Table of Contents - Moreland MATH1/9/2018 Thinking Mathematically, Sixth Edition 1/2

1/9/2018 Thinking Mathematically, Sixth Edition

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6 Algebra: Equations and Inequalities > Chapter Summary, Review, and Test

86.

a. According to the model, what percentage of the U.S. population was foreign-born in 1990? Does the modelunderestimate or overestimate the actual number displayed by the bar graph on the previous page? By how much?

b. If trends shown by the model continue, in which year will 23% of the U.S. population be foreign-born? Round tothe nearest year.

87. If you have not yet done so, read the Blitzer Bonus on page 398. In this exercise, you will use the goldenrectangles shown to obtain an exact value for the ratio of the long side to the short side in a golden rectangle ofany size.

a. The golden ratio in rectangle A, or the ratio of the long side to the short side, can be modeled by Write afractional expression that models the golden ratio in rectangle B.

b. Set the expression for the golden ratio in rectangle A equal to the expression for the golden ratio in rectangle B.Solve the resulting proportion using the quadratic formula. Express as an exact value in simplified radical form.

c. Use your solution from part (b) to complete this statement: The ratio of the long side to the short side in a goldenrectangle of any size is _____ to 1.

Writing in Mathematics88. Explain how to multiply two binomials using the FOIL method. Give an example with your explanation.

89. Explain how to factor

90. Explain how to solve a quadratic equation by factoring. Use the equation in your explanation.

91. Explain how to solve a quadratic equation using the quadratic formula. Use the equation inyour explanation.

92. Describe the trend shown by the data for the percentage of foreign-born Americans in the graph for Exercises 85–86. Do you believe that this trend is likely to continue or might something occur that would make it impossible toextend the model into the future? Explain your answer.

Critical Thinking ExercisesMake Sense? In Exercises 93–96, determine whether each statement makes sense or does not make sense, andexplain your reasoning.

93. I began factoring by finding all number pairs with a sum of

94. It's easy to factor because of the relatively small numbers for the constant term and the coefficient ofx.

95. The fastest way for me to solve is to use the quadratic formula.

96. I simplified to because 2 is a factor of

97. The radicand of the quadratic formula, can be used to determine whether hassolutions that are rational, irrational, or not real numbers. Explain how this works. Is it possible to determine the kindsof answers that one will obtain to a quadratic equation without actually solving the equation? Explain.

In Exercises 98–99, find all positive integers b so that the trinomial can be factored.

98.

99.

100. Factor:

101. Solve:

Chapter Summary, Review, and Test

Table of Contents

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Φ,

.Φ1

Φ

− 5x + 6.x2

+ 6x + 8 = 0x2

+ 6x + 8 = 0x2

− 17x + 72x2 −17.

+ x + 1x2

− x − 2 = 0x2

3+2 3√

23 + 3–√ 2 .3–√

− 4ac,b2 a + bx + c = 0x2

+ bx + 15x2

+ 4x + bx2

+ 20 + 99.x2n xn

+ 2 x − 9 = 0.x2 3–√

Thinking Mathematically, SixthEdition

1 Problem Solving and CriticalThinking

2 Set Theory

3 Logic

4 Number Representation andCalculation

5 Number Theory and the RealNumber System

6 Algebra: Equations andInequalities

7 Algebra: Graphs, Functionsand Linear Systems

8 Personal Finance

9 Measurement

10 Geometry

11 Counting Methods andProbability Theory

12 Statistics

13 Voting and Apportionment

14 Graph Theory

Answers to Selected Exercises

Credits

Subject Index

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Page 401

SUMMARY – DEFINITIONS AND CONCEPTS EXAMPLESSUMMARY – DEFINITIONS AND CONCEPTS EXAMPLES6.1 Algebraic Expressions and Formulasa. An algebraic expression combines variables and numbers using addition, subtraction, multiplication,division, powers, or roots.

b. Evaluating an algebraic expression means finding its value for a given value of the variable or for givenvalues of the variables. Once these values are substituted, follow the order of operations agreement inthe box on page 340.

Ex. 1, p.341; Ex. 2, p.341; Ex. 3, p.341

c. An equation is a statement that two expressions are equal. Formulas are equations that expressrelationships among two or more variables. Mathematical modeling is the process of finding formulas todescribe real-world phenomena. Such formulas, together with the meaning assigned to the variables, arecalled mathematical models. The formulas are said to model, or describe, the relationships among thevariables.

Ex. 4, p.342

d. Terms of an algebraic expression are separated by addition. Like terms have the same variables withthe same exponents on the variables. To add or subtract like terms, add or subtract the coefficients andcopy the common variable.

e. An algebraic expression is simplified when parentheses have been removed (using the distributiveproperty) and like terms have been combined.

Ex. 5, p.344; Ex. 6, p.345; Ex. 7, p.345

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6 Algebra: Equations and Inequalities > Chapter Summary, Review, and Test

6.2 Linear Equations in One Variable and Proportionsa. A linear equation in one variable can be written in the form where a and b are real numbers, and b. Solving a linear equation is the process of finding the set of numbers that makes the equation a true statement. These numbers are the solutions. The setof all such solutions is the solution set.

c. Equivalent equations have the same solution set. Properties for generating equivalent equations are given in the box on page 351. Ex. 1, p.352

d. A step-by-step procedure for solving a linear equation is given in the box on page 352.

Ex. 2, p.352; Ex. 3, p.354; Ex. 4, p.354

e. If an equation contains fractions, begin by multiplying both sides of the equation by the least common denominator of the fractions in the equation, therebyclearing fractions.

Ex. 5, p.355; Ex. 6, p.356

f. The ratio of a to b is written or

g. A proportion is a statement in the form

h. The cross-products principle states that if then Ex. 7, p.358

i. A step-by-step procedure for solving applied problems using proportions is given in the box on page 359.

Ex. 8, p.359; Ex. 9, p.360

j. If a false statement (such as ) is obtained in solving an equation, the equation has no solution. The solution set is the empty set. Ex. 10,p. 360

k. If a true statement (such as ) is obtained in solving an equation, the equation has infinitely many solutions. The solution set is the set of all realnumbers, written

Ex. 11,p. 361

6.3 Applications of Linear Equationsa. Algebraic translations of English phrases are given in Table 6.2 on page 366.

b. A step-by-step strategy for solving word problems using linear equations is given in the box on page 365.

Ex. 1, p.366; Ex. 2, p.368; Ex. 3, p.370; Ex. 4, p.371

c. Solving a formula for a variable means rewriting the formula so that the variable is isolated on one side of the equation.

Ex. 5, p.372; Ex. 6, p.372

6.4 Linear Inequalities in One Variable

A procedure for solving a linear inequality is given in the box on page 379. Remember to reverse the direction of the inequality symbol when multiplying ordividing both sides of an inequality by a negative number, thereby changing the sense of the inequality.

Ex. 2, p.379; Ex. 3, p.379; Ex. 4, p.380; Ex. 5, p.381; Ex. 6, p.382

6.5 Quadratic Equationsa. A quadratic equation can be written in the form

b. Some quadratic equations can be solved using factoring and the zero-product principle. A step-by-step procedure is given in the box on page 393.

Ex. 8, p.393; Ex. 9, p.394

c. All quadratic equations in the form can be solved using the quadratic formula: Ex. 10,p. 394; Ex. 11,p. 395; Ex. 12,p. 397



ax + b = 0, a ≠ 0.

,a

ba : b.

= .a

b

c

d

= ,a

b

c

dad = bc.

−6 = 7 ∅,

−6 = −6{x|x is a real number} .

a + bx + c = 0, a ≠ 0.x2

a + bx + c = 0x2

x = .−b ± − 4acb2− −−−−−−√

2a

















































6 Algebra: Equations and Inequalities > Chapter Summary, Review, and Test > Review Exercises

Review Exercises6.1

In Exercises 1–3, evaluate the algebraic expression for the given value of the variable.

1.

2.

3.

4. The diversity index, from 0 (no diversity) to 100, measures the chance that two randomly selected people are a different race or ethnicity. The diversity index in theUnited States varies widely from region to region, from as high as 81 in Hawaii to as low as 11 in Vermont. The bar graph shows the national diversity index for theUnited States for four years in the period from 1980 through 2010.

dSource: USA Today

The data in the graph can be modeled by the formula

where D is the national diversity index in the United States x years after 1980. According to the formula, what was the U.S. diversity index in 2010? How does thiscompare with the index displayed by the bar graph?

In Exercises 5–7, simplify each algebraic expression.

5.

6.

7.

6.2

In Exercises 8–14, solve each equation.

8.

9.

10.

11.

12.

13.

14.

In Exercises 15–18, solve each proportion.

15.

16.



6x + 9; x = 4

7 + 4x − 5; x = −2x2

6 + 2 ; x = 5(x − 8)3

D = 0.005 + 0.55x + 34,x2

5 (2x − 3) + 7x

3 (4y − 5) − (7y − 2)

2 ( + 5x) + 3 (4 − 3x)x2 x2

4x + 9 = 33

5x − 3 = x + 5

3 (x + 4) = 5x − 12

2 (x − 2) + 3 (x + 5) = 2x − 2

= + 12x

3x

6

7x + 5 = 5 (x + 3) + 2x

7x + 13 = 2 (2x − 5) + 3x + 23

=3x

15

25

=−7

5

91x








Page 403

17.

18.

19. If a school board determines that there should be 3 teachers for every 50 students, how many teachers are needed for an enrollment of 5400 students?

20. To determine the number of trout in a lake, a conservationist catches 112 trout, tags them, and returns them to the lake. Later, 82 trout are caught, and 32 of themare found to be tagged. How many trout are in the lake?

21. The line graph shows the cost of inflation. What cost $10,000 in 1982 would cost the amount shown by the graph in subsequent years.

dSource: U.S. Bureau of Labor Statistics

Here are two mathematical models for the data shown by the graph. In each formula, C represents the cost x years after 1985 of what cost $10,000 in 1982.

a. Use the graph to estimate the cost in 1995, to the nearest thousand dollars, of what cost $10,000 in 1982.

b. Use model 1 to determine the cost in 1995. How well does this describe your estimate from part (a)?

c. Use model 2 to determine the cost in 1995. How well does this describe your estimate from part (a)?

d. Use model 1 to determine in which year the cost will be $28,320 for what cost $10,000 in 1982.


=x+2

345

=5

x+7

3

x+3

►CModel 1

►CModel 2

= 438x + 10, 800

= 0.3 + 430x + 10, 824x2








6 Algebra: Equations and Inequalities > Chapter Summary, Review, and Test > Review Exercises

6.3

22. Destined for Gory. As sequels to horror films increase, so does the body count. Wes Craven's slasher Scream series adheres to that axiom.

d

Whether it's knife to the back, knife to the gut, or knife to the head, the body count in Scream 2 exceeds the departed in Scream by 2. Appropriately, the number ofcharacters killed off in Scream 3 exceeds the departed in Scream by 3. The total body count in the four Scream films shown in the graphic is 33. Find the body countin Scream, Scream 2, and Scream 3.

23. The bar graph shows the average price of a movie ticket for selected years from 1980 through 2010. The graph indicates that in 1980, the average movie ticketprice was $2.69. For the period from 1980 through 2010, the price increased by approximately $0.15 per year. If this trend continues, by which year will the averageprice of a movie ticket be $8.69?

dSources: Motion Picture Association of America, National Association of Theater Owners (NATO), and Bureau of Labor Statistics (BLS)

24. You are choosing between two texting plans. One plan has a monthly fee of $15 with a charge of $0.05 per text. The other plan has a monthly fee of $5 with acharge of $0.07 per text. For how many text messages will the costs for the two plans be the same?

25. After a 20% price reduction, a cordless phone sold for $48. What was the phone's price before the reduction?

26. A salesperson earns $300 per week plus 5% commission on sales. How much must be sold to earn $800 in a week?

In Exercises 27–30, solve each formula for the specified variable.

27.

28.

29.

30.

6.4



Ax − By = C for x

A = bh for h12

A = for BB+C

2

vt + g = s for gt2








Page 404

In Exercises 31–37, solve each inequality and graph the solution set on a number line.

31.

32.

33.

34.

35.

36.

37.

38. To pass a course, a student must have an average on three examinations of at least 60. If a student scores 42 and 74 on the first two tests, what must be earnedon the third test to pass the course?


2x − 5 < 3

> −4x

2

3 − 5x ≤ 18

4x + 6 < 5x

6x − 10 ≥ 2 (x + 3)

4x + 3 (2x − 7) ≤ x − 3

−1 < 4x + 2 ≤ 6








6 Algebra: Equations and Inequalities > Chapter 6 Test

6.5

Use FOIL to find the products in Exercises 39–40.

39.

40.

Factor the trinomials in Exercises 41–46, or state that the trinomial is prime.

41.

42.

43.

44.

45.

46.

Solve the quadratic equations in Exercises 47–50 by factoring.

47.

48.

49.

50.

Solve the quadratic equations in Exercises 51–54 using the quadratic formula.

51.

52.

53.

54.

55. As gas prices surge, more Americans are cycling as a way to save money, stay fit, or both. In 2010, Boston installed 20 miles of bike lanes and New York Cityadded more than 50 miles. The bar graph shows the number of bicycle-friendly U.S. communities, as designated by the League of American Bicyclists, for selectedyears from 2003 through 2011.

dSource: League of American Bicyclists

The formula

models the number of bicycle-friendly communities, B, x years after 2003.

a. Use the formula to find the number of bicycle-friendly communities in 2011. Round to the nearest whole number. Does this rounded value underestimate oroverestimate the number shown by the graph? By how much?

b. Use the formula to determine the year in which 826 U.S. communities will be bicycle friendly.

Chapter 6 Test



(x + 9) (x − 5)

(4x − 7) (3x + 2)

− x − 12x2

− 8x + 15x2

+ 2x + 3x2

3 − 17x + 10x2

6 − 11x − 10x2

3 − 6x − 5x2

+ 5x − 14 = 0x2

− 4x = 32x2

2 + 15x − 8 = 0x2

3 = −21x − 30x2

− 4x + 3 = 0x2

− 5x = 4x2

2 + 5x − 3 = 0x2

3 − 6x = 5x2

B = 1.7 + 6x + 26x2







Page 405

1. Evaluate when

2. Simplify:

In Exercises 3–6, solve each equation.

3.

4.

5.

6.

7. Solve for

8. The bar graph in the next column shows the percentage of American adults reporting personal gun ownership for selected years from 1980 through 2010.

Here are two mathematical models for the data shown by the graph. In each formula, p represents the percentage of American adults who reported personal gunownership x years after 1980.

dSource: General Social Survey

a. According to model 1, what percentage of American adults reported personal gun ownership in 2010? Does this underestimate or overestimate the percentageshown by the graph? By how much?

b. According to model 2, what percentage of American adults reported personal gun ownership in 2010? Does this underestimate or overestimate the percentageshown by the graph? By how much?

c. If trends shown by the data continue, use model 1 to determine in which year 17.7% of American adults will report personal gun ownership.


− 4x3 (x − 1)2x = −2.

5 (3x − 2) − (x − 6) .

12x + 4 = 7x − 21

3 (2x − 4) = 9 − 3 (x + 1)

3 (x − 4) + x = 2 (6 + 2x)

− 2 =x

5x

3

y : By − Ax = A.

►PModel 1

►PModel 2

= −0.3x + 30

= −0.003 − 0.22x + 30x2








6 Algebra: Equations and Inequalities > Chapter 6 Test

In Exercises 9–10, solve each proportion.

9.

10.

11. Park rangers catch, tag, and release 200 tule elk back into a wildlife refuge. Two weeks later they observe a sample of 150 elk, of which 5 are tagged. Assumingthat the ratio of tagged elk in the sample holds for all elk in the refuge, how many elk are there in the park?

12. What's the last word in capital punishment? An analysis of the final statements of all men and women Texas has executed since the Supreme Court reinstated thedeath penalty in 1976 revealed that “love” is by far the most frequently uttered word. The bar graph shows the number of times various words were used in finalstatements by Texas death-row inmates.

dSource: Texas Department of Criminal Justice

The number of times “love” was used exceeded the number of times “sorry” was used by 419. The number of utterances of “thanks” exceeded the number ofutterances of “sorry” by 32. Combined, these three words were used 1084 times. Determine the number of times each of these words was used in final statements byTexas inmates.

13. You bought a new car for $50,750. Its value is decreasing by $5500 per year. After how many years will its value be $12,250?

14. You are choosing between two texting plans. Plan A charges $25 per month for unlimited texting. Plan B has a monthly fee of $13 with a charge of $0.06 per text.For how many text messages, will the costs for the two plans be the same?

15. After a 60% reduction, a jacket sold for $20. What was the jacket's price before the reduction?

In Exercises 16–18, solve each inequality and graph the solution set on a number line.

16.

17.

18.

19. A student has grades on three examinations of 76, 80, and 72. What must the student earn on a fourth examination in order to have an average of at least 80?

20. Use FOIL to find this product:

21. Factor:

22. Solve by factoring:

23. Solve using the quadratic formula:

The graphs show the amount being paid in Social Security benefits and the amount going into the system. All data are expressed in billions of dollars. Amounts from2012 through 2024 are projections.



=5

8x

12

=x+5

8

x+2

5

6 − 9x ≥ 33

4x − 2 > 2 (x + 6)

−3 ≤ 2x + 1 < 6

(2x − 5) (3x + 4) .

2 − 9x + 10.x2

+ 5x = 36.x2

2 + 4x = −1.x2







Page 406

dSource: 2004 Social Security Trustees Report

Exercises 24–26 are based on the data shown by the graphs.

24. In 2004, the system's income was $575 billion, projected to increase at an average rate of $43 billion per year. In which year will the system's income be $1177billion?

25. The data for the system's outflow can be modeled by the formula

where B represents the amount paid in benefits, in billions of dollars, x years after 2004. According to this model, when will the amount paid in benefits be $1177billion? Round to the nearest year.

26. How well do your answers to Exercises 24 and 25 model the data shown by the graphs?


B = 0.07 + 47.4x + 500,x2






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Table of Contents - Moreland MATH1/9/2018 Thinking Mathematically, Sixth Edition 1/2