

Division Property of Equality

If a = b and c ≠ 0, then a c= .bc Distributive Property a(b + c) = ab + ac MATH TIP 100 4− r 100 4− 1 25

(

r .

)

r r 20 75 1004 1 25

( )

. =

(

−

)

. Sample work: r r 20 75 1004 1 25

( )

. =

(

−

)

.

0.375r = (100 − r)0.3125 Simplify each side. 0.375r = 31.25 − 0.3125r Distributive Property

0.6875r = 31.25 Add 0.3125r to each side.

r ≈ 45.5 Divide each side by 0.6875.

continued continued continued

ACTIVITY 1

Differentiating Instruction ELL Support

ACTIVITY

1

Continued

3–5 Activating Prior Knowledge, Simplify the Problem If students are struggling with Item 3, then share this simpler but similar problem: Suppose Jaabir is aware that of the 100 riders, 46 plan on riding slide 1. How many of those riders are planning on riding slide 2? How did you find this? [by using subtraction] Now the only difference is that Jaabir does not know the exact number of riders for slide 1; however, the variable r represents slide 1 riders. Use the same method you used to find slide 2 riders—subtraction. The only difference is you do not know the exact number of each.

It may be beneficial for some students to review the properties of equality using numeric examples, rather than only the algebraic definitions. You can use the following examples to demonstrate the properties numerically:

• Addition Property of Equality: Start with 8 = 8; add 4 to both sides.

8 + 4 = 8 + 4, or 12 = 12✓

• Subtraction Property of Equality: Start with −4 = − 4; subtract 2 from both sides.

−4 − 2 = −4 − 2, or −6 = − 6✓

• Multiplication Property of Equality: Start with 16 = 16; multiply both sides by −3.

16(−3) = 16(−3), or −48 = −48✓ • Division Property of Equality:

Start with −45 = −45; divide both sides by −9. 45 9 459 − − =−− , or 5 = 5✓

6–7 Discussion Groups, Sharing and Responding, Think-Pair-Share After students complete Item 6, they can discuss the process of solving this problem in Item 7. If there are students within a group having trouble getting from one step to the next in the solution, other students may provide an explanation. Additionally, these steps are samples, so if any students approach the problem with a different strategy (for example, they multiply through first by 100 to eliminate some of the decimals Show English language learners

various pictures of water slides. Explain that riders climb up stairs or take an elevator to the top of the slide and then slide down the slide into a pool of water below. Ask students to share if there is any type of ride or activity similar to this in their native countries.

and/or they multiply both sides of the equation by 4 in order to eliminate fractions), they could share this with their peers and demonstrate that there is more than one way to arrive at the correct solution.

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My Notes

8. Make sense of problems. Consider the meaning of the solution

from Item 7.

a. Explain why you should or should not round the value of r to the nearest whole number.

b. How many people out of the 100 riders will take slide 1?

9. Use the expression you wrote in Item 2 to determine how long it will take the number of riders from Item 8b to go through slide 1. a. Evaluate the expression for the appropriate value of r.

b. How many minutes will it take the riders to go through slide 1? Round to the nearest minute.

The rest of the 100 riders will go through slide 2 in about the same amount of time. So, your answer to Item 9b gives an estimate of the number of minutes it will take all 100 riders to go down the Penguin slides.

10. Recall that when a rider reaches the sign, there will be approximately 100 people waiting in front of him or her. What number should Jaabir write to complete the statement on the sign?

From this point, your wait is approximately minutes.

When you evaluate an algebraic expression, you substitute values for the variables and then simplify the expression.

MATH TIP

Because the value of r represents a number of riders, and it doesn’t make sense to have a fraction of a rider, you should round r to the nearest whole number.

46 riders 17 minutes 17 r 20 75 462 0 75 17 25

( )

. =

( )

. = .

continued continued continued ACTIVITY 1

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ACTIVITY

1

Continued

8 Discussion Groups, Construct an Argument Have students work with a partner or in a small group to interpret the solution of r = 45.5. Have them explore whether this solution actually makes sense, given that r represents a number of riders (people). Is this a situation where rounding to the nearest whole number is necessary? If so, then why? Ask students to reflect on whether their results or solutions to a problem make sense.

Developing Math Language

This lesson contains the vocabulary term evaluate. When you evaluate an expression, you replace each variable in the expression by a given value and simplify the result. The term evaluate will surface again in later chapters, when students will be asked to evaluate the value of a function with a given input value.

9 Critique Reasoning, Discussion Groups, Construct an Argument It is important that students understand that the number of minutes written on the sign is an estimate, for several reasons. It is not certain that there will be exactly 100 people in front of the sign. There are different factors that could affect this number. For example, whether people stand close together or farther apart could impact the total number. Also, the size (age and weight) of the people would have an impact on the number of people in line. This problem depends upon approximations and estimations. Furthermore, the answer in Item 9, 17.25 minutes, does not divide evenly by the 1.25-minute timing cycle of slide 2 (neither does 17 minutes; the “common multiples” of 0.75 and 1.25 minutes closest to 17 are 15 and 18.75 minutes). 10–11 Discussion Groups, Construct an Argument, Debriefing Prior to Item 10, students read that it will take about the same amount of time for the remaining riders to go down slide 2. Since the result from Item 9b is 17 minutes, Jaabir should place 17 minutes on the board as the best approximation.

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My Notes

LESSON 1-1 PRACTICE

Use this information for Items 14–15. When full, one of the pools at Sapphire Island will hold 43,000 gallons of water. The pool currently holds 20,000 gallons of water and is being filled at a rate of 130 gallons per minute. 14. Write an equation that can be used to find h, the number of hours it will

take to fill the pool from its current level. Explain the steps you used to write your equation.

15. Solve your equation from Item 14, and interpret the solution.

Use this information for Items 16–18. Sapphire Island is open 7 days a week. The park has 8 ticket booths, and each booth has a ticket seller from 10:00 a.m. to 5 p.m. On average, ticket sellers work 30 hours per week.

16. Model with mathematics. Write an equation that can be used to

find t, the minimum number of ticket sellers the park needs. Explain the steps you used to write your equation.

17. Solve your equation from Item 16, and interpret the solution. 18. The park plans to hire 20 percent more than the minimum number of

ticket sellers needed in order to account for sickness, vacation, and lunch breaks. How many ticket sellers should the park hire? Explain. 12. Suppose that Jaabir needs to place a second sign in the waiting line for

the Penguin slides. When a rider reaches this sign, there will be approximately 250 people in front of him or her. What number should Jaabir write to complete the statement on this sign? Explain how you determined your answer.

From this point, your wait is approximately minutes. 13. Explain the relationships among the terms variable, expression, and

equation.

Check Your Understanding

11. Describe how you could check that your answer to Item 10 is reasonable.

When you interpret a solution, you state the meaning of the solution in the context of the problem or real-world situation.

ACADEMIC VOCABULARY

Sample answer: I could substitute the value of r into the equation and check that it makes the equation true. I could also evaluate the expression from Item 5 to check that I get the same time for slide 2 as for slide 1.

ACTIVITY 1

10–11 (continued) Another way to check the answer would be to take 54 (which is the value of 100 − r, or the approximate number of slide 2 riders) and multiply it by 1.25; 54 × 1.25 = 16.875 ≈ 17 minutes. Alert students that even though each tube on slide 1 will hold 2 passengers and each tube on slide 2 will hold 4 passengers, the water park may not necessarily be filling every spot in each tube.

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to interpreting the solution to an equation. When students answer Item 12, you may also want to have them explain how they arrived at their answer. Answers

12. 43 min; Explanations may vary.

Some students may write and solve an equation to find the answer. Other students may reason that the wait time for 250 riders will be 2.5 times the wait time for 100 riders.

13. Sample answer: A variable is a letter

or symbol that represents an unknown value or values. An expression can include variables, along with numbers and operations. An equation is a statement that two expressions are equal.

ASSESS

Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. LESSON 1-1 PRACTICE

14. 130(60h) + 20,000 = 43,000;

Explanations will vary. Students may note that the number of gallons added to the pool plus the 20,000 gallons already in the pool must equal 43,000 gallons. They may also note that the minutes needed to fill the pool equals the number of hours h times 60.

ACTIVITY

1

Continued

15. h ≈ 2.9; The solution shows that it will take about 3 more hours to fill the pool.

16. 30t = 49(8) or equivalent;

Explanations may vary. Students may note that the minimum number of ticket sellers times 30 hours per week equals the total number of hours worked by the ticket sellers per week. Students may also note that the total number of hours worked by the ticket sellers each week is equal to the number of hours the park is open times the number of booths, or 49(8).

17. t ≈ 13.1; Some students may reason that 13.1 is sufficiently close to 13 that 13 ticket sellers will be enough for the park. Other students will note that the park technically needs more than 13 ticket sellers and will give an answer of 14.

18. 16 or 17, depending on the answer to

Item 17

ADAPT

Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving equations as well as interpreting solutions of equations. If students are having difficulty creating equations that model the situation, have them write out word equations.

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My Notes

Lesson 1-2

Two-Variable Equations

Learning Targets:

•

Create equations in two variables to represent relationships between quantities.

•

Graph two-variable equations.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Summarizing, Paraphrasing, Look for a Pattern, Think-Pair-Share, Create Representations, Interactive Word Wall, Identify a Subtask

At Sapphire Island, visitors can rent inner tubes to use in several of the park’s rides and pools. Maria works at the rental booth and is preparing materials so that visitors and employees will understand the pricing of the tubes. Renting a tube costs a flat fee of $5 plus an additional $2 per hour. As you work in groups on Items 1–7, review the above problem scenario carefully and explore together the information provided and how to use it to create potential solutions. Discuss your understanding of the problems and ask peers or your teacher to clarify any areas that are not clear.

1. Maria started making a table that relates the number of hours a tube is rented to the cost of renting the tube. Use the information above to help you complete the table.

Tube Rentals Hours Rented Cost ($)

1 2 3 4 5

2. Explain how a customer could use the pattern in the table to determine the cost of renting a tube for 6 hours.

Next, Maria wants to write an equation in two variables, x and y, that employees can use to calculate the cost of renting a tube for any number of hours.

3. Reason abstractly. What does the independent variable x represent in this situation? Explain.

Recall that in a relationship between two variables, the value of the independent variable determines the value of the

dependent variable.

MATH TIP

If you need help in describing your ideas during group discussions, make notes about what you want to say. Listen carefully to other group members as they describe their ideas, and ask for clarification of meaning for any words routinely used by group members.

DISCUSSION GROUP TIP

Sample answer: Add $2 to the cost of renting a tube for 5 hours: $15 + $2 = $17.

The number of hours the tube is rented; The time in hours is the independent variable because this value determines the cost of renting the tube.

7 9 11 13 15

ACTIVITY 1

ACTIVITY 1

Continued

Lesson 1-2

PLAN

Pacing: 1 class period

Chunking the Lesson

#1–2 #3–7

#8–13 #14

TEACH

Have students write each equation in the form Ax + By = C, where A, B, and C are integers and A is nonnegative.

1. 4x − 7 = 2y [4x − 2y = 7]

2. 12 = y − 8x [8x − y = −12]

3. 5y = x − 3 [x − 5y = 3]

1–2 Activating Prior Knowledge, Look for a Pattern, Role Play, Group Presentation Have students work in pairs or small groups to complete the second column of the table. Encourage them to find a pattern from one number to the next, going downward in both columns of the table. While they are working on this, circulate around the room and have students tell you the rental fee for some arbitrary number of hours that is not already in the table. After visiting various groups, have the students share their patterns and any other findings with the class as a whole.

3–7 Activating Prior Knowledge, Debriefing, Chunking the Activity, KWL Chart Discuss with students what it means to be independent vs. dependent in the real world. For example, students are dependents of their parents. Construct a KWL Chart by writing Know, Want to Know, and Learn as column headings in one row

across the board. Beneath the Know column, write the terms Dependent

variable and Independent variable, and

ask students to define them in their own words, drawing from what they remember from their previous math courses.

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My Notes

Lesson 1-2 Two-Variable Equations

4. What does the dependent variable y represent in this situation? Explain.

5. Write an equation that models the situation.

6. How can you tell whether the equation you wrote in Item 5 correctly models the situation?

7. Construct viable arguments. Explain how an employee could use

the equation to determine how much to charge a customer.

Maria also thinks it would be useful to make a graph of the equation that relates the time in hours a tube is rented and the cost in dollars of renting a tube.

8. List five ordered pairs that lie on the graph of the relationship between x and y.

Before you can graph the equation, you need to determine the coordinates of several points that lie on its graph. One way to do this is by using pairs of corresponding values from the table on the previous page. You can also choose several values of x and substitute them into the equation to determine the corresponding values of y.

MATH TIP

The cost in dollars of renting the tube; The cost is the dependent variable because this value depends on the number of hours the tube is rented.

Sample answer: The employee should substitute the number of hours the customer rented the tube for x in the equation. Next, the employee should solve the equation for y. This value is the amount to charge the customer in dollars.

Sample answer: Substitute values for x into the equation and check whether they give the correct values for y by comparing the results to the table on the previous page.

y = 5 + 2x

Sample answer: (1, 7), (2, 9), (3, 11), (4, 13), (5, 15)

ACTIVITY 1

Technology Tip

ACTIVITY

1

Continued

3–7 (continued) Beneath the Want to

Know column, write how the dependent and independent variables are

represented in this situation, as well as a linear equation that can be written to model the situation (refer to Bell-Ringer Activity if needed). Beneath the Learn column, write how the items in the first two columns can be tied together to answer Items 6 and 7.

Students can use the table function on a graphing calculator to find how much to charge a customer, y, based upon the number of hours, x, as follows: Press y= to enter the equation 5 + 2x. Press 2nd WINDOW to access TBLSET. Set the Tblstart = 0, Tbl = 1, and Indpnt: to “Ask.” Then press 2nd GRAPH to access the table. At this screen, students can type in any number of hours at the X= prompt. The corresponding charge will appear next to it in the y1 column. This is a

great way to check their work and practice using technology.

For additional technology resources, visit SpringBoard Digital.

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My Notes

Lesson 1-2

9. Use the grid below to complete parts a and b.

a. Write an appropriate title for the graph based on the real-world situation. Also write appropriate titles for the x- and y-axes.

b. Graph the ordered pairs you listed in Item 8. Then connect the points with a line or a smooth curve.

10. Based on the graph, explain how you know whether the equation that models this situation is or is not a linear equation.

11. Reason quantitatively. Explain why the graph is only the first quadrant.

12. What is the y-intercept of the graph? Describe what the y-intercept represents in this situation.

13. What is the slope of the graph? Describe what the slope represents in this situation. x y 1 2 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 5 6

Number of Hours Rented

Cost ($)

Tube Rentals

The y-intercept of a graph is the

y-coordinate of a point where the

graph intersects the y-axis. The slope of a line is the ratio of the change in y to the change in x between any two points.

MATH TIP

Recall that a linear equation is an equation whose graph is a line. A linear equation can be written in

standard form Ax + By = C, where A, B, and C are integers and A is

nonnegative.

MATH TIP

Check students’ answers.

Yes, this is a linear equation. Sample explanation: The graph has a constant rate of change and therefore models a linear equation.

The variable x represents time and the variable y represents cost. It would not make sense for either of these variables to be negative.

5; the fl at fee in dollars for renting a tube

2; the hourly cost in dollars of renting a tube

ACTIVITY 1

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Universal Access

ACTIVITY 1

Continued

8–13 Predict and Confirm, Debriefing

Have students predict what they think the graph will look like based upon the previous items leading up to this set of items. Remind students that a solution of an equation in two variables is an ordered pair of numbers, which can be plotted on a coordinate plane. The easiest way to get these ordered pairs is from the table created in Item 1. The axes should be labeled according to what the x- and y-values represent— hours and cost, respectively. When graphed, the ordered pairs should form a straight line. In this particular case, the graph is in the first quadrant because you cannot rent for a negative number of hours. Item 12 refers to the

y-intercept; explain why there is no x-intercept (at least for this situation).

A common error students make is not connecting, or drawing a line through, the points. Emphasize that there are infinite solutions to equations in two variables, and that every point on the line is a solution.

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My Notes

Lesson 1-2

Use this information for Items 18–22. Some of the water features at Sapphire Island are periodically treated with a chemical that prevents algae growth. The directions for the chemical say to add 16 fluid ounces per 10,000 gallons of water.

18. Make a table that shows how much of the chemical to add for water features that hold 10,000; 20,000; 30,000; 40,000; and 50,000 gallons of water.

19. Write a linear equation in two variables that models the situation. Tell what each variable in the equation represents.

20. Graph the equation. Be sure to include titles and use an appropriate scale on each axis.

21. What are the slope and y-intercept of the graph? What do they represent in the situation?

22. Construct viable arguments. An employee adds 160 fluid ounces of the chemical to a feature that holds 120,000 gallons of water. Did the employee add the correct amount? Explain.

15. Explain why the slope of the line you graphed in Item 9 is positive.

16. Explain how you would graph the equation from Item 14. What quantity and units would be represented on each axis?

17. Is the equation y = −2x + x2_{a linear equation? Explain how you know.}

14. Work with your group. Describe a plausible scenario related to the water park that could be modeled by this equation: y = 40x − 8. In your description, be sure to use appropriate vocabulary, both real-world and mathematical. Refer to the Word Wall and any notes you may have made to help you choose words for your description.

Share your description with your group members and list any details you may not have considered before. If you do not know the exact words to describe your ideas, use synonyms or request assistance from group members to help you convey your ideas. Use nonverbal cues such as raising your hand to ask for clarification of others’ ideas.

DISCUSSION GROUP TIP

Sample answer: The water park mails out coupons for $8 off the total cost of a ticket purchase. Tickets to the park normally cost $40 each. The equation models the cost y in dollars of purchasing x tickets to the water park with a coupon.

10 SpringBoard®_{Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions} continued

continued continued

ACTIVITY 1

ACTIVITY 1

Continued

14 Marking the Text, Debriefing, Discussion Groups, Group

Presentation Write the slope-intercept form of a line (y = mx + b) on the board. Write the equation y = 40x − 8 right below. Highlight that the slope of the equation, m, is 40. Highlight that the

y-intercept, b, is −8. Be sure to stress

this is a negative value this time. Have students spend a few minutes

collaborating in small groups to create a plausible scenario, related to the water park, for which this equation could be modeled. Ask students to share their ideas and have a classroom discussion as to why their scenarios are or are not plausible.

Debrief students’ answers to these items to ensure that they understand concepts related to two-variable linear equations.

Answers

15. The cost of renting a tube increases as the number of hours it is rented increases. So, y increases as

x increases, which indicates a

positive slope.

16. Sample answer: Substitute several values of x into the equation to find the corresponding values of y. Then use the values of x and y to write ordered pairs. Finally, graph the ordered pairs and draw a line through them. Quantities and units will vary depending on the scenario the student wrote for Item 14.

17. No. Sample explanation: The equation cannot be written in the form Ax + By = C. The graph of the equation is a curve, not a line.

ASSESS

Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity.

ADAPT

Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and graphing equations in two variables. If students are having difficulty graphing the equations, review the process of creating a table of values and identifying slope and intercepts.

19. Sample answer: y = 1.6x; y represents the number of fluid ounces of the chemical to add to a water feature, and x represents the amount of water the feature holds, in thousands of gallons.

18. Amount of Chemical to Add Gallons of

Water Fluid Ounces of Chemical

10,000 16

20,000 32

30,000 48

40,000 64

(13)

My Notes

Lesson 1-3

Absolute Value Equations and Inequalities

An absolute value equation is an equation involving the absolute value of a variable expression.

MATH TERMS Learning Targets:

•

Write, solve, and graph absolute value equations.

•

Solve and graph absolute value inequalities.

SUGGESTED LEARNING STRATEGIES: Marking the Text, Interactive Word Wall, Close Reading, Create Representations, Think-Pair-Share, Identify a Subtask, Quickwrite, Self Revision/Peer Revision

You can use the definition of absolute value to solve absolute value equations algebraically. Since ax b+ =−ax b_{ax b}+₊ ax b_{ax b}+ <_{+ ≥}  ( ) if _, if 00

then the equation |ax + b| = c is equivalent to −(ax + b) = c or (ax + b) = c. Since −(ax + b) = c is equivalent to ax + b = −c, the absolute value equation |ax + b| = c is equivalent to ax + b = −c or ax + b = c.

Recall that the geometric interpretation of |x| is the distance from the number x to 0 on a number line.

If |x| = 5, then x = −5 or x = 5 because those two values are both 5 units away from 0 on a number line.

MATH TIP

Example A

Solve 2|x − 1| − 5 = 1. Graph the solutions on a number line. Step 1: Isolate the absolute value

expression. Add 5 to both sides and then divide by 2.

2|x − 1| − 5 = 1 2|x − 1| = 6 |x − 1| = 3 Step 2: Write and solve two equations

using the definition of absolute value.

x − 1 = 3 or x − 1 = −3 x = 4 or x = −2

Solution: There are two solutions: x = 4 and x = −2

Check to see if both solutions satisfy the original equation. Substitute 4 and −2 for x in the original equation.

2|4 − 1| − 5 = 1 2|3| − 5 = 1 2(3) − 5 = 1 6 − 5 = 1 2|−2 − 1| − 5 = 1 2|−3| − 5 = 1 2(3) − 5 = 1 6 − 5 = 1 To graph the solutions, plot points at 4 and −2 on a number line.

Try These A

Solve each absolute value equation. Graph the solutions on a number line. a. |x − 2| = 3 b. |x + 1| − 4 = −2 c. |x − 3| + 4 = 4 d. |x + 2| + 3 = 1 –5 –4 –3 –2 0 1 2 3 4 5 –2 4 –1 x = 5, x = −1 x = 1, x = −3 no solution x = 3

ACTIVITY

1

Continued

Lesson 1-3

PLAN

Pacing: 1 class period

Chunking the Lesson Example A #1 Example B Example C #2

TEACH

Students should recall that an absolute value of a number is its distance from zero on a number line.

Have students evaluate the following: 1. |6| [6]

2. |−6| [6]

Then have students solve the following equation.

3. |x|= 6 [x = 6 or x = −6] Example A Marking the Text, Interactive Word Wall Point out the Math Tip to reinforce why two solutions exist. Work through the solutions to the equation algebraically. Remind students that solutions to an equation make the equation a true statement. This mathematical understanding is necessary for students to be able to check their results.

Developing Math Language

An absolute value equation is an equation involving an absolute value of an expression containing a variable. Just like when solving algebraic equations without absolute value bars, the goal is to isolate the variable. In this case, isolate the absolute value bars because they contain the variable. It should be emphasized that when solving absolute value equations, students must think of two cases, as there are two numbers that have a specific distance from zero on a number line.

21. The slope is 1.6; the slope

represents the number of fluid ounces of the chemical to add to 1000 gallons of water. The y-intercept is 0; the y-intercept shows that no chemical should be added when a feature contains no water.

22. No. Sample explanation:

Substituting 120 for x in the equation and solving for y shows that the employee should have added 192 fluid ounces of the chemical.

LESSON 1-3 PRACTICE (continued) 20. 64 80 96 48 32 16 50 x y 10 20 30 40

Amount of Chemical to Add

Amount of Chemical (fl oz)

(14)

My Notes

Lesson 1-3 Absolute Value Equations and Inequalities

1. Reason abstractly. How many solutions are possible for an absolute

value equation having the form |ax + b| = c, where a, b, and c are real numbers?

Example B

The temperature of the wave pool at Sapphire Island can vary up to 4.5°F from the target temperature of 82°F. Write and solve an absolute value equation to find the temperature extremes of the wave pool. (The temperature extremes are the least and greatest possible temperatures.) Step 1: Write an absolute value equation to represent the situation.

Let t represent the temperature extremes of the wave pool in degrees Fahrenheit.

|t − 82| = 4.5

Step 2: Use the definition of absolute value to solve for t. |t − 82| = 4.5

t − 82 = 4.5 or t − 82 = −4.5 t = 86.5 or t = 77.5

Solution: The greatest possible temperature of the wave pool is 86.5°F, and the least possible temperature is 77.5°F. Both of these temperatures are 4.5°F from the target temperature of 82°F.

Try These B

The pH of water is a measure of its acidity. The pH of the water on the Seal Slide can vary up to 0.3 from the target pH of 7.5. Use this information for parts a–c.

a. Write an absolute value equation that can be used to find the extreme pH values of the water on the Seal Slide. Be sure to explain what the variable represents.

b. Solve your equation, and interpret the solutions.

c. Reason quantitatively. Justify the reasonableness of your answer to part b.

You know that the distance from t to 82°F on a thermometer is 4.5°F. This distance can be modeled with the absolute value expression |t − 82|.

MATH TIP

There are possibly two, one, or zero solutions to an absolute value equation having this form.

|p − 7.5| = 0.3; p represents the extreme pH values of the water.

Sample answer: 7.8 is 0.3 more than 7.5; 7.2 is 0.3 less than 7.5. p = 7.8, p = 7.2; The greatest allowed pH is 7.8, and the least allowed pH is 7.2.

ACTIVITY 1

ELL Support

ACTIVITY

1

Continued

1 Identify a Subtask, Quickwrite When solving absolute value equations, students may not see the purpose in creating two equations. Reviewing the definition of the absolute value function as a piecewise-defined function with two rules may enable students to see the reason why two equations are necessary. Have students look back at Try These A, parts c and d. Have volunteers construct a graph of the two piecewise-defined functions used to write each equation and then discuss how the solution set is represented by the graph.

Example B Marking the Text, Simplify the Problem, Critique Reasoning, Group Presentation Start with emphasizing the word vary in the Example, discussing what it means when something varies. You may wish to present a simpler example such as: The average cost of a pound of coffee is $8. However, the cost sometimes varies by $1. This means that the coffee could cost as little as $7 per pound or as much as $9 per pound. Now have students work in small groups to examine and solve Example B by implementing an absolute value equation. Additionally, ask them to take the problem a step further and graph its solution on a number line. Have groups present their findings to the class.

For those students for whom English is a second language, explain that the word varies in mathematics means changes. There are different ways of thinking about how values can vary. Values can vary upward or downward, less than or greater than, in a positive direction or a negative direction, and so on. However, the importance comes in realizing that there are two different directions, regardless of how you think of it. Also address the word extremes as it pertains to mathematics. An extreme value is a maximum value if it is the largest possible amount (greatest value), and an extreme value is a minimum value if it is the smallest possible amount (least value).

(15)

My Notes

Lesson 1-3

Absolute Value Equations and Inequalities

Solving absolute value inequalities algebraically is similar to solving absolute value equations. By the definition of absolute value, |ax + b| > c, where c > 0, is equivalent to −(ax + b) > c or ax + b > c. Multiplying the first inequality by −1, and then using a similar method for |ax + b| < c, gives these statements:

• |ax + b| > c, c > 0, is equivalent to ax + b < −c or ax + b > c.

• |ax + b| < c, c > 0, is equivalent to ax + b < c or ax + b > −c, which can also be written as −c < ax + b < c.

Example C

Solve each inequality. Graph the solutions on a number line. a. |2x + 3| + 1 > 6

Step 1: Isolate the absolute

value expression. |2x + 3| + 1 > 6_{|2x + 3| > 5}

Step 2: Write two inequalities. 2x + 3 > 5 or 2x + 3 < −5

Step 3: Solve each inequality. x > 1 or x < −4

Solution:

–4 0 1

b. |3x − 1| + 5 < 7

Step 1: Isolate the absolute

value expression. |3x − 1| + 5 < 7_{|3x − 1| < 2} Step 2: Write the compound

inequality. −2 < 3x − 1 < 2

Step 3: Solve the inequality. − < <1

3 x 1

Solution:

–1₃ 1

Try These C

Solve and graph each absolute value inequality.

a. x − >2 3 b. x + − ≤ −2 3 1

c. 5x − + ≥ 2 1 4 d. 2x + − <7 4 1

An absolute value inequality is an inequality involving the absolute value of a variable expression.

MATH TERMS

These properties of real numbers can help you solve inequalities. The properties also apply to inequalities that include <, ≥, or ≤.

Addition Property of Inequality

If a > b, then a + c > b + c. Subtraction Property of Inequality If a > b, then a − c > b − c. Multiplication Property of Inequality If a > b and c > 0, then ca > cb. If a > b and c < 0, then ca < cb.

Division Property of Inequality

If a > b and c > 0, then a_c_{> bc .} If a _{> b and c < 0, then ac <}b_c. MATH TIP –1 5 –4 0 1 5 – 1 –6 –1 x > 5 or x < −1 x ≥ 1 or x ≤− 15 −4 ≤ x ≤ 0 −6 < x < −1

T

EACHER

to T

EACHER

ACTIVITY

1

Continued

Developing Math Language

An absolute value inequality is basically the same as an absolute value equation, except that the equal sign is now an inequality symbol: <, >, ≤, ≥, or ≠. It still involves an absolute value expression that contains a variable, just like before. Use graphs on a number line of the solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related.

Example C Simplify the Problem, Debriefing Before addressing Example C, discuss the following: Inequalities with |A| > b, where b is a positive number, are known as disjunctions and are written as A < −b or A > b.

For example, |x| > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. The solution is x < − 5 or x > 5. See graph A.

This also holds true for |A| ≥ b. Inequalities with |A| < b, where b is a positive number, are known as conjunctions and are written as −b < A < b, or as −b < A and A < b.

For example: |x| < 5; this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. The solution is −5 < x < 5. See graph B.

This also holds true for |A| ≤ b. Students can apply these generalizations to Example C. Point out that they should proceed to solve these just as they would an algebraic equation, except in two parts, as shown above. After they have some time to work through parts a and b, discuss the solutions with the whole class.

Another method for solving inequalities relies on the geometric definition of absolute value |x − a| as the distance from x to a. Here’s how you can solve the inequality in the example:

|2x + 3| + 1 > 6 |2x + 3| > 5 2x− − >23 5 x− − >23 52

Thus, the solution set is all values of x whose distance from 32− is greater than 52. The solution can be represented on a number line and written as x < −4 or x > 1. A –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 –7 –6 B –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 –7 –6

(16)

My Notes

Lesson 1-3 Absolute Value Equations and Inequalities

2. Make sense of problems. Why is the condition c > 0 necessary for

|ax + b| < c to have a solution?

3. Compare and contrast a linear equation having the form ax + b = c with an absolute value equation having the form |ax + b| = c.

4. Critique the reasoning of others. Paige incorrectly solved an

absolute value equation as shown below. −2 |x + 5| = 8

−2(x + 5) = 8 or −2(x + 5) = −8 −2x − 10 = 8 or −2x − 10 = −8

x = −9 or x = −1 a. What mistake did Paige make?

b. How could Paige have determined that her solutions are incorrect? c. Solve the equation correctly. Explain your steps.

5. Explain how to write the inequality |5x − 6| ≥ 9 without using an absolute value expression.

LESSON 1-3 PRACTICE Solve each absolute value equation.

6. |x − 6| = 5 7. |3x − 7| = 12

8. |2x + 9| − 10 = 5 9. |5x − 3| + 12 = 4

10. Model with mathematics. The flow rate on the Otter River Run can

vary up to 90 gallons per minute from the target flow rate of 640 gallons per minute. Write and solve an absolute value equation to find the extreme values of the flow rate on the Otter River Run.

Solve each absolute value inequality. Graph the solutions on a number line.

11. |x − 7| > 1 12. |2x − 5| ≤ 9

13. |3x − 10| − 5 ≥ −1 14. |4x + 3| − 9 < 5

When you compare and contrast two topics, you describe ways in which they are alike and ways in which they are different.

ACADEMIC VOCABULARY

If c = 0 or if c < 0 (c is negative), the inequality would be impossible or trivial because absolute value is the distance from zero on a number line.

ACTIVITY 1

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ACTIVITY

1

Continued

2 Quickwrite, Self Revision/Peer Revision, Debriefing Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving inequalities.

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to absolute value equations. Have groups of students present their solutions to Item 4.

Answers

3. Sample answer: A linear equation of

the form ax + b = c has 1 solution, but an absolute value equation of the form |ax + b| = c may have 0, 1, or 2 solutions depending on the value of c.

4. a. She did not isolate the absolute

value expression first.

b. Sample answer: She could have

substituted the values for x into the original equation to check whether they satisfy the equation.

c. No solution; Sample explanation:

First, isolate the absolute value expression: |x + 5| = −4. The equation |x + 5| = −4 has no solutions because the absolute value of an expression cannot be negative.

5. Sample answer: The inequality has

the form |ax − b| ≥ c, where c ≥ 0, so it can be written as ax − b ≥ c or ax − b ≤ −c. Thus, |5x − 6| ≥ 9 is equivalent to 5x − 6 ≥ 9 or 5x − 6 ≤ −9.

ASSESS

Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity.

ADAPT

Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute value equations and inequalities and graphing the solutions of absolute value equations and inequalities. If students are still having difficulty, review the process of rewriting an absolute value equation or inequality as two equations or inequalities.

LESSON 1-3 PRACTICE 6. x = 1, x = 11

7. _{x 193}= , x= −5₃

8. x = 3, x = −12

9. no solution

10. | f − 640| = 90, where f represents the

extreme flow rates in gallons per minute; f = 730, f = 550; The greatest flow rate is 730 gallons per minute and the least flow rate is 550 gallons per minute. 11. x < 6 or x > 8 0 2 4 6 8 12. −2 ≤ x ≤ 7 –4 –2 0 2 4 6 8 13. _{x ≤ 2 or x 143}≥ –2 0 2 4 6 14. −17₄ < <x 11₄ –4 –2 0 2 4