Maintaining Skills, 61. ▻ CHAPTER STUDY GUIDE. STUDENT HELP. Study Tip
2, 8, 16, 26, 36,. 37, 42, 44, 49. Skills Review 7, 16, 32, 43. Reading Algebra 9 ...
Summer 2013 Algebra 1, Math 1 Course Content Teacher Edition
“Algebra 1 Concepts and Skills” Larson, Boswell, Kanold, Stiff McDougal Littell (2004)
1
Connections to Algebra 䉴 CHAPTER STUDY GUIDE
STUDENT HELP Study Tip 2, 8, 16, 26, 36, 37, 42, 44, 49 Skills Review 7, 16, 32, 43 Reading Algebra 9, 11, 30, 31, 32 Writing Algebra 3 Vocabulary Tip 5, 24, 31, 43 Keystroke Help 13 Test Tip 14, 41, 60
APPLICATION HIGHLIGHTS
INT
Scuba Diving 1, 53 Race Cars 4 Basketball 17 Veterinarians 26 Mach Number 28 Northwest Territory 34 Plant Growth 40 Braking Distance 46 Hot-air Ballooning 49 Chisholm Trail 52 NE ER T
INTERNET
1, 7, 10, 19, 25, 34, 38, 40, 42, 44, 46, 50, 52, 61
2
1.1 Variables in Algebra
3
1.2 Exponents and Powers
9
1.3 Order of Operations
15
QUIZ 1, 21
1.4 Equations and Inequalities 䉴 DEVELOPING CONCEPTS: Finding Patterns, 22
24
1.5 Translating Words into Mathematical Symbols
30
QUIZ 2, 35
1.6 A Problem Solving Plan Using Models
36
1.7 Tables and Graphs
42
1.8 An Introduction to Functions
48
QUIZ 3, 54
ASSESSMENT Chapter Readiness Quiz, 2 Quizzes, 21, 35, 54 Standardized Test Practice, 8, 14, 20, 29, 35, 41, 47, 53 Chapter Summary and Review, 55 Chapter Test, 59 Chapter Standardized Test, 60 Maintaining Skills, 61
Getting Ready
Pre-Course Test
xviii
A diagnostic test on key skills from earlier courses, referenced to the Skills Review (pp. 759–782)
Pre-Course Practice
xx
Additional practice on the skills in the Pre-Course Test, also referenced to the Skills Review
Contents
v
2
Properties of Real Numbers 䉴 CHAPTER STUDY GUIDE
STUDENT HELP Study Tip 66, 86, 93, 94, 101, 102, 108, 109, 113, 114 Skills Review 66, 111 Reading Algebra 65, 71, 107 Writing Algebra 94 Vocabulary Tip 100 Look Back 72, 79, 84, 87, 97, 99, 119, 120 Keystroke Help 80 Test Tip 126
APPLICATION HIGHLIGHTS
INT
Helicopters 63, 75 Nome, Alaska 67 Stars 69 Space Shuttle 73 Planets 75 Golf Scores 82 Stock Market 88 Water Cycle 90 Flying Squirrels 95 Rappelling 97
64
2.1 The Real Number Line
65
2.2 Absolute Value
71
2.3 Adding Real Numbers 䉴 DEVELOPING CONCEPTS: Addition of Integers, 77
78
QUIZ 1, 83
2.4 Subtracting Real Numbers 䉴 DEVELOPING CONCEPTS: Subtraction of Integers, 84
86
2.5 Multiplying Real Numbers 䉴 DEVELOPING CONCEPTS: Multiplication of Integers, 92
93
2.6 The Distributive Property 䉴 DEVELOPING CONCEPTS: The Distributive Property, 99
100
QUIZ 2, 106
2.7 Combining Like Terms
107
2.8 Dividing Real Numbers
113
QUIZ 3, 118 Extension: Inductive and Deductive Reasoning
119
NE ER T
INTERNET
63, 67, 69, 73, 75, 80, 90, 91, 95, 97, 101, 104, 108, 115, 117, 127
vi
Contents
ASSESSMENT Chapter Readiness Quiz, 64 Quizzes, 83, 106, 118 Standardized Test Practice, 70, 76, 82, 91, 98, 105, 112, 117
Chapter Summary and Review, 121 Chapter Test, 125 Chapter Standardized Test, 126 Maintaining Skills, 127
3
Solving Linear Equations 䉴 CHAPTER STUDY GUIDE
STUDENT HELP Study Tip 132, 133, 138, 146, 151, 157, 158, 178, 184 Skills Review 165 Writing Algebra 177 Vocabulary Tip 144 Look Back 131, 152, 171 Test Tip 143, 194
APPLICATION HIGHLIGHTS
INT
Bald Eagles 129, 181 City Parks 136 Newspaper Recycling 142 Thunderstorms 142 Earth’s Temperature 145 Steamboats 155 Cheetahs 155 Moons of Jupiter 162 Cocoa Consumption 167 Bottle-nosed Whales 175 NE ER T
INTERNET
129, 133, 136, 139, 142, 145, 148, 153, 155, 159, 161, 162, 164, 167, 170, 172, 173, 179, 181, 182, 185, 187, 195
130
3.1 Solving Equations Using Addition and Subtraction 䉴 DEVELOPING CONCEPTS: One-Step Equations, 131
132
3.2 Solving Equations Using Multiplication and Division
138
3.3 Solving Multi-Step Equations
144
QUIZ 1, 149
3.4 Solving Equations with Variables on Both Sides 䉴 DEVELOPING CONCEPTS: Variables on Both Sides, 150
151
3.5 More on Linear Equations
157
3.6 Solving Decimal Equations
163
QUIZ 2, 169
䉴
GRAPHING CALCULATOR: Solving Multi-Step Equations, 170
3.7 Formulas
171
3.8 Ratios and Rates
177
3.9 Percents
183
QUIZ 3, 188 Project: Planning a Car Wash
198
ASSESSMENT Chapter Readiness Quiz, 130 Quizzes, 149, 169, 188 Standardized Test Practice, 137, 143, 148, 156, 162, 168, 176, 182, 188 Chapter Summary and Review, 189
Chapter Test, 193 Chapter Standardized Test, 194 Maintaining Skills, 195 Cumulative Practice, Chapters 1–3, 196
Contents
vii
4
Graphing Linear Equations and Functions 䉴 CHAPTER STUDY GUIDE
STUDENT HELP Study Tip 203, 211, 212, 216, 223, 231, 238, 250, 251, 253, 254 Reading Algebra 230, 236, 254 Look Back 218, 252 Keystroke Help 250 Test Tip 264
APPLICATION HIGHLIGHTS
INT
Cable Cars 201, 247 Wing Length 207 Triathlon 214 Boiling Point 214 Mount St. Helens 218 Zoo Fundraising 226 U.S.S. Constitution 234 Gold Bullion 237 Violin Family 240 Monarch Butterflies 257
4.1 The Coordinate Plane
203
4.2 Graphing Linear Equations 䉴 DEVELOPING CONCEPTS: Linear Equations, 209
210
4.3 Graphing Horizontal and Vertical Lines
216
QUIZ 1, 221
4.4 Graphing Lines Using Intercepts
222
4.5 The Slope of a Line 䉴 DEVELOPING CONCEPTS: Investigating Slope, 228
229
4.6 Direct Variation
236
QUIZ 2, 241
4.7 Graphing Lines Using Slope-Intercept Form 䉴 DEVELOPING CONCEPTS: Slope-Intercept Form, 242 䉴 GRAPHING CALCULATOR: Graphing a Linear Equation, 250
243
4.8 Functions and Relations
252
QUIZ 3, 258
NE ER T
INTERNET
201, 204, 207, 214, 215, 217, 224, 226, 227, 232, 237, 245, 248, 250, 256, 265
viii
Contents
202
ASSESSMENT Chapter Readiness Quiz, 202 Quizzes, 221, 241, 258 Standardized Test Practice, 208, 215, 221, 227, 235, 240, 249, 257 Chapter Summary and Review, 259 Chapter Test, 263 Chapter Standardized Test, 264 Maintaining Skills, 265
5
Writing Linear Equations 䉴 CHAPTER STUDY GUIDE
STUDENT HELP Study Tip 270, 271, 273, 278, 279, 285, 292, 293, 295, 298, 299, 307, 308 Vocabulary Tip 274, 291 Look Back 300 Test Tip 318
APPLICATION HIGHLIGHTS
INT
Archaeology 267, 303 Hurdling 273 Old Faithful 274 Water Pressure 282 Airplane Descent 289 The Chunnel 289 Speed of Sound 290 Movie Theaters 298, 299 Car Costs 302 City Street Plan 311 NE ER T
INTERNET
267, 271, 273, 274, 280, 282, 287, 289, 295, 298, 303, 306, 310, 319
268
5.1 Slope-Intercept Form
269
5.2 Point-Slope Form 䉴 DEVELOPING CONCEPTS: Point-Slope Form, 276
278
QUIZ 1, 284
5.3 Writing Linear Equations Given Two Points
285
5.4 Standard Form
291
QUIZ 2, 297
5.5 Modeling with Linear Equations
298
5.6 Perpendicular Lines 䉴 DEVELOPING CONCEPTS: Perpendicular Lines, 305
306
QUIZ 3, 312
ASSESSMENT Chapter Readiness Quiz, 268 Quizzes, 284, 297, 312 Standardized Test Practice, 275, 283, 290, 296, 304, 311 Chapter Summary and Review, 313 Chapter Test, 317 Chapter Standardized Test, 318 Maintaining Skills, 319
Contents
ix
6
Solving and Graphing Linear Inequalities 䉴 CHAPTER STUDY GUIDE
STUDENT HELP Study Tip 324, 330, 337, 338, 342, 343, 344, 348, 349, 350, 356, 362, 363, 368, 369 Skills Review 340 Reading Algebra 323 Writing Algebra 325, 331 Vocabulary Tip 367 Keystroke Help 374 Test Tip 380
APPLICATION HIGHLIGHTS Music 321 Astronomy 325, 359 Mercury 327 Fly-fishing 338 Mountain Plants 343 Steel Arch Bridge 346 Water Temperature 352 Poodles 357 Fireworks 365 Nutrition 371
6.1 Solving Inequalities Using Addition or Subtraction
323
6.2 Solving Inequalities Using Multiplication or Division 䉴 DEVELOPING CONCEPTS: Investigating Inequalities, 329
330
6.3 Solving Multi-Step Inequalities
336
QUIZ 1, 341
6.4 Solving Compound Inequalities Involving “And”
342
6.5 Solving Compound Inequalities Involving “Or”
348
6.6 Solving Absolute-Value Equations 䉴 DEVELOPING CONCEPTS: Absolute-Value Equations, 354
355
QUIZ 2, 360
6.7 Solving Absolute-Value Inequalities
361
6.8 Graphing Linear Inequalities in Two Variables
367
QUIZ 3, 373
䉴
GRAPHING CALCULATOR: Graphing Inequalities, 374
Project: Investigating Springs INT
322
384
NE ER T
INTERNET
321, 327, 332, 337, 340, 346, 347, 349, 359, 365, 369, 371, 372, 374, 381
x
Contents
ASSESSMENT Chapter Readiness Quiz, 322 Quizzes, 341, 360, 373 Standardized Test Practice, 328, 335, 341, 347, 353, 360, 366, 372 Chapter Summary and Review, 375
Chapter Test, 379 Chapter Standardized Test, 380 Maintaining Skills, 381 Cumulative Practice, Chapters 1–6, 382
1
Connections to to Connections Algebra Algebra How much does it cost to rent scuba diving equipment?
1.1
Variables in Algebra How many miles has a race car traveled?
Goal Evaluate variable expressions.
A race car zooms around the Indianapolis Motor Speedway at 180 miles per hour. In Example 3 you will find how many miles the car will travel in 2 hours.
Key Words • • • • •
variable value variable expression numerical expression evaluate
In algebra, you can use letters to represent one or more numbers. When a letter is used to represent a range of numbers, it is called a variable. The numbers are called values of the variable. For example, the distance traveled by the race car in the picture above can be expressed as the variable expression 180t, where t represents the number of hours the car has traveled. ASSIGNING VARIABLES
A variable expression consists of constants, variables, and operations. An expression that represents a particular number is called a numerical expression. For example, the distance traveled by the race car in two hours is given by the numerical expression 180 2.
Student Help WRITING ALGEBRA
The multiplication symbol is usually not used in algebra because of its possible confusion with the variable x.
EXAMPLE
1
Describe the Variable Expression
Here are some variable expressions, their meanings, and their operations. VARIABLE EXPRESSION
MEANING
OPERATION
8y, 8 p y, (8)(y)
8 times y
Multiplication
16 , 16 b b
16 divided by b
Division
4s
4 plus s
Addition
9x
9 minus x
Subtraction
Describe the Variable Expression State the meaning of the variable expression and name the operation. x 1. 10 x 2. 13 x 3. 4. 24x 16
1.1
Variables in Algebra
3
To evaluate a variable expression, you write the expression, substitute a number for each variable, and simplify. The resulting number is the value of the expression. EVALUATING EXPRESSIONS
Write expression.
EXAMPLE
2
Substitute numbers.
Simplify.
Evaluate the Variable Expression
Evaluate the variable expression when y 2. Solution EXPRESSION
SUBSTITUTE
SIMPLIFY
a. 5y
5(2)
10
10 b. y
10 2
5
c. y 6
26
8
d. 14 y
14 2
12
Evaluate the Variable Expression Evaluate the variable expression when x 3. 6. 5 x
5. 7x
12 7. x
8. x 2
The variable expression r times t can be written as rt, r p t, or (r)(t).
Sports
EXAMPLE
3
Evaluate rt to Find Distance
RACE CARS Find the distance d traveled in 2 hours by a race car going an average speed of 180 miles per hour. Use the formula: distance equals rate r multiplied by time t.
Solution d rt RACE CARS The fastest average speed in the Indianapolis 500 is 185.981 miles per hour set by Arie Luyendyk, shown above.
Write formula.
180(2)
Substitute 180 for r and 2 for t.
360
Simplify.
ANSWER 䊳
The distance traveled by the race car was 360 miles.
Evaluate rt to Find Distance 9. Using a variable expression, find the distance traveled by a car moving at an
average speed of 60 miles per hour for 3 hours.
4
Chapter 1
Connections to Algebra
Student Help VOCABULARY TIP
The word perimeter comes from perimeaning around and -meter meaning measure. Perimeter is a measure of the distance around a geometric figure.
EXAMPLE
4
Find the Perimeter
GEOMETRY LINK The perimeter P of a triangle is equal to the sum of the lengths of its sides:
c 17 a8 b 15
Pabc Find the perimeter of the triangle in feet. Solution
Pabc
Write the formula. Substitute the side lengths of 8, 15 and 17.
8 15 17
Add the side lengths.
40
ANSWER 䊳
The triangle has a perimeter of 40 feet.
Find the Perimeter 10. Find the perimeter of a square with each side 12 inches long.
EXAMPLE
5
Estimate the Area
GEOGRAPHY LINK
The area A of a triangle is equal to half the base b times
1 2
the height h: A bh. Use this formula to estimate the area (in square miles) of Virginia. height h right angle
h 200 mi b 410 mi
base b
Solution Write the formula.
1 2
A bh 1 2
Substitute 410 for b and 200 for h.
(410)(200)
Simplify the formula.
41,000
ANSWER 䊳
The area of Virginia is about 41,000 square miles.
h7
Find the Area 11. Find the area of the triangle in square meters.
1.1
b 12 Variables in Algebra
5
1.1
Exercises
Guided Practice Vocabulary Check
Identify the variable or variables. 1. y 15
2. 20 s
b 3. 10
4. rt
5. Complete: You ____ ? an expression by substituting numbers for variables
and simplifying. The resulting number is called the ____ ? of the expression.
Skill Check
State the meaning of the variable expression and name the operation. 5 6. c
7. p 4
8. 5 n
9. (8)(x)
Evaluate the variable expression when k 3. 10. 11 k
11. k 2
12. 7k
k 13. 33
18 14. k
15. 18 p k
Find the perimeter of each triangle.
16. a.
b. 8 cm
6 cm
9 in.
10 cm
15 in.
12 in.
Practice and Applications DESCRIBING EXPRESSIONS Match the variable expression with its
meaning. 17. y 8
18. y 8
A. 8 times y
B. y divided by 8
y 19. 8 C. y plus 8
20. 8y D. y minus 8
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable.
Student Help HOMEWORK HELP
Example 1: Exs. 17–20 Example 2: Exs. 21–32 Example 3: Exs. 33–39 Example 4: Exs. 40–42 Example 5: Exs. 43–46
6
Chapter 1
21. 9 p when p 11
1 22. t when t 2 2
b 23. when b 14 7
d 24. when d 36 12
25. (4)(n) when n 5
26. 8a when a 6
27. 12 x when x 3
28. 9 y when y 8
29. 10r when r 7
30. 13c when c 3
18 31. when x 3 x
63 32. when k 9 k
Connections to Algebra
Student Help
33. DRIVING DISTANCE You are driving across the country at an average speed
of 65 miles per hour. Using an appropriate formula, find the distance you travel in 4 hours.
HOMEWORK HELP NE ER T
INT
Extra help with problem solving in Exs. 34–39 is available at www.mcdougallittell.com
FINDING DISTANCE Find the distance traveled using d rt.
34. A train travels at a rate of 75 miles per hour for 2 hours. 35. An athlete runs at a rate of 8 feet per second for 5 seconds. 36. A horse trots at 8 kilometers per hour for 30 minutes. 37. A racecar driver goes at a speed of 170 miles per hour for 2 hours. 38. A plane travels at a speed of 450 miles per hour for 3 hours. 39. A person walks at a rate of 4 feet per second for 1 minute.
Student Help SKILLS REVIEW
The perimeter of any geometric figure is the sum of all its side lengths. To review perimeter and area formulas, see p. 772.
Find the perimeter of the geometric figure. 40.
41.
13 ft
42.
22 m 10 m
8 ft
10 ft
10 m
12 in.
12 in.
12 in.
12 in.
22 m 12 in.
Find the area of each triangle. 43.
h4m
44.
h 6 mi
45. h 3 yd
46.
b 4 yd
b 10 mi
b5m
To find the area A of a rectangle, you multiply the length times the width: Alpw Use the formula to estimate the area of Wyoming.
275 mi
362 mi
47. CHALLENGE A tsunami is a huge fast-moving series of water waves that
can be caused by disturbances such as underwater earthquakes or volcanic explosions. If a tsunami is traveling at a speed of 500 miles per hour across the Pacific Ocean, how far has it gone in 15 minutes? HINT: Convert 15 minutes to hours. 1.1
Variables in Algebra
7
EXAMPLE
Unit Analysis
Writing the units of measure helps you determine the units for the answer. This is called unit analysis. When the same units occur in the numerator and the denominator, you can cancel them. Use unit analysis to evaluate the expression. The letter h is an abbreviation for hours, while mi stands for miles.
冢
冣
冢
25 mi a. (3 h) 1h
冣
45 mi b. (90 mi) 1h
Solution
冢
Cancel hours. 冣 1h 45 mi b. (90 mi) 冢冣 (90 mi)冢冣 2 h 45 mi 1h
25 mi a. (3 h) 75 mi 1h
Student Help STUDY TIP
When you divide by a fraction, you multiply by the reciprocal. See Skills Review, p. 765.
Cancel miles.
冢
冣 20 mi 49. Evaluate the expression (80 mi) 冢冣. 1h 60 mi 48. Evaluate the expression (4 h) . 1h
Standardized Test Practice
50. MULTIPLE CHOICE How many miles does Joyce travel if she drives for
6 hours at an average speed of 60 miles per hour? A
66 miles
B
180 miles
C
360 miles
D
420 miles
51. MULTIPLE CHOICE The lengths of the sides of a triangle are 4 centimeters,
8 centimeters, and 7 centimeters. What is the perimeter of the triangle? F
Mixed Review
7 cm
G
H
16 cm
19 cm
J
28 cm
OPERATIONS WITH DECIMALS Find the value of the expression.
(Skills Review p. 759) 52. 32.8 4
53. 3.98 5.50
54. 0.1(50)
SIMPLIFYING EXPRESSIONS Simplify the expression without using a calculator. (Skills Review p. 765)
冢 冣
30 55. (5) 10
Maintaining Skills
8
Chapter 1
冢 冣
2 56. (60) 12
冢 冣冢 冣
3 5 57. 15 6
ADDING DECIMALS Add. (Skills Review p. 759)
58. 2.3 4.5
59. 16.8 7.1
60. 0.09 0.05
61. 1.0008 10.15
62. 123.8 0.03
63. 46 7.55
64. 0.32 0.094
65. 6.105 7.3
66. 2.008 1.10199
Connections to Algebra
1.2
Exponents and Powers How much water does the tank hold?
Goal Evaluate a power.
How much water do you need to fill a fish tank? You will use a power to find the answer in Example 5.
Key Words • • • •
power exponent base grouping symbols
An expression like 23 is called a power. The exponent 3 represents the number of times the base 2 is used as a factor. base
exponent
23 ⫽ 2 p 2 p 2 power 3 factors of 2
The expression 2 3 means “multiply 2 by itself 3 times.” The numbers you multiply are factors. In general, a n ⫽ a p a p a p ... p a. n times
EXAMPLE
1
Read and Write Powers
Express the power in words. Then write the meaning. EXPONENTIAL FORM
Student Help READING ALGEBRA
Note that x 1 is customarily written as x with the exponent omitted.
WORDS
MEANING
a. 42
four to the second power or four squared
4p4
b. 53
five to the third power or five cubed
5p5p5
c. x6
x to the sixth power
xpxpxpxpxpx
Write the Power Write the expression in exponential form. 1. 3 squared
2. x to the fourth power 1.2
3. s cubed Exponents and Powers
9
Student Help
EXAMPLE
More examples are available at www.mcdougallittell.com
INT
Evaluate the Power
Evaluate x 4 when x 2.
MORE EXAMPLES NE ER T
2
Solution x4 24
Substitute 2 for x. Write out the factors.
2p2p2p2
Multiply the factors.
16
ANSWER 䊳
The value of the power is 16.
Parentheses ( ) and brackets [ ] are grouping symbols. They tell you the order in which to do the operations. You must do the operations within the innermost set of grouping symbols first:
GROUPING SYMBOLS
First multiply. Then add.
First add. Then multiply.
(3 p 4) 7 12 7 19
3 p (4 7) 3 p 11 33
You will learn more about the order of operations in the next lesson.
EXAMPLE
3
Evaluate Exponential Expressions
Evaluate the variable expression when a 1 and b 2. a. (a2) (b2)
b. (a b)2
Solution a. (a2) (b2) (12) (22)
Substitute 1 for a and 2 for b.
(1 p 1) (2 p 2)
Write factors.
14
Multiply.
5
Add.
b. (a b)2 (1 2)2
Substitute 1 for a and 2 for b.
(3)2
Add within parentheses.
3p3
Write factors.
9
Multiply.
Evaluate Exponential Expressions Evaluate the variable expression when s 2 and t 4.
10
Chapter 1
4. (t s)3
5. (s 2) (t 2)
6. (t s)2
7. (t 2) (s 2)
8. (s 2) t
9. (t 2) s
Connections to Algebra
In Lesson 1.3 you will learn several rules for order of operations. One of those tells us that 2x3 is to be interpreted as 2(x3).
Student Help READING ALGEBRA
Notice that in part (a) of Example 4, the exponent applies to x, while in part (b), the exponent applies to 2x.
4
EXAMPLE
Exponents and Grouping Symbols
Evaluate the variable expression when x 4. a. 2x3
b. (2x)3
Solution a. 2x3 2(43)
Substitute 4 for x.
2(64)
Evaluate power.
128
Multiply.
b. (2x) (2 p 4) 3
3
Substitute 4 for x.
8
Multiply within parentheses.
512
Evaluate power.
3
Exponents can be used to find the area of a square and the volume of a cube. s s s s Area of square: A s 2
s Volume of cube: V s 3
Units of area, such as square feet, ft2, can be written using a second power. Units of volume, such as cubic feet, ft3, can be written using a third power.
EXAMPLE
5
Find the Volume of the Tank
The fish tank has the shape of a cube. Each inner edge s is 2 feet long. Find the volume in cubic feet.
FISH TANKS
2 ft
Solution V s3
Write formula for volume of a cube.
23
Substitute 2 for s.
8
Evaluate power.
ANSWER 䊳
2 ft 2 ft
The volume of the tank is 8 cubic feet.
Find Area and Volume 10. Use the formula for the area of a square to find the area of each side of the
fish tank in Example 5. Express your answer in square feet.
1.2
Exponents and Powers
11
1.2 Exercises Guided Practice Vocabulary Check
Complete the sentence. 1. In the expression 37, the 3 is the ____ ? . 2. In the expression 54, the 4 is the ____ ? . 3. The expression 912 is called a ____ ? . 4. Two kinds of grouping symbols are ____ ? and ____ ? .
Skill Check
Match the power with the words that describe it. 5. 37
A. four to the sixth power
6. 73
B. three to the seventh power
7. 46
C. seven to the third power
4
8. 6
D. six to the fourth power
Evaluate the variable expression when t 3. 9. t 2
10. 1 t 3
11. 4t 2
12. (4t)2
Practice and Applications WRITING POWERS Write the expression in exponential form.
13. two cubed
14. p squared
15. nine to the fifth
power 16. b to the eighth power 19.
17. 3 p 3 p 3 p 3
18. 4x p 4x p 4x
A square painting measures 5 feet by 5 feet. Write the power that gives the area of the painting. Then evaluate the power. 5 ft
5 ft EVALUATING POWERS Evaluate the power.
Student Help
20. 92
21. 24
22. 73
23. 26
24. 54
25. 18
26. 103
27. 06
HOMEWORK HELP
Example 1: Exs. 13–19 Example 2: Exs. 19–39 Example 3: Exs. 40–45 Example 4: Exs. 46–51 Example 5: Exs. 52–57
12
Chapter 1
EVALUATING POWERS WITH VARIABLES Evaluate the expression for the
given value of the variable. 28. w2 when w 12
29. b3 when b 9
30. c4 when c 3
31. h5 when h 2
32. n2 when n 11
33. x3 when x 5
Connections to Algebra
Student Help KEYSTROKE HELP
Your calculator may have a key or a key that you can use to evaluate powers.
EVALUATING POWERS Use a calculator to evaluate the power.
34. 86
35. 135
36. 59
37. 127
38. 66
39. 312
EVALUATING EXPRESSIONS Evaluate the variable expression when
c 4 and d 5. 40. (c d)2
41. (d 2) c
42. (c3) d
43. (d 2) (c2)
44. (d c)7
45. (d 2) d
EXPONENTIAL EXPRESSIONS Evaluate the expression for the given
value of the variable. 46. 2x 2 when x 7
47. 6t 4 when t 1
48. 7b2 when b 3
49. (5w)3 when w 5
50. (4x)3 when x 1
51. (5y)5 when y 2
52. INTERIOR DESIGN The floor of a room is 14 feet long by 14 feet wide.
How many square feet of carpet are needed to cover the floor? 53. VOLUME OF A SAFE A fireproof safe is designed in the shape of a cube.
The length of each edge of the cube is 2 meters. What is the volume of the fireproof safe?
Art
54. ARTISTS In 1997, the artist Jon Kuhn of North Carolina created a cubic
sculpture called Crystal Victory, shown at the left. Each edge of the solid glass cube is 9.5 inches in length. What is the volume of the cubic structure? CRITICAL THINKING Count the number of cubic units along the edges of
the cube. Write and evaluate the power that gives the volume of the cube in cubic units. 55.
ARTIST Jon Kuhn used mathematics when creating the cubic sculpture Crystal Victory. The solid glass cube is made of lead crystal and colored glass powders.
56.
57.
CHALLENGE You are making candles. You melt paraffin wax in the cubic container shown below. Each edge of the container is 6 inches in length. The container is half full.
58. What is the volume of the wax in this container? 6 in.
59. Each edge of a second cubic container is 4 inches in
length. Can this second candle mold hold the same amount of melted wax that is in the candle mold shown at the right? Explain your answer.
6 in. 6 in.
60. Design a third cubic candle mold different from the one given above that will
hold all the melted wax. Draw a diagram of the mold including the measurements. Explain why your mold will hold all the melted wax. 1.2
Exponents and Powers
13
Standardized Test Practice
61. MULTIPLE CHOICE Evaluate the expression 2x 2 when x 5. A
B
20
C
40
D
50
100
62. MULTIPLE CHOICE One kiloliter is equal to 103 liters. How many liters are
in one kiloliter? F
G
10
H
100
J
1000
10,000
63. MULTIPLE CHOICE Sondra bought
Student Help
this trunk to store clothes. What is the volume of the trunk?
TEST TIP
Jotting down the formula for the volume of a cube will help you answer Exercise 63.
A C
B D
9 ft 9 ft3
3 ft
9 ft2 None of these 3 ft 3 ft
Find the perimeter of the geometric figure when x 3.
Mixed Review (Lesson 1.1) x
64.
x
65. x
x
x
66. x
x
2x x
x
2x
x x
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 1.1)
Maintaining Skills
67. 9j when j 5
68. 6 t when t 21
b 69. when b 18 2
70. 25 n when n 3
71. c 4 when c 24
72. (7)(r) when r 11
24 73. when s 8 s
74. 3m when m 7
75. d 13 when d 22
SIMPLIFYING FRACTIONS Simplify. (Skills Review p. 763)
4 76. 8
10 77. 2
15 78. 10
6 79. 20
8 80. 14
18 81. 21
8 82. 6
27 83. 3
25 84. 15
21 85. 7
3 86. 24
28 87. 4
ESTIMATING Estimate the answer. Then evaluate the expression.
(Skills Review p. 774)
14
Chapter 1
88. 2.5 0.5
89. 0.3 0.03
90. 10.35 5.301
91. 3.71 1.054
92. 2.1 0.2
93. 5.175 1.15
Connections to Algebra
1.3 Goal
Order of Operations How many points ahead are you?
Use the established order of operations.
You are playing basketball. You make 8 field goals and 2 free throws. Your friend makes half as many field goals as you and no free throws. You will find how many points ahead you are in Example 5.
Key Words • order of operations • left-to-right rule
In arithmetic and algebra there is an order of operations to evaluate an expression involving more than one operation. ORDER OF OPERATIONS
ORDER OF OPERATIONS STEP
First do operations that occur within grouping symbols.
STEP
Then evaluate powers.
STEP
Then do multiplications and divisions from left to right.
STEP
Finally, do additions and subtractions from left to right.
EXAMPLE
1
Evaluate Without Grouping Symbols
Evaluate the expression 3x 2 1 when x 4. Use the order of operations. Solution 3x 2 1 3 p 42 1
Substitute 4 for x.
3 p 16 1
Evaluate power.
48 1
Multiply 3 times 16.
49
Add.
Evaluate Expressions Without Grouping Symbols Evaluate the variable expression when x 2. Use the order of operations. 1. 2x 2 5
2. 8 x 2
3. 6 3x 3 1.3
4. 20 4x 2 Order of Operations
15
Some expressions have operations that have the same priority, such as multiplication and division or addition and subtraction. The left-to-right rule states that when operations have the same priority, you perform them in order from left to right.
LEFT-TO-RIGHT RULE
Use the Left-to-Right Rule
2
EXAMPLE
Evaluate the expression using the left-to-right rule. a. 24 8 6 (24 8) 6
Work from left to right.
16 6
Subtract 8 from 24.
10
Subtract 6 from 16.
b. 15 p 2 6 (15 p 2) 6
Student Help STUDY TIP
You multiply first in part (c) of Example 2, because multiplication has a higher priority than addition and subtraction.
Work from left to right.
30 6
Multiply 15 times 2.
5
Divide 30 by 6.
c. 16 4 p 2 3 16 (4 p 2) 3
Do multiplication first.
16 8 3
Multiply 4 times 2.
(16 8) 3
Work from left to right.
24 3
Add 16 and 8.
21
Subtract 3 from 24.
12 41
A fraction bar can act as a grouping symbol: (1 2) (4 1) .
Expressions with Fraction Bars
3
EXAMPLE
Evaluate the expression. Then simplify the answer. 7p4 7p4 8 72 1 8 49 1 28 8 49 1
Simplify the numerator.
28 57 1
Work from left to right.
28 56
Subtract.
1 2
Simplify.
Student Help
Evaluate power.
SKILLS REVIEW
For help with writing fractions in simplest form, see p. 763.
Use the Order of Operations and Left-to-Right Rule Evaluate the variable expression when x 1. 5. 4x 2 5 3
16
Chapter 1
Connections to Algebra
6. 5 x 3 1
2x 7. x2 1 5
You need to know if your calculator uses the order of operations or not. If it does not, you must input the operations in the proper order yourself. USING A CALCULATOR
EXAMPLE
4
Use a Calculator
Enter the following in your calculator. Does the calculator display 6 or 1? 10
6
2
1
Solution a. If your calculator uses the order of operations, it will display 6.
10 6 2 1 10 (6 2) 1 (10 3) 1 6 b. If your calculator does not use the order of operations and performs the
operations as they are entered, it will display 1. [(10 6) 2] 1 (4 2) 1 21 1
Sports
EXAMPLE
5
Evaluate a Real-Life Expression
You are playing basketball. You make 8 field goals and 2 free throws. Your friend makes half as many field goals as you and no free throws. How many points ahead of your friend are you?
BASKETBALL SCORES
Solution 8p2 2
16 2
8 p 2 2 p 1 16 2
BASKETBALL SCORES
A field goal is worth 2 points. A free throw is worth 1 point.
ANSWER 䊳
Multiply from left to right.
16 2 8
Divide.
18 8
Add.
10
Subtract.
You are 10 points ahead of your friend.
Evaluate a Real-Life Expression 8. Your friend makes 4 field goals. You make three times as many field goals as
your friend plus one field goal. How many points do you have? Explain the order of operations you followed. 9. Your friend makes 6 field goals and 2 free throws. You make twice as many
field goals as your friend and half the number of free throws. How many points do you have? Explain the order of operations you followed. 1.3
Order of Operations
17
1.3 Exercises Guided Practice Vocabulary Check
1. Place the operations in the order in which you should do them. a. Multiply and divide from left to right. b. Do operations within grouping symbols. c. Add and subtract from left to right. d. Evaluate powers. 2. What rule must be applied when evaluating an expression in which the
operations have the same priority?
Skill Check
Evaluate the expression. 3. 5 p 6 p 2
4. 16 4 2
5. 4 9 1
6. 2 p 82
7. 15 6 3
8. 9 3 p 2
9. 2 p 32 5
10. 23 p 32
Evaluate the variable expression when x 3. 11. x2 5
12. x3 5x
13. x 3x4
27 14. 2 16 x
15 15. 23 10 x
24 16. p 5 x
Practice and Applications NUMERICAL EXPRESSIONS Evaluate the expression.
17. 13 3 p 7
18. 7 8 2
19. 24 5 p 3
20. 62 4
21. 43 9 p 2
5 22. 3 p 2 9
VARIABLE EXPRESSIONS Evaluate the expression for the given value of
the variable.
Student Help HOMEWORK HELP
Example 1: Exs. 17–26 Example 2: Exs. 27–35 Example 3: Exs. 36–41 Example 4: Exs. 43–46 Example 5: Exs. 47–53
18
Chapter 1
23. 6 p 2p2 when p 5
24. 2g p 5 when g 4
25. 14(n 1) when n 2
x 26. 16 when x 14 7
NUMERICAL EXPRESSIONS Evaluate the expression.
27. 23 5 2
28. 4 p 2 15 3
29. 6 3 2 p 7
30. 5 8 p 2 4
31. 16 8 p 22
32. 2 p 32 7
33. 10 3 (2 5)
34. 7 18 (6 3)
35. [(7 p 4) 3] 15
Connections to Algebra
Student Help HOMEWORK HELP NE ER T
INT
Extra help with problem solving in Exs. 36–41 is available at www.mcdougallittell.com
EXPRESSIONS WITH FRACTION BARS Evaluate the expression. Then
simplify the answer. 6p4 36. 4 32 1
13 4 37. 18 42 1
52 p 2 38. 2 1 6 12
21 9 39. 52 40 5
33 8 7 40. 2p7
4 p 25 41. 16 42 1
42. LOGICAL REASONING Which is correct? 92 3 A. 92 3 5 5
92 3 B. [92 3] 5 5
CRITICAL THINKING In Exercises 43–46, two calculators were used
to evaluate the expression. Determine which calculator performed the correct order of operations. 43. 15
6
3
4
Calculator A: 12 44. 15
9
Calculator B: 7
3
7
Calculator A: 19 45. 15
10
Calculator B: 9 5
Calculator A: 21 46. 4
3
4 Calculator B: 9
6
Calculator A: 9
2 Calculator B: 15
FOOTBALL UNIFORMS
In Exercises 47 and 48, use the table showing the costs of parts of a football player’s uniform. 47. A sporting goods company
offers a $2000 discount for orders of 30 or more complete uniforms. Your school orders 35 complete uniforms. Write an expression for the total cost. 48. Evaluate the expression you
wrote in Exercise 47. Part of uniform
Jersey and pants
Shoulder pads
Lower body pads
Knee pads
Cleats
Helmet
Cost
$230
$300
$40
$15
$100
$200
1.3
Order of Operations
19
In Exercises 49 and 50, refer to the squares shown at the right. 49. Write an expression that represents the area of the
1 x 1 2 x 2
shaded region. HINT: Subtract the area of the inner square from the area of the outer square. 50. If x 8, what is the area of the shaded region?
x
x
ADMISSION PRICES In Exercises 51 and 52, use the table below. It
State Fairs
shows the admission prices for the California State Fair. California State Fair Admission Prices Age
ADMISSION PRICES Every
year nearly 1,000,000 people attend the California State Fair.
Admission price
General Admission (13–61 years of age)
$7.00
Seniors (62 years and above)
$5.00
Children (5–12 years)
$4.00
Children (4 years and under)
Free
䊳 Source: Sacramento Bee
51. Write an expression that represents the admission price for a group
consisting of 2 adults, 1 senior, and 3 children. The children’s ages are 12 years, 10 years, and 18 months. 52. Evaluate the expression you wrote in Exercise 51. Then find the total cost
of admission for the group. 53. CHALLENGE At a concert you buy a hat for $10.00, a hot dog for $2.75, and
nachos for $3.50. There is a 6% sales tax on the hat. Your calculator follows the established order of operations. Write a keystroke sequence for the amount you owe. Then find the amount you owe. HINT: 6% 0.06
Standardized Test Practice
54. MULTIPLE CHOICE Evaluate the expression 42 10 2. A
3
B
11
C
13
D
21
55. MULTIPLE CHOICE Evaluate the expression 32 x2 9 when x 2. F
19
G
21
H
37
J
39
56. MULTIPLE CHOICE Which expression has a value of 12? A B C D
3352
F
1
18 6 3 3 7 14 7 4 22 p 3 6 p 2
32 6 5 57. MULTIPLE CHOICE Evaluate the expression . 2p5
20
Chapter 1
Connections to Algebra
G
5
H
7
J
10
Mixed Review
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 1.1) 58. (8)(a) when a 4
24 59. when x 3 x
60. c 15 when c 12
x 61. p x when x 18 2
62. 9t when t 7
63. 25 y when y 14
WRITING POWERS Write the expression in exponential form. (Lesson 1.2)
64. twelve squared
65. z to the sixth power
66. 2b p 2b p 2b
EXPONENTIAL EXPRESSIONS Evaluate the expression for the given
value of the variable. (Lesson 1.2) 67. 9t 2 when t 3
Maintaining Skills
68. (7h)3 when h 1
69. (6w)2 when w 5
FACTORS Determine whether the number is prime or composite. If it is composite, list all of its factors. (Skills Review p. 761)
70. 15
71. 9
72. 13
73. 38
74. 46
75. 50
76. 64
77. 29
Quiz 1 Evaluate the variable expression when x 3. (Lessons 1.1, 1.2) 1. 6x
2. 42 x
3. x 29
4. 12 x
5. 5x 10
6. 10 2x
7. x2 3
8. 2x3
9. (2x)3
Find the distance traveled using d rt. (Lesson 1.1) 10. A car travels at an average speed of 50 miles per hour for 4 hours. 11. A plane flies at 500 miles per hour for 4 hours. 12. A marathon runner keeps a steady pace of 10 miles per hour for 2 hours. Write the expression in exponential form. (Lesson 1.2) 13. six cubed
14. 4 p 4 p 4 p 4 p 4
15. 5y p 5y p 5y
16. 3 p 3 p 3
17. 2x p 2x p 2x p 2x
18. eight squared
19. PACKING BOXES A cubic packing box has dimensions of 4 feet on each edge. What is the volume of the box? (Lesson 1.2) Evaluate the expression. Then simplify the answer. (Lesson 1.3) 7 p 22 20. 7 (23 1)
62 11 22. 2(17 2 p 4)
(32 3) 21. 2p9 1.3
Order of Operations
21
1.4
Equations and Inequalities How much do the ingredients cost?
Goal Check solutions of equations and inequalities.
You can use an equation to solve a real-life problem. In Example 3 you will use an equation to estimate the cost of ingredients for nachos.
Key Words • equation • solution • inequality
Student Help
An equation is a statement formed by placing an equal sign () between two expressions. An equation has a left side and a right side.
VOCABULARY TIP
Equation comes from a Latin word that means “to be equal”.
Left side
Right side
4x 1 9 Equation
When the variable in an equation is replaced by a number, the resulting statement is either true or false. If the statement is true, the number is a solution of the equation.
EXAMPLE
1
Check Possible Solutions
Check to see if 2 and 3 are solutions of the equation 4x 1 9. Solution Substitute the x values 2 and 3 into the equation. If both sides of the equation are equal in value, then the number is a solution. X VALUE
SUBSTITUTE
SIMPLIFY
CONCLUSION
2
4(2) 1 ⱨ 9
99
True, 2 is a solution.
3
4(3) 1 ⱨ 9
13 9
False, 3 is not a solution.
is not equal to ANSWER 䊳
24
Chapter 1
Connections to Algebra
The number 2 is a solution of the equation 4x 1 9, because the statement is true. The number 3 is not a solution, because the statement is false.
Finding all the solutions of an equation is called solving the equation. Some equations are simple enough to be solved with mental math. Later in the book you will learn how to systematically solve more complex equations.
SOLVING EQUATIONS
Student Help
EXAMPLE
MORE EXAMPLES
Solve Equations with Mental Math
To solve equations with mental math, think of the equation as a question.
NE ER T
More examples are available at www.mcdougallittell.com
INT
2
EQUATION
QUESTION
SOLUTION
2x 10
2 times what number gives 10?
2 p 5 10, so x 5
4x3
4 is equal to what number minus 3?
4 7 3, so x 7
2x6
2 plus what number gives 6?
2 4 6, so x 4
x 1 3
What number divided by 3 gives 1?
3 1, so x 3 3
Then check each solution by substituting the number in the original equation. If the statement is true, the number is a solution.
Solve Equations and Check Solutions Use mental math to solve the equation. Then check your solution. 1. 2 6 x
EXAMPLE
Nachos
ips 20 tortilla ch s n ea b s p 1 1/2 cu ed ic 1 cup d tomatoes
ted 1/2 cup gra e chees d 2 1/ cup slice olives
Add ns on chips. ves. Spread bea n eese a d oli ch en th , es tomato inutes. °F for 5 m Bake at 400
2. x 3 11
3
x 3. 5 4
4. 14 2x
Use Mental Math to Solve a Real-Life Equation
You are buying ingredients for nachos. At the market you find that tortilla chips cost $2.99, beans cost $.99, cheese costs $3.99, two tomatoes cost $1.00, and olives cost $1.49. There is no tax. You have $10. About how much more money do you need? Solution Ask: The total cost equals 10 plus what number of dollars? Let x
represent the extra money you need. Use rounding to estimate the total cost. 3 1 4 1 1.5 10 x 10.5 10 x ANSWER 䊳
The total cost is about 10.5 or $10.50, so you need about $.50 more to purchase all the ingredients.
Use Mental Math to Solve a Real-Life Equation 5. Solve the equation in Example 3 if a large bag of chips costs $3.99. About
how much more money would you need to buy the nacho ingredients? 1.4
Equations and Inequalities
25
Student Help
An inequality is a statement formed by placing an inequality symbol, such as 35
20. 8 ≥ 64 b
21. 3b 24 > 0
22. 16 ≤ b2
23. 60 > 7b 3
24. 18 b < 10
25. 37 ≥ 4b
Practice and Applications CHECKING SOLUTIONS OF EQUATIONS Check to see if the given value of the variable is or is not a solution of the equation.
26. 3b 1 13; b 4
27. 5r 10 11; r 5
28. 4c 2 10; c 2
29. 6d 5 31; d 6
30. 5 x2 17; x 3
31. 2y3 3 5; y 1
32. 9 2t 15; t 12
33. n2 5 20; n 5
SOLVING WITH MENTAL MATH Use mental math to solve the equation.
Student Help
34. x 3 8
35. n 6 11
36. p 13 20
37. r 1 7
38. 3y 12
39. 4p 36
40. z 4 5
x 41. 3 7
42. 2b 28
43. 11t 22
44. 29 d 10
45. 3 y 8
46. r 30 70
42 47. 7 x
48. 7m 49
HOMEWORK HELP
Example 1: Exs. 26–33 Example 2: Exs. 34–48 Example 3: Exs. 49, 50 Example 4: Exs. 51–56 Example 5: Exs. 57, 58
1.4
Equations and Inequalities
27
49. TIME MANAGEMENT You have a hair appointment in 60 minutes. It takes
20 minutes to get to the gas station and fill your tank. It takes 15 minutes to go from the gas station to the hair stylist. You wait x minutes before leaving your house and arrive on time for your appointment. Use the diagram to help decide which equation best models the situation. GAS
Hair Salon
2 3 0 0 6 1
x
20
15
A. 20 15 x 60
B. 60 20 15 x
C. 60 20 15 x 60
D. x 20 15 60
50. MENTAL MATH Solve the equation you chose in Exercise 49. CHECKING SOLUTIONS OF INEQUALITIES Check to see if the given value
of the variable is or is not a solution of the inequality. 51. n 2 < 6; n 3
52. a 7 ≥ 15; a 22
53. 6 y ≤ 8; y 3
54. s 5 > 8; s 4
55. 7g ≥ 47; g 7
56. 72 t > 6; t 12
57. SELLING CARDS Your community center is selling cards. Your goal is to
sell $100 worth of cards. Each box sells for $3. Using mental math, solve the inequality 3b ≥ 100 to determine at least how many boxes you must sell to meet your goal. 58. BUYING A GUITAR You are budgeting money to buy a guitar that costs
$150 including tax. If you save $20 per month, will you have enough money in 6 months? Use the inequality 20n ≥ 150 to model the situation, where n represents the number of months.
Science 59.
Mach number is the maximum speed at which a plane can fly divided by the speed of sound. Copy and complete the table. Use the v 660
equation m , where m is the Mach number and v is the speed (in miles per hour) of the aircraft, to find the Mach number for each type of aircraft. Airplane type Speed v Mach number m
Test aircraft
Supersonic
Jet
4620
1320
660
?
?
?
CHUCK YEAGER in 1947
became the first person to fly faster than the speed of sound (Mach 1) or about 660 miles per hour. 28
Chapter 1
Connections to Algebra
Test aircraft
Supersonic aircraft
Jet aircraft
60.
Use mental math to fill in the missing number so that all the equations have the number 6 as a solution.
䡵
䡵
a. ? x 18
Standardized Test Practice
? c. 6 x
b. ? x 30
61. MULTIPLE CHOICE Which is a solution of the equation 5(8 x) 25? A
2
B
3
C
D
4
5
62. MULTIPLE CHOICE For which inequality is x 238 a solution? F H
G J
250 ≥ x 12 250 > x 12
250 < x 12 250 ≤ x 1
63. MULTIPLE CHOICE The width of a soccer field cannot be greater than
100 yards. The area cannot be greater than 13,000 square yards. Which of the following would you use to find the possible length x of a soccer field? A C
Mixed Review
B D
100x ≥ 13,000 100 x ≤ 13,000
100x ≤ 13,000 100x 13,000
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 1.1) 64. b 12 when b 43
65. 12 x when x 4
66. 12n when n 4
y 67. when y 30 15
WRITING POWERS Write the expression in exponential form.
(Lesson 1.2) 68. 3 p 3 p 3 p 3 p 3
69. seven squared
70. y p y p y p y
71. 9 p 9 p 9 p 9 p 9 p 9
72. twelve cubed
73. 8d p 8d p 8d
NUMERICAL EXPRESSIONS Evaluate the expression. Then simplify the
answer. (Lesson 1.3)
Maintaining Skills
74. 9 12 4
75. 7 56 8 2
76. 63 3 p 3
77. 4 p 2 5
78. 3 13 6
79. 49 7 2
80. (28 4) 32
42 2 81. 2
82. 2[(2 3)2 10]
ROUNDING Round the number to the underlined place value.
(Skills Review p. 774) 83. 5.64
84. 0.2625
85. 0.45695
86. 15.295
87. 758.949
88. 32.6582
89. 0.325
90. 26.96
91. 4.0965 1.4
Equations and Inequalities
29
1.5
Translating Words into Mathematical Symbols
Goal Translate words into mathematical symbols.
How long were you on the phone? In Example 6 you will translate words into an algebraic equation to find the length of a long distance phone call.
Key Words • translate • phrase • sentence
To solve real-life problems, you often need to translate words into mathematical symbols. To do this, look for words, such as sum or difference, that indicate mathematical operations.
y Summar Account ges New Char 36.00 ice..........$ rv Basic Se 6.00 nce ........$ Long Dista 42.00 rges .......$ Total Cha 42.00 t Due $ un
o Total Am
EXAMPLE
1
Translate Addition Phrases
Write the phrase as a variable expression. Let x represent the number. PHRASE
TRANSLATION
The sum of 6 and a number
6x
8 more than a number
x8
A number plus 5
x5
A number increased by 7
x7
EXAMPLE
2
Translate Subtraction Phrases
Write the phrase as a variable expression. Let y represent the number. PHRASE
Student Help READING ALGEBRA
Order is important for subtraction. “4 less than a number” means y 4, not 4 y.
TRANSLATION
The difference between 5 and a number
5y
7 minus a number
7y
A number decreased by 9
y9
4 less than a number
y4
Translate Addition and Subtraction Phrases Write the phrase as a variable expression. Let x represent the number. 1. 11 more than a number 30
Chapter 1
Connections to Algebra
2. A number decreased by 10
Student Help VOCABULARY TIP
Quotient comes from a word meaning “how many times.” When you divide you are finding how many times one quantity goes into another.
Notice that order does not matter for addition and multiplication. “The sum of 6 and a number” can be written as either 6 x or x 6. Order is important for n 4
4 n
subtraction and division. “The quotient of a number and 4” means , not .
EXAMPLE
3
Translate Multiplication and Division Phrases
Write the phrase as a variable expression. Let n represent the number. PHRASE
TRANSLATION
The product of 9 and a number
9n
10 times a number
10n
A number multiplied by 3
3n
One fourth of a number
1 n 4
The quotient of a number and 6
n 6
7 divided by a number
7 n
Translate Multiplication and Division Phrases Write each phrase as a variable expression. Let x represent the number. 3. The quotient of 8 and a number
4. The product of 2 and a number
In English there is a difference between a phrase and a sentence. Phrases are translated into variable expressions. Sentences are translated into equations or inequalities.
TRANSLATING SENTENCES
EXAMPLE Student Help
PHRASE
EXPRESSION
SENTENCE
EQUATION OR INEQUALITY
4
Translate Sentences
Write the sentence as an equation or an inequality.
READING ALGEBRA
SENTENCE
TRANSLATION
The word is by itself means “.”
The sum of a number x and 12 is 16.
x 12 16
The words is less than mean “ 50
5. 10 a < 30
6. 40 3a ≥ 50
a 7. ≤ 5 5
80 8. ≥ 5 a
a 9. 2 ≤ 5 5
10.
The rectangle shown at the right has an area of 32 square units. Write an equation to find the width x. Use mental math to solve the equation. HINT: The area of a rectangle equals length times width. (Lesson 1.4)
x 8
Write the sentence as an equation or an inequality. (Lesson 1.5) 11. A number x divided by 9 is less than 17. 12. The product of 10 and a number x is 50. 13. A number y plus 10 is greater than or equal to 57. 14. A number y minus 6 is 15. 1.5
Translating Words into Mathematical Symbols
35
1.6
A Problem Solving Plan Using Models
Goal Model and solve real-life problems.
How much food did you order? In Chinese restaurants the bill is sometimes totaled by counting the number of plates ordered. In Example 1 you will use an algebraic model to find out how many plates of food you ordered.
Key Words • modeling • verbal model • algebraic model
Writing algebraic expressions, equations, or inequalities that represent real-life situations is called modeling. First you write a verbal model using words. Then you translate the verbal model into an algebraic model. Write a verbal model.
EXAMPLE
1
Assign labels.
Write an algebraic model.
Write an Algebraic Model
You and some friends are at a Chinese restaurant. You order several $2 plates of wontons, egg rolls, and dumplings. Your bill is $25.20, which includes tax of $1.20. Use modeling to find how many plates you ordered.
Student Help
Solution
STUDY TIP
Be sure you understand the problem before you write a model. For example, notice that the tax is added after the cost of the plates is figured.
VERBAL MODEL
LABELS
ALGEBRAIC MODEL
Cost per p Number Amount of bill Tax of plates plate Cost per plate 2
(dollars)
Number of plates p
(plates)
Amount of bill 25.20
(dollars)
Tax 1.20
(dollars)
2 p 25.20 1.20 2 p 24 p 12
ANSWER 䊳
36
Chapter 1
Connections to Algebra
Write algebraic model. Subtract. Solve using mental math.
Your group ordered 12 plates of food costing $24.
A PROBLEM SOLVING PLAN USING MODELS VERBAL MODEL
Ask yourself what you need to know to solve the problem. Then write a verbal model that will give you what you need to know.
LABELS
Assign labels to each part of your verbal model. Use the labels to write an algebraic model based on your verbal model.
ALGEBRAIC MODEL SOLVE
Solve the algebraic model and answer the original question.
CHECK
Check that your answer is reasonable.
EXAMPLE
2
Write an Algebraic Model
A football field is about 53 yards wide and 120 yards long. A soccer field has the same area, but is 60 yards wide. How long is the soccer field?
STUDY TIP
Sometimes a diagram can help you see what you know and what you need to find to solve the problem.
Solution Write a verbal model showing that the area (width length) of the soccer field equals the area (width length) of the football field. 10 20 30 40 50 40 30 20 10
Student Help
53 yd
60 yd 10 20 30 40 50 40 30 20 10
120 yd
x yd VERBAL MODEL
Width of p Length of Width of p Length of soccer field soccer field football field football field
LABELS
Width of soccer field 60
(yards)
Length of soccer field x
(yards)
Width of football field 53
(yards)
Length of football field 120
(yards)
ALGEBRAIC MODEL
60 x 53 120 60 x 6360 x 106
ANSWER 䊳
Write algebraic model. Simplify. Solve with mental math. (60 p 106 6360)
The soccer field is 106 yards long.
Write an Algebraic Model 1. You want two rectangular gardens to have equal areas. The first garden is
5 meters by 16 meters. The second garden is 8 meters wide. How long should the second garden be? Apply the problem solving plan to find the answer. 1.6
A Problem Solving Plan Using Models
37
EXAMPLE
Careers
3
Write an Algebraic Model
A jet pilot is flying from Los Angeles to Chicago at a speed of 500 miles per hour. When the plane is 600 miles from Chicago, an air traffic controller tells the pilot that it will be 2 hours before the plane can get clearance to land.
JET PILOTS
a. At what speed would the jet have to fly to arrive in Chicago in 2 hours? b. The pilot knows that at her present altitude, the speed of the jet must be
greater than 322 miles per hour or the plane could stall. Is it reasonable for the pilot to fly directly to Chicago at the reduced speed from part (a) or should the pilot take some other action?
INT
JET PILOTS select a route, an altitude, and a speed that will provide the fastest and safest flight. NE ER T More about jet pilots available at www.mcdougallittell.com
Solution a. You can use the formula (rate)(time) (distance) to write a verbal model. VERBAL MODEL LABELS
ALGEBRAIC MODEL
ANSWER 䊳
Speed of jet p Time Distance to travel Speed of jet x
(miles per hour)
Time 2
(hours)
Distance to travel 600
(miles)
2 x 600 x 300
Write algebraic model. Solve using mental math.
To arrive in 2 hours, the pilot would have to slow the jet down to a speed of 300 miles per hour.
b. It is not reasonable for the pilot to fly at 300 miles per hour, because the
plane could stall. The pilot should take some other action, such as circling in a holding pattern, to use some of the time.
You can ignore information that is not needed to solve a problem. To solve Example 3, you do not need to know that the plane is traveling at 500 miles per hour.
Write an Algebraic Model Use the following information to write and solve an algebraic model.
You are running in a marathon. During the first 20 miles, your average speed is 8 miles per hour. During the last 6.2 miles, you increase your average speed by 2 miles per hour. 2. How long will it take you to run the last 6.2 miles of the marathon? Use the
problem solving plan with models to answer the question. 3. A friend of yours completed the marathon in 3.2 hours. Did you finish ahead
of your friend or behind your friend? Explain. 38
Chapter 1
Connections to Algebra
1.6 Exercises Guided Practice Vocabulary Check
In Exercises 1 and 2, complete the sentence. 1. Writing expressions, equations, or inequalities to represent real-life situations
is called ____ ? . 2. A ____ ? model with labels is used to form an algebraic model. 3. Write the steps of the problem solving plan.
Skill Check
4. ADMISSION PRICES Your family and friends are going to an amusement
park. Adults pay $25 per ticket and children pay $15 per ticket. Your group has 13 children and your total bill for tickets is $370. How many adults are in your group? Choose the verbal model that represents this situation. A.
Cost per Number Cost per Number of Total adult p of adults child p children cost
B.
Cost per Number Cost per Number of Total adult p of adults child p children cost
C.
Cost per Number Cost per Number of Total adult of adults p child children cost
Practice and Applications WALK OR TAKE THE SUBWAY? In Exercises 5–10, use the following
information.
You are one mile from your home. You can walk at a speed of 4 miles per hour. The subway comes by every 15 minutes, and you heard one come by 3 minutes ago. The subway ride takes 8 minutes. 5. How many minutes will it take to get home by subway if you take the next
train? 6. Write a verbal model that relates the time it would take to walk home, your
walking speed, and the distance to your home. ____ ? ____ ? ____ ?
Student Help
7. Assign labels to your verbal model. Use t to represent the unknown value.
HOMEWORK HELP
8. Use the labels to translate your verbal model into an equation.
Example 1: Exs. 5–20 Example 2: Exs. 5–20 Example 3: Exs. 5–20
9. Use mental math to solve the equation. 10. Which will get you home faster, walking or taking the subway? Explain. 1.6
A Problem Solving Plan Using Models
39
In Exercises 11–15, use the following information.
Kudzu is a type of Japanese vine that grows at a rate of 1 foot per day during the summer. On August 1, the length of one vine was 50 feet. What was the length on July 1? HINT: July has 31 days.
Science
11. Use the verbal phrases to complete the verbal model.
Total length Original length Number of days
____ ? ____ ? ____ ? ____ ?
Growth rate 12. Assign labels to the verbal model. Use x to represent the unknown value. PLANT GROWTH Kudzu
was introduced to the United States in 1876. Today kudzu covers over 7 million acres of the southeastern United
13. Choose the algebraic model that best represents the verbal model. A. (x 1) p 31 50
B. x (1 p 31) 50
C. x 50 1
D. x (1 31) 50
14. Use mental math to solve the algebraic model you chose in Exercise 13. 15. Check that your answer is reasonable.
Student Help
BUYING A STEREO In Exercises 16–20, use the following information.
HOMEWORK HELP NE ER T
INT
Extra help with problem solving in Exs. 16–20 is available at www.mcdougallittell.com
An appliance store sells two stereo models. The model without a CD player is $350. The model with a CD player is $480. Your summer job allows you to save $50 a week for 8 weeks. At the end of the summer, you have enough to buy the stereo without the CD player. How much would you have needed to save each week to buy the other model? 16. Write a verbal model that relates the number of weeks worked, the amount
you would have needed to save each week, and the price of the stereo with the CD player. ____ ? ____ ? ____ ? 17. Assign labels to your verbal model. Use m to represent the unknown value. 18. Use the labels to translate your verbal model into an equation. 19. Use mental math to solve the equation. 20. Check that your answer is reasonable. 21. CHALLENGE You are running for class president. By two o’clock on
election day you have 95 votes and your opponent has 120 votes. Forty-five more students will be voting. Let x represent the number of students (of the 45) who vote for you. a. Write an inequality that shows the values of x that will allow you to win
the election. b. What is the smallest value of x that is a solution of the inequality? 40
Chapter 1
Connections to Algebra
Standardized Test Practice
Student Help TEST TIP
In Exercise 22, you can use unit analysis (p. 8) to see which answer choice is correct.
22. MULTIPLE CHOICE A jet
Baltimore
is flying from Baltimore to Orlando at a speed r of 500 miles per hour. The distance d between the two cities is about 793 miles. Which equation can be used to find the time t it takes to make the trip? A
793 500t
B
t
C D
793t 500
500 793
Orlando
t 793(500)
23. MULTIPLE CHOICE Jim lives in a state in which speeders are fined $25 for a
speeding ticket plus $10 for each mile per hour over the speed limit. Jim was given a ticket for $175 for speeding in a 45 mile per hour zone. Which equation can be used to find how fast Jim was driving? F H
Mixed Review
45 (x 45) p 10 175 25 (x 45) p 10 175 25
G J
x 45 p 10 175 25 (x 45) p 10 175 25
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 1.2) 24. x2 2 when x 7 25. (2x)3 when x 5
26. (10 x)2 when x 6
NUMERICAL EXPRESSIONS Evaluate the expression. (Lesson 1.3)
27. 22 42 2
28. 4 8 p 4 1
29. 2 p 4 (7 3)
CHECKING SOLUTIONS Check to see if the given value of the variable is
or is not a solution of the equation or the inequality. (Lesson 1.4) 30. 2x 3 < 15; x 9 31. 3x 4 ≤ 16; x 4
32. 16 x2 4 17; x 2
33. FUNDRAISING Your fundraising group earns 25¢ for each lemonade and
50¢ for each taco sold. One hundred tacos are sold. Your total profit is $100. How many lemonades are sold? Write an equation that models this situation. Solve the equation using mental math. (Lesson 1.5)
Maintaining Skills
FRACTIONS AND MIXED NUMBERS Write the improper fraction as a
mixed number. (Skills Review p. 763) 3 34. 2
7 35. 4
11 36. 3
13 37. 6
16 38. 5
21 39. 9
18 40. 4
15 41. 7
30 42. 8
54 43. 12
84 44. 36
20 45. 3
1.6
A Problem Solving Plan Using Models
41
1.8
An Introduction to Functions What is the altitude of the balloon?
Goal Use four different ways to represent functions.
You are in a hot-air balloon. You rise at a steady rate of 20 feet per minute. In Example 2 you will use the relationship between time and height to find the altitude of the balloon after a given number of minutes.
Key Words • • • • • •
function input output input-output table domain range
A function is a rule that establishes a relationship between two quantities, called the input and the output. For each input, there is exactly one output—even though two different inputs may give the same output. One way to describe a function is to make an input-output table.
EXAMPLE
1
GEOMETRY LINK
Make an Input-Output Table The diagram shows the first six triangular numbers.
a. Using the first six figures, make an input-output table in which the input is
the figure number n and the output is the triangular number T. b. Does the table represent a function? Justify your answer.
Solution a. Use the diagram to make an input-output table, as shown below. Input n
1
2
3
4
5
6
Output T
1
3
6
10
15
21
b. This is a function, because for each input there is exactly one output.
48
Chapter 1
Connections to Algebra
The collection of all input values is the domain of the function and the collection of all output values is the range of the function. The domain of the function in Example 1 is 1, 2, 3, 4, 5, 6; the range of the function is 1, 3, 6, 10, 15, 21.
DOMAIN AND RANGE
When you are given the rule for a function, you can prepare to graph the function by making a table showing numbers in the domain and their corresponding output values.
EXAMPLE
2
Use a Table to Graph a Function
BALLOONING You are at an altitude of 250 feet in a hot-air balloon. You turn on the burner and rise at a rate of 20 feet per minute for 5 minutes. Your altitude h in feet after you have risen for t minutes is given by
h ⫽ 250 ⫹ 20t, where t ≥ 0 and t ≤ 5. a. Use the function to find the output h in feet for several inputs. Then
organize the data into an input-output table. b. Use the data in the table to draw a graph that represents the function.
Solution a. Find the outputs for t ⫽ 0, 1, 2, 3, 4, and 5. Then make a table. INPUT (MINUTES)
FUNCTION
OUTPUT (FEET)
t⫽0
h ⫽ 250 ⫹ 20(0)
h ⫽ 250
t⫽1
h ⫽ 250 ⫹ 20(1)
h ⫽ 270
t⫽2
h ⫽ 250 ⫹ 20(2)
h ⫽ 290
t⫽3
h ⫽ 250 ⫹ 20(3)
h ⫽ 310
t⫽4
h ⫽ 250 ⫹ 20(4)
h ⫽ 330
t⫽5
h ⫽ 250 ⫹ 20(5)
h ⫽ 350
Input t Output h b.
Student Help STUDY TIP
To plot the first point (t, h) find t ⫽ 0 on the horizontal axis. Then find h ⫽ 250 on the vertical axis. Mark the point (0, 250).
0
1
2
3
4
5
250
270
290
310
330
350
Let the horizontal axis represent the input t (in minutes). Label the axis from 0 to 5. Let the vertical axis represent the output h (in feet). Label the axis from 0 to 400. Plot the data points given in the table. Finally, connect the points. The graph shows that as the time increases, the height of the balloon increases. The graph represents the function h ⫽ 250 ⫹ 20t, where t ≥ 0 and t ≤ 5.
1.8
An Introduction to Functions
49
MORE EXAMPLES
Write an Equation to Represent a Function
As you dive deeper and deeper into the ocean, the pressure of the water on your body steadily increases. The pressure at the surface of the water is 14.7 pounds per square inch (psi). The pressure increases at a rate of 0.445 psi for each foot you descend. Write an equation to represent the pressure P as a function of the depth d for every 20 feet you descend until you reach a depth of 60 feet.
SCUBA DIVING
NE ER T
More examples are available at www.mcdougallittell.com
INT
3
EXAMPLE
Student Help
Solution Pressure at Diving Pressure Rate of change p given depth depth in pressure at surface
VERBAL MODEL
LABELS
(psi)
Pressure at surface 14.7
(psi)
Rate of change in pressure 0.445
(psi per foot of depth)
Diving depth d
(feet)
P 14.7 0.445 d where d ≥ 0 and d ≤ 60
ALGEBRAIC MODEL
ANSWER 䊳
Pressure at given depth P
The function can be represented by the equation P 14.7 0.445d, where d ≥ 0 and d ≤ 60.
Represent a Function Use the algebraic model from Example 3. 1. Make an input-output table for the function. Use d 0, 20, 40, and 60. 2. Draw a graph that represents the function.
SUMMARY Four Ways to Represent Functions INPUT-OUTPUT TABLE
50
Input n
Output P
1
1
2
3
3
6
4
10
5
15
6
21
Chapter 1
WORDS
You are in a hot-air balloon at a height of 250 feet. You begin to rise higher at a rate of 20 feet per minute for a period of 5 minutes.
Connections to Algebra
EQUATION
h 250 20t, where t ≥ 0 and t≤5
GRAPH
1.8 Exercises Guided Practice Vocabulary Check
Complete the sentence. 1. A function is a relationship between two quantities, called the ____ ? and
the ____ ? . 2. The collection of all input values is the ____ ? of the function. 3. The collection of all output values is the ____ ? of the function.
Skill Check
CAMPING In Exercises 4–6, use the following information.
You are going camping. The cost for renting a cabin at Shady Knoll Campground is $65.00 plus $12.00 per person. The cost in dollars is C 65 12n, where n is the number of people. 4. Copy and complete the input-output table. Input n
1
2
3
4
5
6
Output C
?
?
?
?
?
?
5. Draw a graph that is made up of isolated points representing the cost of
renting a cabin. 6. Determine the range of the function from the given input values in the
input-output table.
Practice and Applications INPUT-OUTPUT TABLES Make an input-output table for the function. Use
0, 1, 2, 3, 4, and 5 as values for x. 7. y 6x 5
8. y 26 2x
9. y (x 3) p 7
10. y 85 15x
11. y 5(15 x)
12. y 2(6x 10)
LINE GRAPHS Draw a line graph to represent the function given by the
input-output table.
Student Help
13.
HOMEWORK HELP
Example 1: Exs. 7–12, 15–20, 23, 24 Example 2: Exs. 13–15, 21, 23–25 Example 3: Exs. 23, 24
14.
Input x
1
2
3
4
5
6
Output y
14
12
10
8
6
4
Input x
1
2
3
4
5
6
Output y
8
11
14
17
20
23
1.8
An Introduction to Functions
51
15.
Student Help
CRITICAL THINKING Determine whether the table represents a function.
HOMEWORK HELP
16.
NE ER T
Extra help with problem solving in Exs. 16–19 is available at www.mcdougallittell.com
INT
The distance d (in miles) that sound travels in air in time t (in seconds) is represented by the function d 0.2t. Make a table of the input t and the output d. Use t values of 0, 5, 10, 15, 20, 25, and 30. Use your table to help you draw the graph of the function.
18.
Input
Output
1
17.
Input
Output
3
1
2
2
4
3
3
3
5
3
4
Input
Output
Input
Output
2
2
1
3
3
4
1
4
4
6
2
5
19.
CAR RACING In Exercises 20–22, use the following information.
The fastest winning speed in the Daytona 500 is about 178 miles per hour. In the table below, calculate the distance traveled d (in miles) after time t (in hours) using the equation d 178t. 20. Copy and complete the input-output table. Time (hours)
0.25
0.50
0.75
1.00
1.25
1.50
?
?
?
?
?
?
Distance traveled (miles) 21. Use the data to draw a graph.
History
22. For what values of t does the formula d 178t correspond to the situation
being modeled? 23.
In 1866 Texas cowhands used the Chisholm Trail to drive cattle north to the railroads in Kansas. The average rate r that the cattle could be moved along the trail was 11 miles per day. a. Write an equation, where d is
distance and t is time in days. b. Make a table of input t and
output d for t 7, 14, and 28. Then graph the data. JESSE CHISHOLM , the person for whom the Chisholm Trail is named, was a trader who was part Cherokee. 52
Chapter 1
c. The distance d from San Antonio
to Abilene was about 1100 miles. How long did it take to drive cattle the entire length of the trail? Connections to Algebra
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