4 units 4 units Scaffolding Questions - Ms. Astetemsastete.com/yahoo_site_admin1/assets/docs/AnswersforChpt1-4... · Lesson 1-4 Solving Absolute Value Equations 27 Scaffolding Questions

Lesson 1-4 Solving Absolute Value Equations 27

Scaffolding QuestionsHave students read the Why? section of the lesson.Ask:• In ⎪E - 323.1⎥ = 0.5, what does E

represent? the actual distance to shore

• What is the meaning of the number 0.5 in the equation? the degree of accuracy of the range finder

• What would be the equation for the distance to shore estimated at 962.3 yards? ⎪E - 962.3⎥ = 0.5

• How can this be shown on a number line?

962.8961.8 962.3

E

1 FOCUS1 FOCUS

2 TEACH2 TEACH

Vertical Alignment

Before Lesson 1-4Solve equations using properties of equality.

Lesson 1-4Evaluate expressions involving absolute values. Solve absolute value equations.

After Lesson 1-4 Solve absolute value inequalities.

LessonNotes

Resource Approaching-Level On-Level Beyond-Level English LearnersTeacher Edition • Differentiated Instruction, p. 29 • Differentiated Instruction, pp. 29, 32 • Differentiated Instruction, p. 32 • Differentiated Instruction, p. 29

Chapter ResourceMasters

• Study Guide and Intervention, pp. 24–25

• Skills Practice, p. 26• Practice, p. 27• Word Problem Practice, p. 28


• Skills Practice, p. 26• Practice, p. 27• Word Problem Practice, p. 28• Enrichment, p. 29• Spreadsheet Activity, p. 30

• Practice, p. 27• Word Problem Practice, p. 28• Enrichment, p. 29


• Skills Practice, p. 26• Practice, p. 27• Word Problem Practice, p. 28

Transparencies • 5-Minute Check Transparency 1-4 • 5-Minute Check Transparency 1-4 • 5-Minute Check Transparency 1-4 • 5-Minute Check Transparency 1-4

Other • Study Notebook • Study Notebook • Study Notebook • Study Notebook

Lesson 1-4 Resources


Solving Absolute Value EquationsWhy?Sailors sometimes use a laser range finder to determine distances. Suppose one such range finder is accurate to within ±0.5 yard. This means that if a sailor estimating the distance to shore reads 323.1 yards on the laser range finder, the distance to shore might actually be as close as 322.6 or as far away as 323.6 yards. These extremes can be described by the equation ⎪E - 323.1⎥ = 0.5.

Absolute Value Expressions The absolute value of a number is its distance from 0 on the number line. Since distance is nonnegative, the absolute value of a number is always nonnegative. The symbol |x| is used to represent the absolute value of a number x.

Key Concept

Absolute Value

Words For any real number a, if a is positive or zero, the absolute value of ais a. If a is negative, the absolute value of a is the opposite of a.

Symbols For any real number a, ⎪a⎥ = a if a ≥ 0, and ⎪a⎥ = -a if a < 0.

Model ⎪-4⎥ = 4 and ⎪4⎥ = 4

5-5 4321-4 -3 -2 -1 0

4 units 4 units

When evaluating expressions, absolute value bars act as a grouping symbol. Perform any operations inside the absolute value bars first.

EXAMPLE 1 Evaluate an Expression with Absolute Value

Evaluate 8.4 - ⎪2n + 5⎥ if n = -7.5.

8.4 - ⎪2n + 5⎥ = 8.4 - ⎪2(-7.5) + 5⎥ Replace n with -7.5.

= 8.4 - ⎪-15 + 5⎥ Multiply 2 and -7.5.

= 8.4 - ⎪-10⎥ Add -15 and 5.

= 8.4 - 10 ⎪-10⎥ = 10

= -1.6 Subtract 10 from 8.4.

Guided Practice✓ 1A. Evaluate ⎪4x + 3⎥ - 3 1 _

2 if x = -2. 1 1 _

2 1B. Evaluate 1 1 _

3 - ⎪2y + 1⎥ if y = - 2 _

3 . 1

Personal Tutor glencoe.com

ThenThenYou solved equations using properties of equality. (Lesson 1-3)

NowNow Evaluate expressions

involving absolute values.

Solve absolute value equations.

NGSSS

Reinforcement of MA.912.A.3.6 Solve and graph the solutions ofabsolute value equations and inequalities with one variable.

New VocabularyNew Vocabularyabsolute valueempty setextraneous solution

FL Math Online

glencoe.com

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28 Chapter 1 Equations and Inequalities

Additional Examples also in Interactive Classroom PowerPoint® Presentations

Absolute Value ExpressionsExample 1 shows how to evaluate expressions that contain absolute values.

Formative AssessmentUse the Guided Practice exercises after each example to determine students’ understanding of concepts.

Tips for New Teachers

Reading Students may find it helpful to read the first absolute value bar as “the distance of” and the last absolute value bar as “from zero, without regard to direction.” So, the expression ⎪6 - 2x⎥ would be read as “the distance of the value of 6 - 2x from zero, without regard to direction.”

Absolute Value EquationsExamples 2–4 show how to solve absolute value equations with two solutions, no solution, and one solution.

✓ ✓

Additional Example

Evaluate 2.7 + ⎪6 - 2x⎥ if x = 4. 4.7

11

Additional Example

Solve ⎪y + 3⎥ = 8. Check your solutions. {–11, 5}

22

Additional ExamplesAdditional Examples parallel the

examples in the text exactly.

Step-by-step solutions for these

examples are included in

Interactive Classroom.Interactive Classroom.

INTERACTIVEWHITEBOARDREADY

IWBIWB


Originally, players used leather gloves to hit tennis balls. Soon after, the glove was placed at the end of a stick to extend the reach of the “hand.” Source: The Cliff Richard Tennis Foundation

Absolute Value Equations Some equations contain absolute value expressions. The definition of absolute value is used in solving these equations. For any real numbers a and b, where b ≥ 0, if |a| = b, then a = b or -a = b. This second case is often written as a = -b.

EXAMPLE 2 Solve an Absolute Value Equation

TENNIS A standard adult tennis racket has a 100-square-inch head, plus or minus 20 square inches. Write and solve an absolute value equation to determine the least and greatest possible sizes for the head of an adult tennis racket.

Understand We need to determine the greatest and least possible sizes for the head of a tennis racket given the middle size and the range in sizes.

Plan When writing an absolute value equation, the middle or central value is always placed inside the absolute value symbols. The range is always placed on the other side of the equality symbol.

central value range

⎪x - c⎥ = r

Solve ⎪x - c⎥ = r Absolute value equation

⎪x - 100⎥ = 20 c = 100, and r = 20

Case 1 a = b

x - 100 = 20

x - 100 + 100 = 20 + 100

x = 120

Case 2 a = -b

x - 100 = -20

x - 100 + 100 = -20 + 100

x = 80

Check ⎪x – 100⎥ = 20

⎪120 – 100⎥ � 20

⎪20⎥ � 20

20 = 20 ✔

⎪x - 100⎥ = 20

⎪80 - 100⎥ � 20

⎪-20⎥ � 20

20 = 20 ✔

On a number line, you can see that both solutions are 20 units away from 100.

120100 1109080

20 units 20 units

The solutions are 120 and 80. The greatest size is 120 square inches and the least is 80 square inches.

Guided Practice✓Solve each equation. Check your solutions.

2A. 9 = ⎪x + 12⎥ {-21, -3} 2B. 8 = ⎪y + 5⎥ {-13, 3}


Because the absolute value of a number is always positive or zero, an equation like |x| = -4 is never true. Thus, it has no solution. The solution set for this type of equation is the empty set, symbolized by { } or ∅.

Problem-SolvingTip

Write an Equation Frequently, the best way to solve a problem is to use the given information to write and solve an equation.

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Additional Examples

Solve ⎪6 - 4t⎥ + 5 = 0. �

Solve ⎪8 + y⎥ = 2y - 3. Check your solutions. {11}

3344

Watch Out!

Preventing Errors Remind students to think about the meaning of the mathematical sentence before they begin their calculations and again when they evaluate the reasonableness of their solution. Explain to students that they can solve verbal problems when they ask questions about words they do not understand, take time to read, understand, and plan, using a sketch to help.

Differentiated Instruction

If students found anything from the lesson confusing,

Then ask them to record two or three of the confusing items separately on an index card. Have them write an explanation or example for each item in their own words. This will help them review in the future.

Focus on Mathematical Content

Absolute Value The absolute value of a number is the distance of that number from 0 on a number line. Therefore, the statement “the absolute value of x is always x” is not true. For example, if x is –3 then the absolute value of –3 is 3.

AL OL ELL

INTERACTIVE WHITEBOARD On the board, work through several examples solving absolute value equations. Save your work to a file and send it to your students so they can use it as an additional reference.


EXAMPLE 3 No Solution

Solve ⎪3x - 2⎥ + 8 = 1.

⎪3x - 2⎥ + 8 = 1 Original equation

⎪3x - 2⎥ + 8 - 8 = 1 - 8 Subtract 8 from each side.

⎪3x - 2⎥ = -7 Simplify.

This sentence is never true. The solution set is ∅.


3A. -2 ⎪3a⎥ = 6 ∅ 3B. ⎪4b + 1⎥ + 8 = 0 ∅


It is important to check your answers when solving absolute value equations. Even if the correct procedure for solving the equation is used, the answers may not be actual solutions to the original equation. Such a number is called an extraneous solution.

EXAMPLE 4 One Solution

Solve ⎪x + 10⎥ = 4x - 8. Check your solutions.

Case 1 a = b

x + 10 = 4x - 8

10 = 3x - 8

18 = 3x

6 = x

Case 2 a = -b

x + 10 = -(4x - 8)

x + 10 = -4x + 8

5x + 10 = 8

5x = -2

x = - 2 _ 5

There appear to be two solutions, 6 and - 2 _ 5 .

CHECK Substitute each value in the original equation.

⎪x + 10⎥ = 4x - 8

⎪6 + 10⎥ � 4(6) - 8

⎪16⎥ � 24 - 8

16 = 16 ✔

⎪x + 10⎥ = 4x - 8

⎪- 2 _ 5 + 10⎥ � 4 (- 2 _

5 ) - 8

⎪9 3 _ 5 ⎥ � -1 3 _

5 - 8

9 3 _ 5 ≠ -9 3 _

5 ✘

Because 9 3 _ 5 ≠ -9 3 _

5 , the only solution is 6. The solution set is {6}.


4A. 2 ⎪x + 1⎥ - x = 3x - 4 3 4B. 3 ⎪2x + 2⎥ - 2x = x + 3 -1


StudyTip

Absolute Value It is possible for an absolute value equation to have only one solution. Remember to set up two cases. Then check your solutions.

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Formative AssessmentUse Exercises 1–13 to check for understanding.

Use the chart at the bottom of this page to customize assignments for your students.

Multiple Representations In Exercise 44, students use a number line, information organized in a table, and algebra to analyze inequalities.

Additional Answers

43. ⎪x - 100⎥ = 245; maximum: 345 ft above sea level; minimum: –145 ft below sea level. No, the maximum is reasonable but the minimum is not. Florida’s lowest point should be at sea level where Florida meets the Atlantic Ocean and the Gulf of Mexico.

44a. Sample answer:

-2-3 3-1 210

44c. Sample answer: If A is less than B, then any number added to or subtracted from A will be less than the same number added to or subtracted from B. If B is greater than A, then any number added to or subtracted from B is greater than the same number added to or subtracted from A.

3 PRACTICE3 PRACTICE

✓ ✓

Differentiated Homework OptionsLevel Assignment Two-Day Option

AL Basic 14–34, 45, 47–74 15–33 odd, 52–55 14–34 even, 45, 47–51, 56–74

OL Core 15–43 odd, 44, 45, 47–74 14–34, 52–55 35–45, 47–51, 56–74

BL Advanced 35–68, (optional: 69–74)

Watch Out!

Error Analysis For Exercise 45, students should see that Ana and Ling have differences in their solutions. They must decide which person checked the solutions correctly.


✓ Check Your Understanding

Evaluate each expression if x = -4 and y = -9. 4. Example 1p. 27

1. ⎪x - 8⎥ 12 2. ⎪7y⎥ 63 3. -3 ⎪xy⎥ -108 4. -2 ⎪3x + 8⎥ - 4 -12

5. FISH Most freshwater tropical fish thrive if

ºF

697071727374757677

7978

80

the water is within 2°F of 78°F.

a. Write an equation to determine the least and greatest optimal temperatures. ⎪x - 78⎥ = 2

b. Solve the equation you wrote in part a.

c. If your aquarium’s thermometer is accurate to within plus or minus 1°F, what should the temperature of the water be to ensure that it reaches the minimum temperature? Explain. 77°F; This would ensure a minimum temperature of 76°F.

Solve each equation. Check your solutions. Examples 2–4pp. 28–29

6. ⎪x + 8⎥ = 12 {4, -20} 7. ⎪y - 4⎥ = 11 {15, -7}

8. ⎪a - 5⎥ + 4 = 9 {10, 0} 9. ⎪b - 3⎥ + 8 = 3 ∅

10. 3 ⎪2x - 3⎥ - 5 = 4 {3, 0} 11. -2 ⎪5y - 1⎥ = -10 { 6 _ 5 , - 4 _

5 }

12. ⎪a - 4⎥ = 3a - 6 2.5 13. ⎪b + 5⎥ = 2b + 3 2

Practice and Problem Solving= Step-by-Step Solutions begin on page R20.

Extra Practice begins on page 947.

Evaluate each expression if a = -3, b = -5, and c = 4.2. Example 1p. 27

14. ⎪-3c⎥ 12.6 15. ⎪5b⎥ 25 16. ⎪a - b⎥ 2 17. ⎪b - c⎥ 9.2

18. ⎪3b - 4a⎥ 3 19. 2 ⎪4a - 3c⎥ 49.2 20. - ⎪3c - a⎥ -15.6 21. - ⎪abc⎥ -63

22. FOOD To make cocoa powder, cocoa beans are roasted. The ideal temperature for roasting is 300°F, plus or minus 25°. Write and solve an equation describing the maximum and minimum roasting temperatures for cocoa beans. |x - 300| = 25; maximum: 325°F; minimum: 275°F

Solve each equation. Check your solutions.Examples 2–4pp. 28–29

23. ⎪z - 13⎥ = 21 {34, -8} 24. ⎪w + 9⎥ = 17 {8, -26}

25. 9 = ⎪d + 5⎥ {4, -14} 26. 35 = ⎪x - 6⎥ {-29, 41}

27. 5 ⎪q + 6⎥ = 20 {-2, -10} 28. -3 ⎪r + 4⎥ = -21 {3, -11}

29. 3 ⎪2a - 4⎥ = 0 2 30. 8 ⎪5w - 1⎥ = 0 1 _ 5

3131 2 ⎪3x - 4⎥ + 8 = 6 ∅ 32. 4 ⎪7y + 2⎥ - 8 = -7 {- 1 _ 4 , - 9 _

28 }

33. -3 ⎪3t - 2⎥ - 12 = -6 ∅ 34. -5 ⎪3z + 8⎥ - 5 = -20 {- 5 _ 3 , - 11 _

3 }

35. MONEY The U.S. Mint produces quarters that weigh about 5.67 grams each. After the quarters are produced, a machine weighs them. If the quarter weighs 0.02 gram more or less than the desired weight, the quarter is rejected. Write and solve an equation to find the heaviest and lightest quarters the machine will approve. ⎪x - 5.67⎥ = 0.02; heaviest: 5.69 g; lightest: 5.65 g

Evaluate each expression if q = -8, r = -6, and t = 3.

36. 12 - t ⎪3r + 2⎥ -36 37. 2q + ⎪2rt + q⎥ 28 38. -5t - q|8r - t| 393

5b. least: 76°F, greatest: 80°F

B

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Chapter 1 28 Glencoe Algebra 2

1-4 Word Problem PracticeSolving Absolute Value Equations

1. LOCATIONS Identical vacation cottages, equally spaced along a street, are numbered consecutively beginning with 10. Maria lives in cottage #17. Joshua lives 4 cottages away from Maria. If n represents Joshua’s cottage number, then |n - 17| = 4. What are the possible numbers of Joshua’s cottage?

10 12 14 16 18 20 2422

Maria’s

2. HEIGHT Sarah and Jessica are sisters. Sarah’s height is s inches and Jessica’s height is j inches. Their father wants to know how many inches separate the two. Write an equation for this difference in such a way that the result will always be positive no matter which sister is taller.

3. AGES In 2005, 24.8% of all Americans were under 18 years old. Rhonda conducts a survey of the ages of students in eleventh grade at her school. On November 1, she finds the average age is 200 months. She also finds that two-thirds of the students are within 6 months of the average age. Write and solve an equation to determine the age limits for this group of students. How many months will it be till the first of these students turn 18?

4. TOLERANCE Martin makes exercise weights. For his 10-pound dumbbells, he guarantees that the actual weight of his dumbbells is within 0.1 pound of 10 pounds. Write and solve an equation that describes the minimum and maximum weight of his 10-pound dumbbells.

5. WALKING Jim is walking along a straight line. An observer watches him. If Jim walks forward, the observer records the distance as a positive number, but if he walks backward, the observer records the distance as a negative number. The observer has recorded that Jim has walked a, then b, then c feet.

a. Write a formula for the total distance that Jim walked.

b. The equation you wrote in part a should not be T = |a + b + c|. What does |a + b + c| represent?

c. When would the formula you wrote in part a give the same value as the formula shown in part b?

13 or 21

d = ⎪s - j⎥ or d = ⎪j - s⎥

⎪a - 200⎥ = 3

a = 197 or 203

⎪w - 10⎥ = 0.1

minimum weight: 9.9 pounds

maximum weight: 10.1 pounds

T = ⎪a⎥ + ⎪b⎥ + ⎪c⎥

The distance Jim ends up from where he started.

They will be equal only if Jim walks in the same direction each time giving a, b, and c all the same sign.

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1-4 PracticeSolving Absolute Value Equations

Evaluate each expression if a = -1, b = -8, c = 5, and d = -1.4.

1. |6a| 2. |2b + 4|

3. - |10d + a| 4. |17c| + |3b - 5|

5. -6 |10a - 12| 6. |2b - 1| - |-8b + 5|

7. |5a - 7| + |3c - 4| 8. |1 - 7c| - |a|

9. -3|0.5c + 2| - |-0.5b| 10. |4d| + |5 - 2a|

11. |a - b| + |b - a| 12. |2 - 2d| - 3|b|

Solve each equation. Check your solutions.

13. |n - 4| = 13 14. |x - 13| = 2

15. |2y - 3| = 29 16. 7|x + 3| = 42

17. |3u - 6| = 42 18. |5x - 4| = -6

19. -3 |4x - 9| = 24 20. -6|5 - 2y| = -9

21. |8 + p| = 2p - 3 22. |4w - 1| = 5w + 37

23. 4 |2y - 7| + 5 = 9 24. -2|7 - 3y| - 6 = -14

25. 2 |4 - s| = -3s 26. 5 - 3|2 + 2w| = -7

27. 5 |2r + 3| - 5 = 0 28. 3 - 5|2d - 3| = 4

29. WEATHER A thermometer comes with a guarantee that the stated temperature differs from the actual temperature by no more than 1.5 degrees Fahrenheit. Write and solve an equation to find the minimum and maximum actual temperatures when the thermometer states that the temperature is 87.4 degrees Fahrenheit.

30. OPINION POLLS Public opinion polls reported in newspapers are usually given with a margin of error. For example, a poll with a margin of error of ±5% is considered accurate to within plus or minus 5% of the actual value. A poll with a stated margin of error of 63% predicts that candidate Tonwe will receive 51% of an upcoming vote. Write and solve an equation describing the minimum and maximum percent of the vote that candidate Tonwe is expected to receive.

6 12

-15 114

-132 -52

23 33

-17.5 12.6

14 -19.2

{-9, 17} {11, 15}

{-13, 16} {-9, 3}

{-12, 16} ∅

{11} {-38}

{3, 4} {1, 3 2 − 3 }

{-8} {-3, 1}

{-2, -1}

� x - 87.4 � ≤ 1.5; or 85.9 ≤ x ≤ 88.9

� x - 51 � ≤ 3 or 48 ≤ x ≤ 54

∅

∅

{1.75, 3.25}

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1-4 Study Guide and InterventionSolving Absolute Value Equations

Absolute Value Expressions The absolute value of a number is its distance from 0 on a number line. The symbol ⎪x⎥ is used to represent the absolute value of a number x.

Absolute Value

• Words For any real number a, if a is positive or zero, the absolute value of a is a.

If a is negative, the absolute value of a is the opposite of a.

• Symbols For any real number a, ⎪a⎥ = a, if a ≥ 0, and ⎪a⎥ = -a, if a < 0.

Evaluate � -4 � - � -2x � if x = 6.

⎪-4⎥ - ⎪-2x⎥ = ⎪-4⎥ - ⎪-2 � 6⎥ = ⎪-4⎥ - ⎪-12⎥ = 4 - 12 = -8

ExercisesEvaluate each expression if w = -4, x = 2, y = 1 −

2 , and z = -6.

1. ⎪2x - 8⎥ 2. ⎪6 + z⎥ - ⎪-7⎥ 3. 5 + ⎪w + z⎥

4. ⎪x + 5⎥ - ⎪2w⎥ 5. ⎪x⎥ - ⎪y⎥ - ⎪z⎥ 6. ⎪7 - x⎥ + ⎪3x⎥

7. ⎪w - 4x⎥ 8. ⎪wz⎥ - ⎪xy⎥ 9. ⎪z⎥ - 3 ⎪5yz⎥

10. 5 ⎪w⎥ + 2 ⎪z - 2y⎥ 11. ⎪z⎥ - 4 ⎪2z + y⎥ 12. 10 - ⎪xw⎥

13. ⎪6y + z⎥ + ⎪yz⎥ 14. 3 ⎪wx⎥ + 1 − 4 ⎪4x + 8y⎥ 15. 7 ⎪yz⎥ - 30

16. 14 - 2 ⎪w - xy⎥ 17. ⎪2x - y⎥ + 5y 18. ⎪xyz⎥ + ⎪wxz⎥

19. z ⎪z⎥ + x ⎪x⎥ 20. 12 - ⎪10x - 10y⎥ 21. 1 − 2 ⎪5z + 8w⎥

22. ⎪yz - 4w⎥ -w 23. 3 − 4 ⎪wz⎥ + 1 −

2 ⎪8y⎥ 24. xz - ⎪xz⎥

Evaluate � 2x - 3y � if x = -4 and y = 3.

⎪2x - 3y⎥ = ⎪2(-4) - 3(3)⎥ = ⎪-8 - 9⎥ = ⎪-17⎥ = 17

Example 1 Example 2

4 -7 15

-1 -4 1 − 2 11

12 23 -39

34 -40 2

6 27 -9

4 6 54

-32 -3 31

17 20 -24

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Study Guide and Intervention pp. 24–25 AL OL ELL

Practice p. 27 AL OL BL ELL

Word Problem Practice p. 28 AL OL BL ELL

c.

1-4 Enrichment

Considering All Cases in Absolute Value Equations

You have learned that absolute value equations with one set of absolute value symbols have two cases that must be considered. For example, | x + 3 | = 5 must be broken into x + 3 = 5 or -(x + 3) = 5. For an equation with two sets of absolute value symbols, four cases must be considered.

Consider the problem | x + 2 | + 3 = | x + 6 |. First we must write the equations for the case where x + 6 ≥ 0 and where x + 6 < 0. Here are the equations for these two cases:

| x + 2 | + 3 = x + 6

| x + 2 | + 3 = -(x + 6)

Each of these equations also has two cases. By writing the equations for both cases of each equation above, you end up with the following four equations:

x + 2 + 3 = x + 6 x + 2 + 3 = -(x + 6) -(x + 2) + 3 = x + 6 -x - 2 + 3 = -(x + 6)

Solve each of these equations and check your solutions in the original equation,

| x + 2 | + 3 = | x + 6 |. The only solution to this equation is - 5 − 2 .

E i

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Enrichment p. 29 OL BL


Solve each equation. Check your solutions.

39. 8x = 2 ⎪6x - 2⎥ {1, 1 _ 5 } 40. -6y + 4 = ⎪4y + 12⎥ - 4 _

5

41. 8z + 20 = - ⎪2z + 4⎥ - 8 _ 3 42. -3y - 2 = ⎪6y + 25⎥ {-3, - 23 _

3 }

4343 SEA LEVEL Florida is on average 100 feet above sea level. This level varies by as much as 245 feet depending on precipitation and your location. Write and solve an equation describing the maximum and minimum sea levels for Florida. Is this solution reasonable? Explain. See margin.

44. MULTIPLE REPRESENTATIONS Draw a number line. a. See margin. a. GEOMETRIC Label any 5 integers on the number line points A, B, C, D, and F.

b. TABULAR Fill in each blank in the table with either > or < using the points from the number line.

A ____ BA + C ____ B + CA + D ____ B + DA + F ____ B + F

A ____ BA - C ____ B - CA - D ____ B - DA - F ____ B - F

B ____ AB + C ____ A + CB + D ____ A + DB + F ____ A + F

B ____ AB - C ____ A - CB - D ____ A - DB - F ____ A - F

c. VERBAL Describe the patterns in the table. See margin.

d. ALGEBRAIC Describe the patterns algebraically, using the variable x to replace C, D, and F. If A < B, then A + x < B + x. If A < B, then A - x < B - x. If B > A, then B + x > A + x. If B > A, then B - x > A - x.

H.O.T. Problems Use Higher-Order Thinking Skills

45. ERROR ANALYSIS Ana and Ling are solving ⎪3x + 14⎥ = -6x. Is either of them correct? Explain your reasoning.

Ana| 3x + 14| = –6x

3x + 14 = –6x or 3x + 14 = 6x 9x = –14 14 = 3x x = – 14

_ 9 ✔ x = 14 _ 3 ✔

Ling|3x + 14| = –6x

3x + 14 = –6x or 3x + 14 = 6x 9x = –14 14 = 3x x = – 14 _ 9 ✗ x = 14 _ 3 ✔

46. CHALLENGE Solve ⎪2x - 1⎥ + 3 = ⎪5 - x⎥ . List all cases and resulting equations. (Hint: There are four possible cases to examine as potential solutions.) See Chapter 1 Answer Appendix.

REASONING If a, x, and y are real numbers, determine whether each statement is sometimes, always, or never true. Explain your reasoning.

47. If ⎪a⎥ > 7, then ⎪a + 3⎥ > 10.

48. If ⎪x⎥ < 3, then ⎪x⎥ + 3 > 0.

49. If y is between 1 and 5, then ⎪y - 3⎥ ≤ 2.

50. OPEN ENDED Write an absolute value equation of the form ⎪ax + b⎥ = cx + d that has no solution. Assume that a, b, c, and d ≠ 0.

51. WRITING IN MATH Explain step by step how you solve an absolute value equation of the form a ⎪x - b⎥ + c = d for x. See Chapter 1 Answer Appendix.

45. Ling; Ana included an extraneous solution. She would have caught this error if she had checked to see if her answers were correct by substituting the values into the original equation.47. Sometimes; this is only true for certain values of a. For example, it is true for a = 8; if 8 > 7, then 11 > 10. However it is not true for a = -8; if 8 > 7, then 5 ≯ 10.48. Always; if ⎪x⎥ < 3, then x is between -3 or 3. Adding 3 to the absolute value of any of the numbers in this set will produce a positive number.49. Always; starting with numbers between 1 and 5 and subtracting 3 will produce numbers between -2 and 2. These all have an absolute value less than or equal to 2.50. Sample answer: ⎪2x + 1⎥ = x - 3, or ⎪3x + 10⎥ = x - 5, or ⎪x - 1⎥ = 1 _

2 x - 4

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A Study Guide and Study Guide and

Intervention, Intervention,

Practice, Word Practice, Word

Problem Practice, Problem Practice, and

Enrichment Master Enrichment Master

are shown for every

lesson. These masters

can be found in the

Chapter Resource

Masters.



Crystal Ball Have students write how they think what they have learned in Lessons 1-3 and 1-4 will connect with Lesson 1-5, Solving Inequalities.

4 ASSESS4 ASSESS

Tips for New Teachers

Looking Ahead Lesson 1-5 presents solving inequalities using steps similar to those for solving equations. Exercises 69–74 should be used to determine students’ familiarity with solving equations.

Differentiated Instruction

Extension For equations with one set of absolute value symbols, two cases must be considered. For an equation with two sets of absolute value symbols, four cases must be considered. How many cases must be considered for an equation containing three sets of absolute value symbols? 8

OL BL

New teachers, or teachers new

to teaching mathematics, may

especially appreciate the Tips for Tips for

New Teachers.New Teachers.


Spiral Review

Solve each equation. Check your solution. (Lesson 1-3)

56. 4x + 6 = 30 6 57. 5p - 10 = 4(7 + 6p) -2 58. 3 _

5 y – 7 = 2 _

5 y + 3 50

59. MONEY Nhu is saving to buy a car. In the first 6 months, his savings were $80 less

than 3 _ 4 the price of the car. In the second six months, Nhu saved $50 more than 1 _

5 the

price of the car. He still needs $370. (Lesson 1-3)

a. What is the price of the car? $6800

b. What is the average amount of money Nhu saved each month? $535.83

c. If Nhu continues to save the average amount each month, in how many months will he be able to afford the car? 1 mo

Name the property illustrated by each equation. (Lesson 1-2)

60. (1 + 8) + 11 = 11 + (1 + 8) Comm. (+) 61. z(9 - 4) = z · 9 - z · 4 Distributive

Simplify each expression. (Lesson 1-2)

62. 7a + 3b - 4a - 5b 3a - 2b 63. 3x + 5y + 7x - 3y 10x + 2y

64. 3(15x - 9y) + 5(4y - x) 40x - 7y 65. 2(10m - 7a) + 3(8a - 3m) 11m + 10a

66. 8(r + 7t) - 4(13t + 5r) -12r + 4t 67. 4(14c - 10d) - 6(d + 4c) 32c - 46d

68. GEOMETRY The formula for the surface area of a rectangular prism is

12 in.5 in.

7 in.SA = 2�w + 2�h + 2wh, where � represents the length, w represents the width, and h represents the height. Find the surface area of the rectangular prism at the right. (Lesson 1-1) 358 in2

Skills Review

Solve each equation. (Lesson 1-3)

69. 15x + 5 = 35 2 70. 2.4y + 4.6 = 20 ≈6.417 71. 8a + 9 = 6a - 7 -8

72. 3(w - 1) = 2w - 6 -3 73. 1 _

2 (2b - 4) = 2 + 8b - 4 _

7 74. 1

_ 3 (6p - 24) = 18 + 3p -26

52. If 4x - y = 3 and 2x + 3y = 19, what is the value of y? D

A. 2

B. 3

C. 4

D. 5

53. GRIDDED RESPONSE Two male and 2 female students from each of the 9th, 10th, 11th, and 12th grades comprise the Student Council. If a Student Council representative is chosen at random to attend a board meeting, what is the probability that the student will be either an 11th grader or male? 5/8

54. Which equation is equivalent to 4(9 - 3x) = 7 - 2(6 - 5x)? G

F. 8x = 41

G. 22x = 41

H. 8x = 24

I. 22x = 24

55. SAT/ACT A square with side length 4 units has one vertex at the point (1, 2). Which one of the following points cannot be diagonally opposite that vertex? C

A. (-3, -2) C. (5, -3)

B. (-3, 6) D. (5, 6)

912.A.3.14, 912.P.1.2, 912.A.3.1, 912.G.1.1NGSSS PRACTICE

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Documents

4 units 4 units Scaffolding Questions - Ms. Astetemsastete.com/yahoo_site_admin1/assets/docs/AnswersforChpt1-4... · Lesson 1-4 Solving Absolute Value Equations 27 Scaffolding Questions