How Should I Study? - Weebly

Lesson 1-5 Solving Inequalities 39

59. Writing in Math Use the information about phone rate plans on page 33 to explain how inequalities can be used to compare phone plans. Include an explanation of how Kuni might determine when Plan 2 might be cheaper than Plan 1 if she typically uses more than 400 but less than 650 minutes.

Solve each equation. Check your solutions. (Lesson 1-4)

62. x - 3 = 17 63. 84x - 3 = 64 64. x + 1 = x

.56 E-COMMERCE On average, by how much did the amount spent on online purchases increase each year from 2000 to 2004? Define a variable, write an equation, and solve the problem. (Lesson 1-3)

Name the sets of numbers to which each numberbelongs. (Lesson 1-2)

66. 31 67. -4.2 68. √7

.96 BABY-SITTING Jenny baby-sat for 51_2

nosruoh

Friday night and 8 hours on Saturday. Shecharges $4.25 per hour. Use the DistributiveProperty to write two equivalent expressionsthat represent how much money Jenny earned.(Lesson 1-2)

PREREQUISITE SKILL Solve each equation. Check your solutions. (Lesson 1-4)

70. x = 7 71. x + 5 = 18 72. 5y - 8 = 12

73. 14 = 2x - 36 74. 10 = 2 w + 6 75. x + 4 + 3 = 17

60. ACT/SAT If a < b and c < 0, which ofthe following are true?

I. ac > bc

II. a + c < b + c

III. a - c > b - c

A I only

B II only

C III only

D I and II only

61. REVIEW What is the completesolution to the equation8 - 4x = 40?

F x = 8; x = 12

G x = 8; x = -12

H x = -8; x = -12

J x = -8; x = 12

Mid-Chapter QuizLessons 1-1 through 1-4

CHAPTER

1Evaluate each expression if a = -2, b = 1_

3dna,

c = -12. (Lesson 1-1)

1. a3 + b(9 - c) 2. b(a2 - c)

3. 3ab_c 4. a - c_

a + c

5. a3 - c_b2

6. c + 3_ab

7. ELECTRICITY Find the amount of current I (in amperes) produced if the electromotive forceE is 2.5 volts, the circuit resistance R is 1.05 ohms, and the resistance r within a battery is0.2 ohm. Use the formula I = E_

R + r.

(Lesson 1-1)

Name the sets of numbers to which eachnumber belongs. (Lesson 1-2)

8. 3.5 9. √100

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

10. bc + (-bc) = 0

11. (4_7)(13_

4) = 1

12. 3 + (x - 1) = (3 + x) + (-1)

Name the additive inverse and multiplicativeinverse for each number. (Lesson 1-2)

13. 6_7

14. -4_3

15. Simplify 4(14x - 10y) - 6(x + 4y). (Lesson 1-2)

Write an algebraic expression to represent eachverbal expression. (Lesson 1-3)

16. twice the difference of a number and 11

17. the product of the square of a number and 5

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

18. -2(a + 4) = 2

19. 2d + 5 = 8d + 2

20. 4y - 1_10

= 3y + 4_5

21. Solve s = 1_2

gt2 for g. (Lesson 1-3)

22. MULTIPLE CHOICE Karissa has $10 per month to spend text messaging on her cell phone. The phone company charges $4.95 for the first 100 messages and $0.10 for each additional message. How many text messagescan Karissa afford to send each month?(Lesson 1-3)

A 50 C 150

B 100 D 151

23. GEOMETRY Use the information in the figure to find the value of x. Then state the degree measures of the three angles of the triangle. (Lesson 1-3)

Solve each equation. Check your solutions.(Lesson 1-4)

24. a + 4 = 3 25. 3x + 2 = 1

26. 3m - 2 = -4 27. 2x + 5 - 7 = 4

28. h + 6 + 9 = 8 29. 5x - 2 - 6 = -3

30. CARNIVAL GAMES Julian will win a prize if the carnival worker cannot guess his weight to within 3 pounds. Julian weighs 128 pounds. Write an equation to find the highest and lowest weights that the carnival guesser can guess to keep Julian from winning a prize. (Lesson 1-4)

32 Chapter 1 Equations and Inequalities

54 Chapter 1 Equations and Inequalities

CHAPTER

1

Question 1 To solve equations or inequalities, you can replace the variables in the question with the values given in each answer choice. The answer choice that results in true statements is the correct answer choice.

California Standards PracticeChapter 1

Read each question. Then fill in thecorrect answer on the answer documentprovided by your teacher or on a sheetof paper.

1 What is the complete solution to the equation2 - 4x = 14?A x = 3; x = 4B x = -3; x = 4C x = 3; x = -4D x = -3; x = -4

2 Lucas determined that the total cost to rent acar C for the weekend could be representedby the equation C = 0.35m + 125, where m isthe number of miles that he drives. If thetotal cost to rent the car was $363, how manymiles did he drive?F 125G 238H 520

680

3. Solve the equation 4x - 5 = 2x + 5 - 3x for x.

4. Evaluate the following expression if x = 3,y = -4, and z = 2.

x(y + z)3_xy + 3z

A -4B -2C 2D 4

5. Evaluate t2 − 4uv_(v + 2)2

if t = 12, u = 2, and v = 6.

F 4_3

H 2

G 3_2

J 4

6. The profit p that Selena’s Shirt store makes ina day can be represented by the inequality10t + 200 < p < 15t + 250, where t representsthe number of shirts sold. If the store sold45 shirts on Friday, which of the following isa reasonable amount that the store made?A $200.00B $625.00C $850.00D $950.00

7. Which of the following represents thesolution to the inequality 6b - 12 ?42<F

G

H

J

8. Which inequality represents an algebraicexpression for the verbal expression below?

Half the difference of a number and 3 is noless than four times the number decreasedby 10.

A 1_2

(n − 3) > 4n − 10

B 1_2

(n − 3) < 4n − 10

C 1_2

(n − 3) ≥ 4n − 10

D 1_2

(n − 3) ≤ 4n − 10

California Standards Practice at ca.algebra1.com

Throughout the year, you may be required to take several tests, and you may have some questions about them. Here are some answers to help you get ready.

How Should I Study?The good news is that you’ve been studying all along—a little bit every day. Here are some of the ways your textbook has been preparing you.

• Every Day Each lesson had practice questions that cover the California Standards.

• Every Week The Mid-Chapter Quizzes and Practice Tests had several multiple-choice practice questions.

• Every Month The California Standards Practice pages at the end of each chapter had even more questions similar to those on the test.

Are There Other Ways to Review?Absolutely! The following pages contain even more practice for each California standard.

Tips for Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CA1

Multiple-Choice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . CA2

Practice by Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . CA4–CA43

Prepare• Go to bed early the night before the test. You will think more clearly after a

good night’s rest.

• Become familiar with common formulas and when they should be used.

• Think positively.

During the Test• Read each problem carefully. Underline key words and think about different

ways to solve the problem.

• Watch for key words like NOT. Also look for order words like least, greatest, first, and last.

• Answer questions you are sure about first. If you do not know the answer to a question, skip it and go back to that question later.

• Check your answer to make sure it is reasonable.

• Make sure that the number of the question on the answer sheet matches the number of the question on which you are working in your test booklet.

Whatever you do…• Don’t try to do it all in your

head. If no figure is provided, draw one.

• Don’t rush. Try to work at a steady pace.

• Don’t give up. Some problems may seem hard to you, but you may be able to figure out what to do if you read each question carefully or try another strategy.

RELAX!Just do your best.

CA1

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Multiple-Choice QuestionsMany tests have multiple-choice questions. You are asked to choose the best answer from four or five possible answers.

To record a multiple-choice answer, you will be asked to shade in a bubble that is a circle. Always make sure that your shading is dark enough and completely covers the bubble.

White chocolate sells for $3.25 per pound and dark chocolate sells for $2.50 per pound. How many pounds of white chocolate are needed for a 10-pound mixture of both kinds that sells for $2.80 per pound?

A 2 lb B 4 lb C 6 lb D 10 lb

Let w be the number of pounds of white chocolate and let d be the number of pounds of dark chocolate.

Write a system of equations.

w + d = 10 There is a total of 10 pounds of chocolate.

3.25w + 2.50d = 2.80(10) The price is $2.80 × 10 for the mixed chocolate.

Solve the first equation for d.

w + d = 10 First equation

d = 10 – w Subtract w from each side.

Use substitution to solve.

3.25w + 2.50d = 2.80(10) Original equation

3.25w + 2.50(10 - w) = 28 Substitute.

3.25w + 25 - 2.5w = 28 Distributive Property

0.75w = 3 Simplify.

w = 4 Divide each side by 0.75.

The answer is B.

Using the following tips will help you to be successful.

• Since every question is worth the same number of points, answer the easy questions first. Skip over the harder questions for now, but mark them in your test booklet so you can return to them later.

• As you work on a problem, check the answer choices and cross out any choices that you know are wrong.

• If you think it will be faster, substitute the answer choices and see which one works.

• If you are torn between two answers, it is better to guess than not to answer.• When you’re done, check your work. Look for careless mistakes.

STRATEGY

Reasonableness Check to see that your answer is reasonable with the given information.

CA2 California Standards Review

California Standards Review

Use the tips you have learned to solve the following example.

On a recent test, Alisa wrote the equation x2 - 10x + 25

__ x - 5

= x - 5.

Which of the following statements is correct about the equation she wrote?

A The equation is always true.

B The equation is always true, except when x = 5.

C The equation is never true.

D The equation is true only when x = 5.

• Are any of the choices obviously wrong? Choice C is not the correct choice. Since x2 - 10 + 25 = (x - 5)(x - 5),

the equation is true at least some of the time.

• Can you substitute any of the answer choices to see which one works? Choices B and D involve cases in which x = 5. If we replace x with 5

in the equation, the result would cause division by 0. Thus, choices A and D cannot be correct.

Therefore, the answer is B.

Sometimes a question does not provide you with a figure that represents the situation. Drawing a diagram may help you to solve the problem.

Josh throws a baseball upward at a velocity of 105 feet per second, releasing the baseball 5 feet above the ground. The height of the ball t seconds after being thrown is given by h(t) = -16 t 2 + 105t + 5. Find the time at which the baseball reaches its maximum height.

F 1.0 s G 3.3 s H 6.6 s J 177.3 s

Graph the equation. The ball is at its maximum height at the vertex of the graph.

The graph indicates that the maximum height is achieved between 3 and 4 seconds after launch.

The answer is G.

STRATEGY

Test-Taking Learn and use effective strategies to be a better test-taker.

STRATEGY

Diagrams Drawing a diagram for a situation may help you to answer the question.

Multiple-Choice Questions CA3

Hei

gh

t (f

t) (

s)

0

100

200

Time (s) 1 2 3 4 5 6

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Practice by StandardStandard 1.0 Students solve equations and inequalities involving absolute

values.

DIRECTIONS Choose the best answer.

Practice on Your Own Standard 1.0

2 Which of the following best describes the solution set to the equation |8x - 12| + 16 = 4?

F infi nitely many solutions

G two solutions

H one solution

J no solutions

3 Which graph represents the solution to |3x + 6| < 15?

A

B

C

D

4 What is the complete solution to the equation |x + 2| = 4x + 5? F x = -1.4

G x = -1

H x = -1.4; x = -1

J No solution

5 Which of the following is the correct solution set for |1 - 2x| ≥ 3? A -1 ≤ x ≤ 2B x ≤ -1 or x ≥ 2

C x ≤ 1 or x ≥ 2

D 1 ≤ x ≤ 2

6 Payton is training for a triathlon. Over the weekend she plans to run for x hours at a pace of 6 miles per hour and bike for y hours at a pace of 18 miles per hour. She plans to run and bike for no more than 50 miles. Which inequality represents this solution? F xy(6 + 18) ≤ 50

G 6x + 18y ≤ 50

H (x + y)(6 + 18) ≤ 50

J (6 + x)(18 + y) ≤ 50

1 What is the complete solution to the equation |9 - 2x | = 17?

A x = 4; x = 13

B x = 4; x = -13

C x = -4; x = 13

D x = -4; x = -13

STRATEGY An absolute value equation must be split into two separate equations.

The expression inside the absolute value bars may be equal to either 17 or -17.

For more help with solving absolute value equations, see page 27.



Practice by StandardStandard 2.0: Students solve systems of linear equations and inequalities

(in two or three variables) by substitution, with graphs, or with matrices.


1 A school group is planning an outing to an amusement park. A total of 32 adults and students will be going to the park. The admission fee for adults is $25 and for students, the cost is $15. If the total of all the admission fees paid was $560, how many students went to the amusement park?

A 8

B 16

C 20

D 24

STRATEGY A situation like this requires you to set up a system of linear equations.

Let the variables a and s represent the numbers of adults and students who are going on the outing. Then write two equations with these variables based on the given information and choose a method to solve.

For more help with solving systems of linear equations, see page 123.

2 What is the solution to the system of equations graphed below?

F (0, 0)

G (0, 3)

H (2, 1)

J no solution

READING HINT The solution to a system of equations is the point of intersection of the two lines.

Find the point at which the graphs intersect. If the graphs do not intersect, then there is no solution. If the graphs of the lines coincide, then there are infinitely many solutions.

For more help with solving systems of equations by graphing, see page 116.

Practice by Standard: Standard 2.0 CA5

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3 What is the solution to the system of equations shown below?

3x − y + 2z = 12

−x + 5y − 2z = −4

−6x + 2y − 4z = 24

A (2, 0, 3)

B (4, 2, 5)

C (4, 0, 0)

D no solution

4 Which of the following ordered pairs is a possible solution to the system of inequalities shown?

x + y > −2

3x ≥ y − 2

F (−1, −1)

G (0, 4)

H (1, −4)

J (2, 0)

5 What is the solution to the system of equations shown below?

6x − 8y = 14

−3x + 4y = −4

A (1, −1)

B (2, − 1_4)

C infi nitely many solutions

D no solution

6 Marilyn spent $185 on supplies to make necklaces and bracelets. She made a total of 10 necklaces and bracelets. If each necklace costs $20 to make and each bracelet costs $15 to make, how many necklaces did she make?

F 3

G 5

H 7

J 9

7 Which system of inequalities is shown in the graph below?

A 2y − x < 5x + y > 5

B 2y − x ≥ 4

x + y < 5

C 2y − x ≤ 4

x + y > 5

D 2y − x > 4x + y ≤ 5

8 Ms. Lopez is receiving a shipment of calculators for the math department. The graphing calculators are $90 each, and the scientific calculators are $18 each. The total listed on the invoice was $4500 for all 122 calculators. How many of them were graphing calculators?

F 32 H 61

G 45 J 75

9 What is the solution to the system of equations graphed below?

A (0, 2) C all real numbers

B (2, 0) D no solution



Practice by StandardStandard 3.0: Students are adept at operations on polynomials, including

long division.


1 Which polynomial is equivalent to (2x - 3)(4x2 - x + 5)?

A 8x3 - 2x2 + 10x - 15

B 8x3 - 14x2 + 13x - 15

C 8x3 + 10x2 + 7x - 15

D 8x3 + 10x2 + 13x - 15

STRATEGY Use the Distributive Property to multiply two polynomials.

Make sure that each term of the first polynomial is multiplied with each term of the second polynomial.

For more help with multiplying polynomials, see page 322.

2 Simplify the following expression.

(3x2 + 7x - 11) - 3(8x2 - 3x - 6)

F -21x2 - 2x - 29

G -21x2 + 16x + 7

H -21x2 + 4x - 17

J -21x2 + 13x - 20

STRATEGY The rules governing the order of operations must be followed to correctly simplify the expression.

Make sure you distribute before combining the like terms.

For more help with order of operations and simplifying expressions, see page 321.

3 Which of the following is a trinomial with degree 4?

A x4 + x2 + 5x − 1

B 2x3 − 9x2 − 14x

C 3x3 + 5x2 + 8x − 11

D 7x4 + 2x − 3

READING HINT Be sure to know your definitions. A trinomial must have exactly three terms.

The degree of a polynomial is the highest of the degrees of its terms. The degree of a term is found by adding the exponents of all of the variables in that term.

For more help with polynomial definitions, see page 320.


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4 2x + 1 � �� 6x4 + x3 - 9x2 + 10 F 3x3 + 2x2 − 5x + 4 + 6 _

2x + 1

G 3x3 − x2 − 4x + 4 + 6 _ 2x + 1

H 3x3 − x2 − 4x + 2 + 8 _ 2x + 1

J 3x3 + 2x2 − 4x + 2 + 8 _ 2x + 1

5 Simplify (4x2y)3.

A 64x6y3

B 64x5y3

C 16x5y3

D 12x6y3


(5x2 + 2x − 7) − (6x + 4) + (2x2 + 4x + 3)

F 7x2 + 12x − 8

G 7x2 − 8

H 7x2

J 3x2 − 8

7 Simplify (2x2 + 5)2.

A 4x4 + 25

B 4x4 + 20x2 + 25

C 4x4 + 10x2 + 25

D 4x2 + 10

8 A cube has sides of length (x − 3). Which of the following expressions is equivalent to its volume?

x 3 x 3

x 3

F x2 − 6x + 9

G x2 − 9

H x3 − 9x2 + 27x − 27

J x3 − 27


(3x + 4)(x − 5)

A 3x2 − 19x − 20

B 3x2 − 11x − 20

C 3x2 − 20

D 4x − 1

10 (15x3 − 2x2 − 5x + 2) ÷ (3x + 2) =

F 5x2 − 4x + 1

G 5x2 + 1

H 5x2 − 4x − 1 + 4_3x + 2

J 5x2 − 4x + 3 − 4_3x + 2

11 Which expression best represents the

simplification of (6a2b3)

2 _

18ab5 ?

A a2_9b4

B 2a3b

C a3b_2

D a_3b2

12 Which expression is equivalent

to (25x4) 3 _ 2 ?

F 5x6

G 125x 8 _ 3

H 125x6

J 25x 8 _ 3

13 (−3x2 + 9x + 2) − 2 (2x2 − 5x − 1) =

A −x2 + 4x + 1

B −x2 + 14x + 3

C −7x2 + 4x + 1

D −7x2 + 14x + 3



Practice by StandardStandard 4.0: Students factor polynomials representing the difference of squares,

perfect square trinomials, and the sum and difference of cubes.


1 The area of a rectangle is given as 8x2 - 10x - 3. Which of the following shows expressions that are possible values for its base and height?

A (2x - 3) and (4x + 1)

B (2x + 3) and (4x - 1)

C (8x - 3) and (x + 1)

D (8x + 3) and (x - 1)

READING HINT The formula for the area of a rectangle is A = bh.

Try to eliminate any of the pairs that do not have a product equal to the area of the rectangle, or factor the polynomial as one of the pairs listed.

For more help with factoring polynomials, see page 349.

2 What is the greatest common factor of the terms in the polynomial 18x3y2 - 12x2y2 + 24xy2?

F 3xy2

G 6x2y

H 6xy2

J 12xy2

READING HINT The greatest common factor (GCF) of a polynomial is the greatest monomial that is a factor of each term of the original polynomial.

Find the GCF for the coefficients first, then for each variable.

For more help with greatest common factors, see page 350.

3 Factor the following expression.

x3 + 125

A (x - 5) (x2 - 5x - 25)

B (x + 5) (x2 - 5x + 25)

C (x - 5) (x2 + 5x - 25)

D (x + 5) (x2 + 5x + 25)

READING HINT Factoring an expression results in a product of two polynomials.

Since 125 = 53, this expression is a sum of two cubes.

For more help with factoring a sum or difference of two cubes, see page 350.


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64x4 − 9y2

F (4x2 + 3y)(4x2 − 3y)

G (8x2 − 3y)(8x2 − 3y)

H (8x2 + 3y)(8x2 + 3y)

J (8x2 + 3y)(8x2 − 3y)

5 Completely factor 8x2 − 24x + 18.

A 2(2x − 3)2

B (2x − 9)(4x − 2)

C (4x − 6)(2x − 3)

D (8x − 9)(x − 2)


8m3 − n3

F (2m + n) (4m2 − 2mn − n2)

G (2m + n) (4m2 + 2mn − n2)

H (2m − n) (4m2 − 2mn + n2)

J (2m − n) (4m2 + 2mn + n2)

7 Which of the following expressions is not a factor of x3 − x2 − 6x?

A x

B x − 1

C x + 2

D x − 3


12x2 + 19x − 18

F (2x + 6)(6x − 3)

G (3x + 9)(4x − 2)

H (4x − 9)(3x + 2)

J (4x + 9)(3x − 2)

9 Which of the following expressions is considered PRIME?

A 22x2 − 18

B 8x3 − 1

C 5x2 − 28x − 12

D 4x2 + 9

10 Which of the following expressions is not a factor of 12x4 + 21x3 − 45x2?

F x2

G 3x − 5

H 3x + 3

J 4x − 5

11 Completely factor the following expression.

27x2 − 147

A 3(3x + 7)(3x + 7)

B 3(3x + 7)(3x − 7)

C 3(3x − 7)2

D prime

12 The area of a rectangular plot of land is given as 18x2 + 9x − 20. Which of the following shows expressions that are possible values for its length and width?

F (3x − 4) and (6x + 5)

G (3x + 4) and (6x - 5)

H (4x − 5) and (6x - 3)

J (4x + 5) and (6x + 3)

13 What is the greatest common factor of

15a4b5 - 30a3b2 + 45a2b2?

A 3ab

B 5a2b

C 5a2b2

D 15a2b2



Practice by StandardStandard 5.0: Students demonstrate knowledge of how real and complex numbers

are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.


1 If i = √ �� −1 , which point shows the location of 4 − 3i on the plane?

AB

y

xO

C

D

Imaginary

Rea

l

A Point A C Point C

B Point B D Point D

READING HINT An expression in the form a + bi is a complex number. a is called the real part and bi is the imaginary part. i is the imaginary unit.

Graph the real part on the real axis and the imaginary part on the imaginary axis.

For more help with complex numbers, see page 259.


2 Simplify √ �� −72 .

F −6 √ � 2

G 6i √ � 2

H −3 √ � 8

J −3i √ � 8

3 If i = √ �� −1 , what is the value of i6?

A 1

B −1

C i

D − i

4 Simplify √ �� −80x4 .

F x2i √ � 80

G 2x2i √ � 20

H 4x2i √ � 5

J 16x2i √ � 5

5 If i = √ �� −1 , what is the value of i47?

A 1

B −1

C i

D −i


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1 Simplify the following expression.(8 − 3i) + (6 + 5i)

A 16i

B 14 + 2i

C 14 + 8i

D 2 + 2i

STRATEGY Adding complex numbers is like adding any other binomials; that is, add like terms together.

Add the real part of each complex number to get the real part of the sum. Then add the imaginary part of each to get the imaginary part of the sum.

2 Simplify the following expression completely.

(3i)(−5i)(−2i)

F −30i

G 10i

H 30i

J 30i3

STRATEGY Multiply these complex numbers like you would multiply any variable terms, except the imaginary unit i should never have an exponent other than 1.

First multiply the coefficients; then multiply the imaginary unit i. Remember, a complex number is not simplified when i has an exponent other than 1. Since i = √ �� −1 , i2 = −1, and i3 = − i.

3 What is the product of the complex numbers (5 − 3i) and (5 + 3i)?

A −34

B 16

C 25 − 9i

D 34

STRATEGY To multiply complex numbers, use the FOIL method.

Two complex numbers of the form a + bi and a − bi are called conjugates. The product of two conjugates is a real number.

For more help with multiplying complex numbers, see page 263.

Practice by StandardStandard 6.0: Students add, subtract, multiply, and divide complex numbers.





4 What is the product of the complex numbers (8 + 3i) and (2 − 5i)?

F 16 − 15i

G 16 − 15i2

H 31

J 31 − 34i


(5 − 4i) − (2 − 7i)

A 3 − 11i

B 3 + 3i

C 7 − 11i

D 7 + 3i

6 Which of the following is an equivalent

form of 6_2 + 4i

?

F −1 − 2i

G 3_10

H 3_5

−24_5

i

J 3_5

−6_5

i


(15 − 9i) − (6 − 4i) + (11 + 2i)

A −2 − 11i

B 20 − 3i

C 20 − 11i

D 32 − 11i


14 − 3i_2 + 8i

F 1_17

−59i_34

G 4 − 118i

H 17 − 59i_34

J 68


(3i)3 � (2i)2

A 6i2

B 36i5

C 36i

D 108i

10 What is the product of the complex numbers (3 + i) and (4 − 6i)?

F 6 − 14i

G 6 − 22i

H 18 − 14i

J 18 − 22i

11 What is the product of the complex numbers (3 + 2i) and (3 − 2i)?

A 5

B 9

C 9 − 4i

D 13

12 The impedance in one part of a series circuit is 4 + 4j ohms, and the impedance in the other part of the circuit is 6 - 2j. Add these complex numbers to find the total impedance in the circuit.

F 3 − 10j

G 3 + 3j

H 10 − 3j

J 10 + 3j

13 What is 2 + 3i_4 - 2i

A 1_10

+4_5

i

B 4_5

+1_10

i

C 9_10

D 9_10

i


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1 A triangle has an area of 3x2 + 2x − 21. If the base of the triangle measures 3x − 7, which of the following is the correct expression for its height?

3x 7

hA 3x 2 2x 21

A x + 3

B 2x + 6

C 3x − 7

D 6x − 14

READING HINT The formula for the area of a

triangle is A = 1 _ 2 bh.

Substitute 3x − 7 for b and 3x2 + 2x − 21 for A in the formula above and solve for h. You will need to simplify a rational expression to get a proper value for h.

For more help with simplifying rational expressions, see page 442.

2 For what value(s) of x is the following expression undefined?

5x−7 __ 2x2 + 9x − 18

F −6

G −6 and 3 _ 2

H 0

J 6 and − 3 _ 2

STRATEGY Try to use the process of elimination to arrive at the correct answer.

A rational expression is undefined if the denominator is equal to zero. Either factor the denominator to find what values make it equal zero, or substitute the numbers in the choices to see if the denominator is zero.

For more help with finding undefined values in rational expressions, see page 443.

Practice by StandardStandard 7.0: Students add, subtract, multiply, divide, reduce, and evaluate rational

expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.




3 Which of the following is a simplified form

of 18x3 y2 z−3

_ (3xy−1 z2 )2

?

A 6x2 y3

_ z5

B 6xy4

_ z7

C 3xy4

_ z7

D2xy4 _

z7

4 x + 6_x + 4

+24_

x2 + x − 12=

F x2 + 3x − 18__x2 + x − 12

G x2 + 2x − 18__x2 + x − 12

H x2 + 3x_x2 + x − 12

J x2 + 3x + 6_x2 + x − 12

5 What is x2 − x − 6_

x − 3 ÷ x

2 − 4 _ x + 4

?

A x + 4_x − 2

B x − 2_x + 4

C x + 2

D x2 + 2x − 8

6 Simplify 24a3 b2 c_7a2 b−3

�42ac−2 _36ab3 c

.

F 4a2 c2 _b4

H 4a_b2 c2

G 4ab2 _c2

J 4ab2 c2

7 Simplify x2 + 6x − 7 _

x2 − 1 .

A x + 7_x + 1

C x + 7_x − 1

B x − 7_x + 1

D x + 1_x − 1

8 Simplify (8x2 yz)(2x)_

(3x3 y)−1 .

F 16z_3

G 48yz

H 48x9 y2 z

J 48x6 y2 z

9 For what value(s) of x is the following expression undefined?

x2 + 16__x2 + 20x + 75

A −15

B −15 and −5

C 0

D 5

10 A rectangle has an area of 4x2 + 22x + 10. If the length of the rectangle measures (4x + 2), which of the following is the correct expression for its width?

F (x + 5)

G (2x + 3)

H (2x + 5)

J (3x + 1)

11 x + 4_x – 6

+ x + 1_x2 – 5x – 6

=

A 2x + 5_x2 – 5x – 6

B x2 + 4x + 3_x2 – 5x – 6

C x2 + 6x + 5_x2 – 5x – 6

D x2 + 5x + 4_x2 – 5x – 6



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Practice by StandardStandard 8.0: Students solve and graph quadratic equations by factoring,

completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.


1 The base of a rectangle is 9 inches longer than its height. Its area is 136 square inches. What is its length?

A 8 inches

B 9 inches

C 15 inches

D 17 inches

READING HINT The formula for the area of a rectangle is A = bh.

Let b represent the base of the rectangle and let h = b − 9. The area equation then becomes 136 = b(b − 9). Solve.

For more help with solving quadratic equations, see page 253.

2 Two consecutive odd positive integers have a product of 323. Find the greater of the two.

F 15

G 17

H 19

J 21

STRATEGY Let n represent the first of the two consecutive odd positive integers. The second integer would then be n + 2.

The equation should then be n(n + 2) = 323. Solve for n, but remember that the question asks for the greater of the two integers.

For more help with quadratic equations, see page 253.

3 Kim needs to solve the following equation.

x2 − 14x = 32

What does she need to do first in order to complete the square?

A Add 196 to each side.

B Divide –14 by 2.

C Factor the left side of the equation.

D Subtract 32 from each side.

READING HINT Remember that completing the square is a useful method to solve quadratic equations when the coefficient of the linear term is an even number.

First make sure the coefficient of the quadratic term is 1. To complete the square, you must identify the coefficient b of the

linear term. Add the square of b _ 2 to each side

to complete the square.

For more help with completing the square, see page 268.



4 What are the solutions to the following equation?

x2 + 6x + 10 = 0

F x = 6 + 2i; x = 6 − 2i

G x = −6 + 2i; x = −6 − 2i

H x = 3 + i; x = 3 − i

J x = −3 + i; x = −3 − i

5 What are the roots of the quadratic equation graphed below?

A 1

B 1 and 5

C 3

D 3 and −4

6 Mallory has a rectangular in-ground pool in her backyard. She wants to have a concrete border of uniform width installed around it. The pool is 8 meters wide and 12 meters long and the area to be covered by the border is equal to the area of the pool itself. How wide is the concrete border?

F 2 m

G 2.5 m

H 3 m

J 4 m

7 Solve the following equation.

x2 − 4x + 1 = 0

A x = 2

B x = −2 + √�3 ; x = −2 − √�3

C x = 4 + 2 √�3 ; x = 4 − 2 √�3

D x = 2 + √�3 ; x = 2 − √�3

8 For which value of c will the roots of x2 − 10x + c = 0 be real and equal?

F 5

G 25

H 50

J 100

9 Jenny tells you that she is thinking of two numbers whose sum is 15 and whose product is 60. What are the numbers?

A 5 and 12

B 6 and 10

C 9 and 6

D No such real numbers exist.

10 What are the solutions to the following equation?

x2 + 4 = 4x

F x = 2

G x = 2; x = −2

H x = −2

J no solution

11 An object is shot straight upward into the air with an initial speed of 800 feet per second. The height h that the object will be after t seconds is given by the equation h = −16t2 + 800t. When will the object reach a height of 10,000 feet?

A 10 seconds C 100 seconds

B 25 seconds D 625 seconds

12 What are the roots of the quadratic equation 3x2 + x = 4?

F (−1, 4_3)

G (− 4_3

, 1)

H (−2, 2_3)

J (− 2_3

, 2)



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1 Which of the following statements most accurately describes the translation of the graph of y = (x − 5)2 + 1 to the graph of y = (x + 1)2?

A down 1 and left 5

B down 1 and left 6

C up 1 and right 6

D up 1 and right 5

READING HINT A translation slides a graph in any direction on the plane, but does not change its shape.

The vertex form of an equation for a parabola is y = a(x − b)2 + c. Changes to the values of b and c will translate the graph up/down and left/right.

For more help with graphs of parabolas, see page 286.

Practice by StandardStandard 9.0: Students demonstrate and explain the effect that changing a

coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x − b)2 + c.



2 Which of the following equations would produce the narrowest parabola?

F y = 5x2 + 5

G y = 3x2 + 1

H y = 0.25x2 − 2

J y = 0.1x2 − 7

3 What is the effect of changing the equation y = 5x2 − 2 to y = 5x2 + 1?

A translation 3 units right

B translation 3 units up

C second graph is narrower

D second graph is wider

4 Which of the following sentences is true about the graphs of the following equations?

y = 1 _ 2 (x + 3)2 + 4

y = 1 _ 2 (x + 3)2 − 4

F The graphs have different shapes and different vertices.

G Their vertices are maximums.

H The graphs have the same shape, but with different vertices.

J One parabola opens up, and the other opens down.



1 What are the x-intercepts of the graph of y = 2x2 + 9x − 35?

A 5 and 7

B −5 and −7

C 5 _ 2 and −7

D − 5 _ 2 and 7

READING HINT The x-intercepts of the graph of a quadratic equation are the zeros (or roots) of the equation.

Let y = 0 and find the values of x that satisfy the equation.

For more help with finding the roots of a quadratic equation, see page 246.

2 Find the maximum or minimum value of y = −x2 + 6x − 4.

F −4

G 4

H 5

J 23

STRATEGY When a quadratic equation is in the form y = ax2 + bx + c, the graph opens up if a > 0 and down if a < 0.

Since a < 0, this graph opens down and the function will have a maximum value, which is the y-coordinate of the vertex.

For more help with maximum and minimum values of quadratic equations, see page 236.

3 Jim is trying to graph a parabola and has put the equation into vertex form, y = 2(x − 3)2 + 6. He wants to begin by plotting the vertex. Which of the following ordered pairs should be the first point that he graphs?

A (−3, 6)

B (2, 6)

C (3, 6)

D (4, 8)

STRATEGY The vertex form of a parabola is y = a(x − h)2 + k. The vertex of the parabola is the ordered pair (h, k).

Be careful of the signs. Since the (x − h) part of the equation is (x − 3), h equals 3, not −3.

For more help with finding the vertex of a parabola, see page 236.

Practice by StandardStandard 10.0: Students graph quadratic functions and determine the

maxima, minima, and zeros of the function.



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4 Which equation is represented by the graph shown below?

F y = x2 − 4x + 5

G y = x2 + 4x + 5

H y = 2(x − 2)2 + 1

J None of the above

5 What are the roots of the function y = 6x2 + 11x + 4?

A − 4_3 and −

1_2

B 4_3 and 1_

2C 6 and 11

D 11 and 4

6 Which equation is represented by the graph shown below?

F y = −4x2

G y = −(x − 4)2

H y = −x2 + 4

J y = −(x + 4)2

7 Which of the following is the graph of the equation y = 2x2 − 4x − 6?

A

B

C

D



1 If log8 x = 2, then what is the value of x?

A x = 1 _ 64

B x = 1 _ 16

C x = 16

D x = 64

READING HINT In the function x = by, y is the logarithm, base b, of x. So y = logb x.

Using the above definition, y = logb x can be

rewritten as x = by.

For more help with logarithms, see page 509.

2 Which of the following equations is incorrect?

F log2 8 = 3

G log3 9 = 2

H log9 3 = 1 _ 2

J log9 3 = 2

STRATEGY Use the process of elimination to narrow your choices.

By using the definition of logarithm to test each of the four choices, you can determine which of the equations is incorrect.


3 What is the solution to the following equation?

3x = 8

A x =log10 8_log10 3

B x = log10 8 − log10 3

C x = 3 (log10 8)

D x =log10 3_ log10 8

STRATEGY Rewrite the exponential equation as a logarithmic equation.

Use the Change of Base Formula to convert from base 3 into base 10.

For more help with the Change of Base Formula, see page 530.

Practice by StandardStandard 11.1: Students understand the inverse relationship between

exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.



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4 Which of the following equations is equivalent to log5 100 = x?

F 5x = 100

G 1005 = x2

H x5 = 100

J x = 20

5 What is the solution to the following equation?

4x = 12A x = log12 4

B x =log10 4_

log10 12

C x =log10 12_log10 4

D x = log10 12 − log10 4

6 Which of the following equations is equivalent to the equation below?

16 1 _ 2 = x

F log16 1_2

= x

G logx1_2

= 16

H logx 16 = 1_2

J log16 x =1_2

7 If log9 x =3_2 , what is the value of x?

A 38.44

B 27

C 13.5

D 6

8 If log 2 x = −3, then what is the value of x?

F −6

G 1_8

H 8

J 9

9 Which of the following equations is incorrect?

A log2 4 = 2

B log2 16 = 4

C log4 8 = 2

D log4 16 = 2

10 If log3 x = 4, then what is the value of x?

F 64

G 81

H 12

J 7

11 What is the first step in solving log5 14 + log5 12 = ?

A use the product property of logarithms

B use the power property of logarithms

C use the change of Base Formula

D use the inverse property of exponents and logarithms

12 log3 12 =

F log12 10

_ log3 10

G log10 12

_ log10 3

H log10 3

_ log10 12

J log12 12

_ log3 3

13 What is the solution to the equation8x = 24?

A x = 3

B x = log10 3

C x = log10 24 – log10 8

D x = log10 24

_ log8 24



1 Anita, Brian, and Carlos are solving the equation log2 x = 8 and have all gotten different answers. Which, if any, of the three is correct?

A Ann

B Brian

C Carlos

D None are correct.

READING HINT If logb x = y , then by the definition of logarithm, x = by.

Decide whether any of the three correctly applied the definition of logarithm and solved the equation properly.


2 Which is the first incorrect step in simplifying the following rational expression?

Step 1: 4xy2 _

(−3x2y)−2 = (4xy2)(−3x2y)2

Step 2: = (4xy2)(−9x4y2)

Step 3: = −36x5y4

F Step 1

G Step 2

H Step 3

J Each step is correct.

STRATEGY Try to simplify it yourself and compare your work to what is shown.

Use the proper order of operations by applying the exponent before multiplying the expressions inside the parentheses.

For more help with simplifying rational expressions, see page 442.

Practice by StandardStandard 11.2: Students judge the validity of an argument according to

whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.


Ann

log2 x = 8

2x = 8

2x = 23

x = 3

Brian

log2 x = 8

x = 82

x = 64

Carlos

log2 x = 8

x = 28

x = 256


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3 Tim, Eli, Jordan, and Shannon are all working on the same math problem on the chalkboard. The teacher said that only one of them has the correct solution. Each student’s work is shown below.

Tim

(−2a2b)3 _

(3ab)−2

= (−2a2b)3(3ab)2

= (8a6b3)(9a2b2)

= 72a8b5

Eli

(−2a2b)3

_ (3ab)−2

= (−2a2b)3(3ab)2

= (−8a5b3)(9a2b2)

= −72a7b5

Jordan

(−2a2b)3 _

(3ab)−2

= (−2a2b)3(3ab)2

= (−8a6b3)(9a2b2)

= −72a8b5

Shannon

(−2a2b)3 _

(3ab)−2

= (−2a2b)3(3ab)2

= (−8a5b3)(9a2b2)

= −72a10b6

Which student simplified correctly?

A Tim

B Eli

C Jordan

D Shannon

4 Which is the first incorrect step in solving the following equation?

2x = 54

Step 1: x = log2 54

Step 2: x = log3 54

_ log3 2

Step 3: x = log3 27

Step 4: x = log3 (33)

Step 5: x = 3

F Step 2

G Step 3

H Step 4

J Step 5

5 Raheem and Ron are simplifying two different rational expressions. Who has simplified correctly?

Raheem

3x _

5y ·

10y3

_ (6x)2

= 3x _

5y ·

10y3

_ 6x2

= y2

_ x

Ron

5x − 15 _

−5x − 5 · x

2 + 2x + 1 _

4x − 12

= 5(x − 3) _

−5(x + 1) · (x + 1)(x + 1)

__ 4(x − 3)

= −x – 1

_ 4

A Raheem is correct.

B Ron is correct.

C Both are correct.

D Neither is correct.



1 Which of the following expressions is

equivalent to 8 √� 16 _ 6 √� 2

?

I. 3 √ � 2

II. √ � 8

III. 2 1 _ 3

A I only

B II only

C I and III only

D II and III only

STRATEGY First, you should convert the radical expressions into expressions with fractional exponents.

Then use the rules of simplifying expressions with fractional exponents to decide which of the three are equivalent to the given expression.

For more help with simplifying expressions with fractional exponents, see page 415.

2 The population of a certain city is growing exponentially as shown in the table below.

Year Population1980 90,0001990 113,0002000 141,800

You determine that the population is increasing by about 2.3% per year. Which of the following choices is the most reasonable estimate for the city’s population in the year 2015?

F 170,600

G 200,000

H 283,600

J 1.26 × 108

STRATEGY You should always look for any choices that can be eliminated because they are not reasonable. The population grew by 28,800 in the 10 years from 1990 to 2000. Choice F is 28,800 more than the population in the year 2000. Clearly, the population would have grown more in 15 years. Also, Choice J is 126 million, which is much too high to be correct.

Use the formula for exponential growth, y = a(1 + r)t, where a is the initial amount, r is the rate of growth, and t is the time frame (years in this case).

For more help with exponential growth and decay, see page 500.

Practice by StandardStandard 12.0: Students know the laws of fractional exponents, understand

exponential functions, and use these functions in problems involving exponential growth and decay.



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3 If the equation y = ( 1 _ 5 )

x is graphed, which

of the following values of x would produce a point closest to the x-axis?

A x = −4

B x = 0

C x = 1 _ 3

D x = 5 _ 3

4 A certain car depreciates at a rate of 12% per year. If you bought the car 6 years ago for $24,000, what is its value today?

F $11,146

G $12,666

H $17,280

J $21,120

5 A sample of a radioactive element decays to y grams over time according to the equation

y = a (1_2)

t _ 200

,

where a = the initial number of grams present and t = the amount of time in years. If, after 800 years, there are only 75 grams of the element remaining, how much was there originally?

A 1200 grams

B 450 grams

C 300 grams

D 150 grams

6 Which of the following expressions is

equivalent to 4 √ � 27 _

4 √ � 9

?

F 3 1 _ 4

G 3 1 _ 2

H 3 3 _ 4

J 3

7 Which of the following is a graph of the equation y = 3x?

A

B

C

D



1 Which of the following expressions is equivalent to log6 √ � 32 ?

A log10 32 − log10 6

B log10 32

_ 2 log10 6

C 6(log10 32)

D log10 32

_ log10 6

STRATEGY Use the Change of Base Formula to convert from a logarithm in one base to a logarithm in another base.

The Change of Base Formula states that

loga n = logb n

_ logb a

, where a ≠ 1 and b ≠ 1.

For more help with the Change of Base Formula, see page 530.

Practice by StandardStandard 13.0: Students use the definition of logarithms to translate between

logarithms in any base.



2 The expression log10 81

_ log10 3

is equivalent to

which of the following expressions?

I. log81 3

_ 10

II. log3 81

III. 4

F I only

G III only

H I and III only

J II and III only

3 The value of log4 15 is equivalent to which of the following?

A log10 15 − log10 4

B 415

C log10 15 + log10 4

D log10 15

_ log10 4

4 Solve the following equation.

log4 8 + log4 x = 3 log4 6

F 9

G 27

H 81

J 108

5 log8 18 =

A log10 18

_ log10 8

B log10 18 − log10 8

C (log10 8)(log10 18)

D log10 8 + log10 18


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Practice by StandardStandard 14.0: Students understand and use the properties of logarithms to

simplify logarithmic numeric expressions and to identify their approximate values.


1 What is the value of log4 64?

A 3

B 4

C 5

D 6

STRATEGY Try to rewrite the number 64 as the base of the logarithm to a power.

Replace 64 with 43 and simplify.

For more help with evaluating logarithms, see page 510.

2 Given that log2 5 ≈ 2.322, what is the approximate value of log2 40?

F 4.644

G 5.322

H 6.322

J 18.576

STRATEGY Use the Product Property of Logarithms to simplify the expression.

Since 40 can be rewritten as a product of 5 and a power of 2, you can use the Product Property of Logarithms to simplify the problem.

For more help with using the Product Property, see

3 Given that log9 15 ≈ 1.232 and that log9 12 ≈ 1.131, what is the

approximate value of log9 4 _ 5 ?

A 1.089

B 0.918

C 0.101

D −0.101

STRATEGY Rewrite the fraction using the values of 12 and 15 and then apply the Quotient Property of Logarithms to simplify the expression.

The Quotient Property states that

logb ( m _ n ) = logb m − logb n.

For more help with using the Quotient Property of Logarithms, see page 521.



4 Use the properties of logarithms to solve the following equation for x.

2 log 4 5 − 1 _ 3 log 4 8 = log 4 x

F x = 8

G x = 12.5

H x = 16

J x = 23


A 0

B 1

C 1_2

D 6

6 Use the properties of logarithms to solve the following equation for x.

log3 (x + 3) + log3 (x − 5) = 2

F x = 6

G x = −4

H x = 6; x = −4

J x = −6; x = 4

7 Given that log10 24 ≈ 1.380 and that log10 4 ≈ 0.602, solve the following equation for x.

4x = 24

A x ≈ 0.778

B x ≈ 0.831

C x ≈ 1.982

D x ≈ 2.292


F 3

G 4

H 5

J 6

9 Given that log5 3 ≈ 0.683 and that log5 7 ≈ 1.209, what is the approximate value of log5 189?

A 3.258

B 2.575

C 1.892

D 0.526

10 The slope m of a beach is related to the length in millimeters of the diameter d of the sand particles found on the beach by the following equation.

m = 0.159 + 0.118(log10 d)

If the value for log10 2 ≈ 0.301, find the approximate slope of a beach where the average sand particle has a diameter of1_8

millimeter.

F m ≈ 0.02

G m ≈ 0.05

H m ≈ 0.27

J m ≈ 0.30

11 What is the value of log41_2 ?

A 1_4

B − 1_8

C − 1_4

D − 1_2

12 If log3 ≈ 0.477 and log5 ≈ 0.699, what is the approximate value of log 45?

F 0.159

G 0.333

H 1.176

J 1.653



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1 Given the equation y = 2x and that the domain for x is any real number, which of the following statements is true about the possible real values of y?

I. y can be any real number except zero.

II. y > 0

III. y < 0

A I only

B II only

C III only

D None of these

READING HINT The domain is the set of possible x-values. This question is asking about the range, which is the set of possible y-values.

Try converting this exponential equation into a logarithmic equation to determine which of the three statements (if any) are true.

For more help with solving logarithmic equations, see page 512.

2 Jimmy found the following equation on his test.

8x − 2 _ 2 = 4x − 1

Which of the following statements is correct about this equation?

F The equation is true for all values of x.

G The equation is only true for x = 2.

H The equation is always true except

when x = 1 _ 4 .

J The equation is never true.

READING HINT The question is asking you to find any undefined values for x.

A rational expression is undefined if the denominator is equal to zero. Under what circumstances can the denominator in this equation be zero?

For more help with rational expressions and undefined values, see page 479.

Practice by StandardStandard 15.0: Students determine whether a specific algebraic statement involving

rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.




3 If x can be any real number, which statement best describes the possible values of y in the following equation?

y = |x| − 2

A y can be any real number.

B y ≥ 0

C y ≤ −2

D y ≥ −2

4 If x is a real number, for what values of x is the following equation true?

x2 − x − 6_x + 2

= x − 3

F all values of x

G some values of x

H no values of x

J impossible to determine

5 The following equation is given.

√��3x + 7 = x + 3

Larry solves the problem and writes on the board that x = −1. Which of the following statements best applies to Larry’s conclusion?

A He found the only solution.

B He found one of two solutions.

C He found an incorrect solution.

D It is not possible to determine if Larry is correct or not.

6 Which of the following statements is true about the domain and range of the equation y = log3 (x − 2)?

F Both the domain and range are all real numbers.

G The domain is all real numbers, but the range is y > 0.

H The range is all real numbers, but the domain is x > 2.

J The domain is x > 2, and the range is y > 0.

7 Which of the following statements is true about the following system of linear equations?

5x − 2y = 4

4y = 10x − 8

A It has one solution.

B It has no solution.

C It has infi nitely many solutions.

D Impossible to determine.

8 If x can be any real number in the equation

y = (1_4)

x+ 3, which of the following

statements is true?

F y can be any real number.

G y > 0

H y ≥ 3

J y > 3

9 Given the equation y =x2 − 3x − 28__ x2 + 2x − 63

, which

of the following statements are true?

A -4 is an asymptote.

B -9 is an asymptote.

C 7 is an asymptote.

D Both B and C are true.

10 Determine the validity of the equation

3 log5 ( x) =log10 x3_ log10 5

.

F false

G true for all values of x

H true for all positive values of x

J impossible to determine

11 Given the equation y =n√�x , where x > 0

and n > 0, which statement is valid for real values of y?

A y < 0 C y > 0

B y = 0 D y > 1



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1 The graph of x2 _

144 −

y2 _

25 = 1 is a

hyperbola. Which set of equations represents the asymptotes of the hyperbola’s graph?

A y = 5 _ 12

x; y = - 5 _ 12

x

B y = 12 _ 5 x; y = - 12 _

5 x

C y = 144 _ 25

x; y = - 144 _ 25

x

D y = 25 _ 144

x; y = - 25 _ 144

x

STRATEGY The standard form of the equation for a hyperbola with its center at the origin is

x2 _

a2 − y2

_ b2 = 1.

The equations for the asymptotes are

y = b _ a x and y = - b _ a x.

For more help with hyperbolas, see page 590.

Practice by StandardStandard 16.0: Students demonstrate and explain how the geometry of the graph of a

conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.



2 The graph of x2_

4+

y2_16

= 1 is an ellipse.

Which of the following sets of ordered pairs names the two foci of the graph of the ellipse?

F (0, 2 √�3 ) and (0, −2 √�3 )

G (2 √�3 , 0) and (−2 √�3 , 0)

H (0, 2 √�5 ) and (0, −2 √�5 )

J (2 √�5 , 0) and (−2 √�5 , 0)

3 What is the center of a circle with the following equation?

x2 + y2 − 6x + 8y − 11 = 0

A (−6, 8)

B (−3, 4)

C (3, −4)

D (6, −8)

4 The graph of x2 = 16y is a parabola. Which of the following ordered pairs represents the focus of the parabola?

F (0, 0)

G (0, −4)

H (0, 4)

J (4, 0)

5 Which equation represents the ellipse that has an eccentricity closest to zero?

A x2_9

+ y2_16

= 1

B x2_4

+ y2_16

= 1

C x2_20

+ y2_4

= 1

D x2_16

+ y2_36

= 1



1 The graph below is the graph for which of the following equations?

A 9x2 + 4y2 - 36x - 16y - 90 = 0

B 9x2 + 4y2 - 36x - 16y + 16 = 0

C 9x2 + 4y2 + 36x + 16y + 16 = 0

D 9x2 - 4y2 - 36x + 16y - 16 = 0

STRATEGY The graph shows an ellipse. Use the standard form for an ellipse to write the equation of the ellipse shown.

The standard form of the equation for a

vertical ellipse is y2

_ a2

+ x2 _

b2 = 1, where a is

one-half the length of the major axis and b is one-half the length of the minor axis.

For more help with ellipses, see page 581.

Practice by StandardStandard 17.0: Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0,

students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can graph the equation.



2 Which type of conic section is the graph of the following equation?

9x2 - y2 + 54x + 10y + 55 = 0

F Circle

G Ellipse

H Hyperbola

J Parabola

3 Which type of conic section is the graph of the following equation?

4x2 + 4y2 - 40x - 8y + 40 = 0

A Circle

B Ellipse

C Hyperbola

D Parabola


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1 One type of California automobile license plate contains a number, followed by three letters, followed by 3 more numbers as shown below.

9 XYZ 876California

How many different plates of this type can be issued, assuming that letters and numbers can be repeated on one plate?

A 10,000,000

B 17,576,000

C 78,624,000

D 175,760,000

STRATEGY Use the Fundamental Counting Principle to determine the total number of license plates that could be created.

The choices for the letters and numbers to be put on a license plate are independent events. The choice for one digit or letter has no bearing on the choices for the others.

For more help with using the Fundamental Counting Principle, see page 684.

2 Elliott is deciding on a password for his access to the Internet. He has decided that he will use the letters of his first name arranged in random order. How many different ways can the letters be arranged?

F 5040

G 2520

H 1260

J 630

STRATEGY Use the rules for permutations to calculate the number of possible arrangements of the letters of his name.

This example is a permutation because the order of the letters is an important factor. Keep in mind that since two letters are repeated, this will have an effect on your calculations.

For more help with permutations, see page 690.

Practice by StandardStandard 18.0: Students use the fundamental counting principles to compute

combinations and permutations.




3 A local pizzeria offers 5 different meat toppings and 6 different vegetable toppings. You have decided to get two vegetable toppings and one meat topping. How many different types of pizzas can you order?

A 60

B 75

C 120

D 150

4 There are 24 students serving on Student Council, 14 of which are girls and 10 are boys. A 5-person Spirit Committee is to be formed and the principal has decided to choose 3 girls and 2 boys from student council to serve on the committee. How many different ways could he assemble the committee?

F 840

G 3360

H 16,380

J 20,168

5 There are 12 runners in the final heat of the 1600-meter event at the track and field championships. In how many different ways can three of the runners finish to win the gold, silver, and bronze medals?

A 220

B 1000

C 1320

D 1728

6 A standard deck of playing cards has 52 cards of varying rank and suit. No two cards are exactly alike. How many different 5-card hands could be dealt?

F 260

G 2,598,960

H 12,994,800

J 311,875,200

7 The cafeteria at Billy’s school offers 2 types of salads, 5 main dishes, 3 side dishes, and 3 desserts. How many different meals could Billy create if he chooses one salad, one main dish, one side dish, and one dessert?

A 13

B 60

C 90

D 8640

8 Of the 18 people who have tried out for the academic team, six will be chosen to represent the school. How many different teams could be assembled?

F 108

G 360

H 720

J 18,564

9 In how many ways can a president and vice president be chosen from a class with 80 students?

A 160

B 3160

C 6320

D 6400

10 In how many ways can an 8-character pass code be created if the first 4 characters are numbers that can be repeated and the last 4 characters are letters that cannot be repeated?

F 3,588,000,000

G 2,679,487,000

H 1,808,352,000

J 1,230,951,000



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1 Krista and Robyn are playing a game that uses the spinner shown below.

To win the game, Krista needs to spin at least a 7 on her next turn. What is the probability that she will do so?

A 1 _ 2

B 3 _ 8

C 1 _ 3

D 1 _ 4

READING HINT Probability is the ratio of successes to the sum of all successes and failures possible. This ratio may be either expressed as a fraction or a percent.

For Krista to win on her next turn, she must spin a 7 or an 8, so there are 2 successful possibilities and 8 possible outcomes.

For more help with probability, see page 697.

2 There are 11 people on the board of directors of your company. Four of the 11 are women, and the other 7 are men. A subcommittee of 2 is to be chosen at random to study corporate expansion. What is the probability that the two selected will both be women?

F P ≈ 5.5%

G P ≈ 10.9%

H P ≈ 18.2%

J P ≈ 36.4%

STRATEGY Use the rules for combinations to calculate the number of successes and the total number of possible combinations.

There are C(4, 2) combinations of choosing 2 of the 4 women. There are C(11, 2) ways of choosing 2 of the 11 on the board. Set up the ratio and convert it into a percentage rounded to the nearest tenth.

For more help with combinations, see page 690.

Practice by StandardStandard 19.0: Students use combinations and permutations to compute

probabilities.




3 The five letters of the name JENNY are to be randomly chosen one at a time. What is the probability that the letters will be chosen in alphabetical order?

A P = 20%

B P ≈ 8.33%

C P ≈ 1.67%

D P ≈ 0.83%

4 There are 12 songs on your MP3 player. The player is set to shuffle the songs to play them in random order without repeating any. What is the probability that any two of the four country songs will be the first two played?

F P ≈ 5.56%

G P ≈ 8.33%

H P ≈ 9.09%

J P ≈ 16.67%

5 Number cubes (or dice) each have six sides with each side marked with a number 1 through 6 (indicated with dots). If you toss two dice, what is the probability that the sum of the numbers showing face up will be 7 or 11?

A 1_36

C 1_6

B 1_18

D 2_9

6 A standard deck of playing cards has 52 cards of varying rank and suit. No two cards are exactly alike. What is the probability that the first randomly chosen card will be a spade AND the second will be a heart if the second card is chosen without the first card being replaced?

F P ≈ 5.8%

G P = 6.25%

H P ≈ 6.4%

J P = 50%

7 There are 8 students on the Academic Team, but only three of the students compete in any one match. If the students are randomly assigned to compete, what is the probability that two of the three chosen for the upcoming match will be Susan and her friend Carlos?

A P = 18.75%

B P ≈ 10.71%

C P ≈ 8.33%

D P ≈ 5.36%

8 Two cards are chosen in succession from a standard 52-card deck of playing cards. What is the probability that both will be face cards (either a jack, queen, or king in rank)?

F 11_221

G 3_13

H 12_169

J 33_676

9 Having a royal flush in poker is having the ace, king, queen, jack, and 10 all of the same suit. What is the probability of getting a royal flush in a standard 52-card deck?

A 4_2,598,960

C 2_2,598,960

B 3_2,598,960

D 1_2,598,960

10 In a 45-ball lottery, 6 balls are drawn. A person wins if they match all 6 numbers in any order. What is the probability of winning the lottery?

F 6_45

H 1_75,624,840

G 1_8,145,060

J 1__5,864,443,200



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1 Expand (2x + 5)4.

A 16x4 + 160x3 + 1200x2 + 1000x + 625

B 16x4 + 160x3 + 600x2 + 1000x + 625

C 16x4 + 40x3 + 100x2 + 250x + 625

D 16x4 + 40x3 + 1200x2 + 250x + 625

STRATEGY Use the Binomial Theorem to raise a binomial expression to a positive integer power.

The Binomial Theorem states that the binomial expansion of (a + b)n = nC0anb0 + nC1an − 1b1 + nC2an − 2b2 + ... + nCna0bn. In this example, a = 2x, b = 5, and n = 4.

For more help with using the Binomial Theorem, see page 664.

Practice by StandardStandard 20.0: Students know the binomial theorem and use it to expand binomial

expressions that are raised to positive integer powers.



2 Expand (x − y)3.

F x3 − 3x2y − 3xy2 − y3

G x3 − 3x2y − 3xy2 + y3

H x3 + 3x2y − 3xy2 + y3

J x3 − 3x2y + 3xy2 − y3

3 What is the coefficient of x3 in the binomial expansion of (2x − 1)8?

A −448 B −112

C 112 D 448

4 What is the coefficient of x5 in the binomial expansion of (3x + 5)7?

F 1575 G 5103

H 6075 J 127,575

5 Expand (x − 2)5.

A x5 − 10x4 + 40x3 − 80x2 + 80x − 32

B x5 − 5x4 + 10x3 − 10x2 + 5x − 32

C x5 + 10x4 + 40x3 + 80x2 + 80x + 32

D x5 − 10x4 + 20x3 − 20x2 + 10x − 32

6 The polynomial x6 − 12x5 + 60x4 − 128x3 + 240x2 − 192x + 64 is the expansion of which binomial expression?

F (x + 2)6

G (x – 2)6

H (x – 4)6

J (x + 4)6



1 Is the following statement true for all positive integers n?

4 n - 1 is divisible by 3.

A yes, true for all positive integers n

B no, since 48 = 65,536

C no, since 43 = 64

D impossible to tell

STRATEGY Try to use the method of mathematical induction to prove that the statement is true. If you are unable to do so, decide on a counterexample to prove that it is false.

The first step in using mathematical induction is to prove that the statement is true for some integer n. Since the statement is true when n = 1, Step 1 is complete. The next step is to assume that the statement is true for some positive integer k. The final step is to use this assumption to prove that the statement is true for the next integer k + 1.

For more help with mathematical induction, see page 670.

Practice by StandardStandard 21.0: Students apply the method of mathematical induction to prove

general statements about the positive integers.



2 Which of the following values of n is a counterexample for the following statement?

4n2 + 4n - 1 is a prime number.

F n = 6

G n = 7

H n = 8

J There is no counterexample because the statement is always true.

3 Is the following statement true for all positive integers n?

7n - 1 is divisible by 8.

A yes, true for all positive integers n

B no, n = 2 is a counterexample

C no, n = 3 is a counterexample

D impossible to tell


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1 What is the next term in the following geometric sequence?

8, −12, 18, −27, ____

A −40.5

B −36

C 40.5

D 36

READING HINT A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a constant r, which is called the common ratio.

To find the common ratio for this geometric sequence, set up the ratios of one term to the previous term and reduce. If the sequence is indeed a geometric sequence, they should all reduce to the same fraction. That fraction is the common ratio.

For more help with geometric sequences, see page 636.

Practice by StandardStandard 22.0: Students find the general term and the sums of arithmetic

series and of both finite and infinite geometric series.



2 What is the sum of the following geometric sequence?

4 + 2 + 1 + 1 _ 2 + 1 _

4 +...

F 8

G 12

H 16

J The sum is infi nite.

3 Write an equation for the nth term of the following geometric sequence.

4, 20, 100, 500, ...

A an = 5 (4) n − 1

B an = 5 (2) n − 1

C an = 4 (5) n − 1

D an = 2 (5) n − 1

4 What is the sum of the infinite geometric series?

1 _ 3 + 1 _

6 + 1 _

12 + 1 _

24 +...

F 1 _ 3 H 2 _

3

G 1 _ 2 J 3 _

2

5 What is the next term in the following arithmetic sequence

-8, -2, 4, 10, 16, ...?

A 20

B 22

C 24

D 32



1 Write an equation for the nth term of the following arithmetic sequence.

1, 5, 9, 13, ...

A an = n + 4

B an = 1 + 4(n + 1)

C an = −3 + 4n

D none of the above

READING HINT An arithmetic sequence is a sequence in which each term after the first is found by adding a constant to the previous term. This constant is called the common difference.

The formula for an arithmetic sequence is an = a1 + (n − 1)d , where a1 is the first term and d is the common difference. Find the common difference d by subtracting two consecutive terms.

For more help with arithmetic sequences, see page 622.

Practice by StandardStandard 23.0: Students derive the summation formulas for arithmetic series

and for both finite and infinite geometric series.



2 What is the sum of the first 10 terms of the following sequence?

2, 6, 18, 54, ...

F 6560

G 19,682

H 59,048

J 59,050

3 What is the sum of the first 40 terms of the sequence 8, 14, 20, 26, ...?

A 125

B 500

C 5000

D 10,000

4 The table shows the cost of renting a video game for one, two, three, or four nights. If the sequence continues, how much would it cost to rent a video game for 9 nights?

Nights Cost ($)1 1.502 33 4.504 6

F $9

G $10.50

H $12

J $13.50


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1 Given: f(x) = 3x−1 and g(x) = 9x − 3. Which of the following expressions would represent [f g](x)?

A 27x2 − 9

B 18

C 27 _ x − 3

D 1 _ 3x − 1

READING HINT In function notation, [f g](x) denotes the composition of functions.

In a composition of functions, a function is evaluated, and then a second function is evaluated on the result of the first function.

For more help with composition of functions, see page 385.

Practice by StandardStandard 24.0: Students solve problems involving functional concepts, such as

composition, defining the inverse function, and performing arithmetic operations on functions.



2 Which of the following functions is a linear function?

F f(x) = 9x2 + 3x − 7

G f(x) = |x| − 3

H f(x) = 1_x

J f(x) = − 11 + 2x

3 Given: f(x) = 3x and g(x) = x + 2. If h(x) = (f − g)(x), which of the following would represent h(x)?

A h(x) = 2x − 2

B h(x) = 2x + 2

C h(x) = 4x − 2

D h(x) = 4x + 2

4 Which of the following functions is the

inverse function for f(x) = 3x − 2_

4 ?

F g(x) = 4_3

x +2_3

G g(x) = 4_3x − 2

H g(x) = 4_3

x − 2

J g(x) = 4_3x + 2

5 Given: f(x) = 2x2 + 8 and g(x) = 3x2 − 4.Which of the following expressions would represent [g f ](x)?

A 12x4 + 96x2 + 196

B 12x4 + 96x2 + 188

C 12x4 + 188

D 18x4 − 48x2 + 40



1 If f(x) = 9x2 + 2 and g(x) = −2x + 6, what is the value of [f g](5)?

A −460

B −448

C 146

D 2306

READING HINT In function notation, [f g](x) denotes the composition of functions.

In a composition of functions, a function is evaluated, and then a second function is evaluated on the result of the first function.

For more help with composition of functions, see page 385.

Practice by StandardStandard 25.0: Students use properties from number systems to justify steps in

combining and simplifying functions.



2 If f(x) = 8 − 2x2 and g(x) = 1 _ 3 x + 2, what is

the value of [f g](12)?

F − 562 _

3

G −136

H − 274 _

3

J −64

3 If f(x) = 3x2 + 1, which expression would be f(y − 3)?

A 3y2 − 8

B 3y2 + 10

C 3y2 − 18y + 28

D 3y2 − 18y − 8

4 If f(x) = x2 + 2x – 1 and g(x) = 3x – x2, which is an equivalent form of f(x) + g(x)?

F 5x − 1 H 4x2 + x − 1

G 2x2 + 5x − 1 J 4x2 + 5x − 1

5 If f(x) = 4x2 − 1 and g(x) = 2x + 1, which of the following expressions would

represent ( f _ g ) (x)?

A 2x − 1; x ≠ 1 _ 2

B 2x − 1; x ≠ − 1 _ 2

C 2x + 1; x ≠ 1 _ 2

D 2x + 1; x ≠ − 1 _ 2

6 Given: f(x) = 3x + 1 and g(x) = 9x + 4. Which is an equivalent form of f(x) – g(x)?

F 12x − 3

G 12x + 3

H –6x − 3

J –6x + 3


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