Name: ________________________ Class: ___________________ Date: __________ ID: A

1

Final Exam Review

Short Answer

1. Use x = −3, –2, 0, 1, 2 to graph the function f x( ) = 2x. Then graph its inverse. Describe the domain and

range of the inverse function.

2. Graph the inverse of the relation. Identify the domain and range of the inverse.

x −1 1 3 5 7

y 4 2 1 0 1

3. Tell whether the function y = 4 3( )x shows growth or decay. Then graph the function.

4. Write the exponential equation 33= 27 in logarithmic form.

5. Evaluate log 0.0001 by using mental math.

6. Simplify the expression log5125.

7. A initial investment of $10,000 grows at 11% per year. What function represents the value of the investment after t years?

8. Write the logarithmic equation log99 = 1 in exponential from.

9. Nadav invests $6,000 in an account that earns 5% interest compounded continuously. What is the total amount of her investment after 8 years? Round your answer to the nearest cent.

10. Simplify log7 x3

− log7 x .

11. Solve 3x + 4= 9x

.

12. The amount of money in a bank account can be expressed by the exponential equation A = 300(1.005)12t

where A is the amount in dollars and t is the time in years. About how many years will it take for the amount in the account to be more than $900?

Name: ________________________ ID: A

2

13. Graph f(x) = −ex

− 3.

14. Solve 32x= 6561.

15. Solve log5x11

− log5x4

= 33.

16. Simplify lne−3x

.

17. Multiply 8x

4y

2

3z3

⋅9xy

2z

6

4y4

. Assume that all expressions are defined.

18. Find the least common multiple for 8(x + 1)2 (x − 4)2 and 14(x + 1)8 (x − 4)6

.

19. The number of lawns l that a volunteer can mow in a day varies inversely with the number of shrubs s that need to be pruned that day. If the volunteer can prune 6 shrubs and mow 8 lawns in one day, then how many lawns can be mowed if there are only 3 shrubs to be pruned?

20. Distance that sound travels through air d varies directly as time t, and d = 1,675 ft when t = 5 s. Find t when d = 8,375 ft.

21. Simplify 10 − x

2− 3x

x2

− x − 2. Identify any x-values for which the expression is undefined.

22. Subtract −6x

2− 3x + 5

2x2

+ 18−

4x2

− 3

2x2

+ 18. Identify any x-values for which the expression is undefined.

23. Add x + 6

x − 5+

−8x − 26

x2

− 4x − 5. Identify any x-values for which the expression is undefined.

24. Simplify

−1

x − 4+

x − 6

8

x − 5

x − 4

. Assume that all expressions are defined.

25. Identify the zeros and vertical asymptotes of g(x) =x

2+ 5x + 4

x + 2. Then graph.

26. Identify the zeros and asymptotes of f(x) =3x

2− 12

x2

− 16. Then graph.

27. Identify holes in the graph of f x( ) =x

2+ 8x + 12

x + 2. Then graph.

Name: ________________________ ID: A

3

28. Identify the asymptotes, domain, and range of the function g(x) =1

x + 2− 3.

29. Solve the equation x − 2 =3

x.

30. Solve the equation 3x

x − 2=

x + 4

x − 2.

31. Solve the equation 2x

x2

− 7x − 18=

6x

x2

+ x − 2.

32. Solve x

x − 6≥ −1 algebraically.

33. Graph the function f(x) = 5 x + 33, and identify its domain and range.

34. Using the graph of f x( ) = x as a guide, describe the transformation and graph g x( ) = 2 x + 5 .

35. Simplify the expression 81x124

. Assume that all variables are positive.

36. Write the expression 16

3

4 in radical form, and simplify. Round to the nearest whole number if necessary.

37. The function g is a translation 4 units right and 7 units down of f(x) = x + 7 . Write the function g(x).

38. Simplify the expression (27)

1

3⋅ (27)

2

3.

39. Solve the equation −5 + x − 3 = 5.

40. Solve 9x = 2 x + 5 .

41. Solve 2x + 8( )1

2 = x.

42. Solve 5x − 4 ≤ 8.

Name: ________________________ ID: A

4

43. An experiment consists of spinning a spinner. The table shows the results. Find the experimental probability

that the spinner does not land on orange. Express your answer as a fraction in simplest form.

Outcome Frequency

orange 8

purple 5

yellow 9

44. A person is selected at random. What is the probability that the person was not born on a Monday? Express your answer as a percent. If necessary, round your answer to the nearest tenth of a percent.

45. The table shows the distribution of the labor force in the United States in the year 2000. Suppose that a worker is selected at random. Find the probability that a female works in the Industry field. Express your answer as a decimal, and round to the nearest thousandth.

Agriculture Industry Services

Male 3,132,000 25,056,000 50,112,000

Female 667,000 8,004,000 57,362,000

46. A grab bag contains 6 football cards and 4 basketball cards. An experiment consists of taking one card out of the bag, replacing it, and then selecting another card. Determine whether the events are independent or dependent. What is the probability of selecting a football card and then a basketball card? Express your answer as a decimal.

47. A poll of 50 senior citizens in a retirement community asked about the types of electronic communication they used.The table shows the joint and marginal frequencies from the poll results.If you are given that one of the people polled uses text messaging, what is the probability that the person is also using e-mail? Express your answer as a decimal. If necessary, round your answer to the nearest hundredth.

Uses text messaging

Uses e − mail

Yes No Total

Yes 0.16 0.64 0.8

No 0.08 0.12 0.2

Total 0.24 0.76 1

48. Joyce asked 100 randomly-selected students at her school whether they have one or more brothers or sisters.

The table shows the results of Joyce’s poll.Make a table of the joint and marginal relative frequencies. Express percentages in decimal form.

Brother(s) No Brothers

Sister(s) 18 32

No Sisters 31 19

Name: ________________________ ID: A

5

49. Joel owns 12 shirts and is selecting the ones he will wear to school next week. How many different ways can

Joel choose a group of 5 shirts? (Note that he will not wear the same shirt more than once during the week.)

50. There are 5 singers competing at a talent show. In how many different ways can the singers appear?

51. Find the probability of rolling a 5 or an odd number on a number cube. Express your answer as a fraction in

simplest form.

52. An experiment consists of rolling a number cube. What is the probability of rolling a number greater than 4?

Express your answer as a fraction in simplest form.

53. A factory produces nails whose lengths have a mean of 2 inches and a standard deviation of 0.05 inches.

Lengths of 18 nails are shown. Do the data appear to be normally distributed? Explain.Nail Lengths (inches)

2.01 2.06 2.01 2.07 1.99 2.11

1.99 1.96 1.93 1.93 2.04 1.98

2.03 1.97 1.98 2.01 1.94 2.02

54. Constellations are made up of more than one star. The table shows the number of stars that make up various

constellations. Find the mean, median, and mode of the data set.

Constellation Number Number of Stars in Constellation

Constellation 1 29

Constellation 2 23

Constellation 3 22

Constellation 4 29

Constellation 5 21

55. The table shows the probability distribution for the number of people who contract a disease in a scientific study. Find the expected number of people who contract the disease. Round your answer to the nearest tenth.

Number of People 2 3 4 5 6

Probability 0.20 0.32 0.288 0.1536 0.0384

56. The data {5, 1, 0, 4, 0} represent a random sample of the number of days absent from school for five students

at Monta Vista High. Find the mean and the standard deviation of the data.

57. The heights of adult males in the United States are approximately normally distributed. The mean height is

70 inches (5 feet 10 inches) and the standard deviation is 3 inches.

Use the table to estimate the probability that a randomly-selected male is more than 71.5 inches tall. Express

your answer as a decimal.

Name: ________________________ ID: A

6

58. Make a box-and-whisker plot of the data. Find the interquartile range.

7,9,11,12,13,15,12,17,18,12,9,7,12,15,18,10

59. At a school carnival, you can win tickets to trade for prizes. A particular game has 5 possible outcomes. What

is the expected number of tickets won?

Tickets won 17 30 45 71 90

Probability 0.32 0.27 0.18 0.15 0.08

60. The heights of adult males in the United States are approximately normally distributed. The mean height is

70 inches (5 feet 10 inches) and the standard deviation is 3 inches.

Use the table to estimate the probability that a randomly-selected male is between 70 and 74.5 inches tall.

Express your answer as a decimal.

ID: A

1

Final Exam Review

Answer Section

SHORT ANSWER

1.

The domain of f−1

x( ) is x x > 0|ÏÌÓ

ÔÔÔÔ

¸˝˛

ÔÔÔÔ , and the range is all real numbers.

Graph f x( ) = 2x using a table of values.

x –3 –2 0 1 2

f x( ) = 2x 1

8

1

41 2 4

To graph the inverse f−1

x( ) = log2 x , reverse each ordered pair.

x1

8

1

41 2 4

f−1

x( ) = log2 x–3 –2 0 1 2

The domain of f−1

x( ) is x x > 0|ÏÌÓ

ÔÔÔÔ

¸˝˛

ÔÔÔÔ and the range is all real numbers.

ID: A

2

2.

Domain: {x | 0 ≤ x ≤ 4};

Range: {y | − 1 ≤ y ≤ 7}

For the inverse relation, switch the x and y-values in each ordered pair.

x 4 2 1 0 1

y −1 1 3 5 7

Graph each point and connect the points.The inverse is the reflection of the original relation across the line y = x .

Domain: {x | 0 ≤ x ≤ 4}; Range: {y | − 1 ≤ y ≤ 7}

ID: A

3

3. This is an exponential growth function.

Step 1 Find the value of the base: 3.The base is greater than 1. So, this is an exponential growth function.

Step 2 Choose several values of x and generate ordered pairs. Then, graph the ordered pairs and connect with a smooth curve.

4. log327 = 3

The base of the exponent becomes the base of the logarithm.The exponent is the logarithm.

33= 27 becomes log327 = 3.

5. –4

10?= 0.0001 The log is the exponent.

10−4= 0.0001 Think: What power of the base is the number?

log 0.0001 = –4

6. 3

Factor 125. Then write it in the form of 53, and apply the Inverse Properties of Logarithms and Exponents.

7. f(t) = 10000(1.11) t

The investment follows an exponential growth of 11% per year with an initial value of $10,000. Using the

formula f(t) = P(1 + r) t, substitute the given values.

f(t) = 10000(1 + 11%) t

f(t) = 10000(1 + 0.11) t

f(t) = 10000(1.11) t

8. 91= 9

Logarithmic form: log99 = 1

Exponential form: 91= 9

The base of the logarithm becomes the base of the power, and the logarithm is the exponent.

ID: A

4

9. $8950.95

A = Pert Substitute 6,000 for P, 0.05 for r, and 8 for t.

A = 6000e0.05 8( ) Use the [e^x] key on a calculator.

A ≈ 8950.95

The total amount after 8 years is $8950.95.

10. 2log7x

log7 x3

− log7 x

= 3log7x − log7 x Use the Power Property of Logarithms.

= 2log7x Simplify.

11. x = 4

3ÊËÁÁ

ˆ¯˜̃

x + 4

= 32ÊËÁÁÁ

ˆ¯˜̃̃

x

Rewrite each side as powers of the same base.

3x + 4= 32x To raise a power to a power, multiply the exponents.

x + 4 = 2x The bases are the same, so the exponents must be equal.x = 4

The solution is x = 4.

12. 19 years

900 = 300(1.005)12t Write 900 for the amount.

3 = 1.00512t Divide both sides by 300.

log3 = log1.00512t Take the log of both sides.

log3 = (12t) log1.005 Use the Power Property.log 3

12 log(1.005)= t Divide by 12log(1.005).

t = 18.36 Evaluate with a calculator.

t ≈ 19 years Round to the next year.

ID: A

5

13.

Make a table. Because e is irrational, round the values.

x −3 −2 −1 0 1 2 3

f(x) = −ex

− 3 –3.05 –3.14 –3.37 –4 –5.72 –10.39 –23.09

ID: A

6

14. x = 4

Use a graphing calculator. Enter 32x as Y1 and 6561 as Y2. Use the table to locate the value of x where Y1 =

Y2.

X Y1 Y2

1 9 6561

2 81 6561

3 729 6561

4 6561 6561

5 59049 6561

←

The graph shows x = 4 as the point of intersection of Y1 and Y2.

15. x = 5

33

7

log5

x11

x4

= 33 Apply the Quotient Property.

log5 x7

= 33 Simplify.

7log5x = 33 Use the Power Property.

log5x =33

7Divide.

5log 5 x

= 5

33

7 Use 5 as the base for both sides.

x = 5

33

7 Use inverse properties.

16. –3x

lne−3x

= −3x lne = −3x(1) = −3x

ID: A

7

17. 6x5z

3

Arrange the expressions so like terms are together: 8 ⋅ 9(x4

⋅ x)(y2⋅ y

2 )z6

3 ⋅ 4 ⋅ z3y

4.

Multiply the numerators and denominators, remembering to add exponents when multiplying: 72x

5y

4z

6

12z3y

4.

Divide, remembering to subtract exponents: 6x5y

0z

3.

Since y0

= 1, this expression simplifies to 6x5z

3.

18. 56(x + 1)8 (x − 4)6

List the factors for each polynomial.

8(x + 1)2 (x − 4)2 = 4 ⋅ 2 ⋅ (x + 1)2

⋅ (x − 4)2 and 14(x + 1)8 (x − 4)6

= 7 ⋅ 2 ⋅ (x + 1)8⋅ (x − 4)6

If the polynomials have common factors, use the highest power of each common factor.

The LCM is 4 ⋅ 7 ⋅ 2 ⋅ (x + 1)8(x − 4)6 = 56(x + 1)8 (x − 4)6

.

19. 16 lawns

One method is to use s1 l1 = s2 l2 .

(6)(8) = (3)l2 Substitute given values.

48 = 3l2 Simplify.

16 = l2 Divide.

20. 25 sec

r = 335 ft per sec Find the constant of variation r.

d = 335t Write the direct variation function.8,375 = 335t Substitute.t = 25 Solve.

It would take 25 seconds for sound to travel 8,375 feet.

21. −x − 5

x + 1; The expression is undefined at x = 2 and x = −1.

−1(x2+ 3x − 10)

x2

− x − 2Factor −1 from the numerator and reorder the terms.

= −1(x + 5)(x − 2)

(x + 1)(x − 2)Factor the numerator and denominator.

= −x − 5

x + 1Divide the common factors and simplify.

The expression is undefined at those x-values, 2 and −1, that make the original denominator 0.

ID: A

8

22. −10x

2− 3x + 8

2x2

+ 18; The expression is always defined.

−6x2

− 3x + 5

2x2

+ 18−

4x2

− 3

2x2

+ 18

= −6x

2− 3x + 5 − 4x

2+ 3

2x2

+ 18Subtract the numerators. Distribute the negative sign.

= −10x

2− 3x + 8

2x2

+ 18Combine like terms.

There is no real value of x for which 2x2 + 18 = 0; the expression is always defined.

23. x + 4

x + 1;

The expression is undefined at x = −1.

x + 6

x − 5+

−8x − 26

(x + 1)(x − 5)Factor the denominators. The LCD is (x + 1)(x − 5).

= x + 1

x + 1

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

x + 6

x − 5+

−8x − 26

(x + 1)(x − 5)Multiply by

x + 1

x + 1

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ .

= x

2+ 7x + 6

(x + 1)(x − 5)+

−8x − 26

(x + 1)(x − 5)

= x

2− x − 20

(x + 1)(x − 5)Add the numerators.

= (x + 4)(x − 5)

(x + 1)(x − 5)Factor the numerator.

= x + 4

x + 1Divide the common factor.

To determine where the expression is undefined, solve for x + 1 = 0.

ID: A

9

24. x

2− 10x + 16

8 x − 5( )

Method 1 Write the complex fraction as division.

−1

x − 4+

x − 6

8

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ ÷

x − 5

x − 4Divide.

= −1

x − 4+

x − 6

8

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ ⋅

x − 4

x − 5Multiply by the reciprocal.

= −1

x − 4

8

8

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ +

x − 6

8

x − 4

x − 4

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃̃ ⋅

x − 4

x − 5Add by finding the LCD.

= x

2− 10x + 16

8 x − 4( )⋅

x − 4

x − 5The common factor (x − 4) cancels.

= x

2− 10x + 16

8 x − 5( )or

x2

− 10x + 16

8x − 40Simplify.

Method 2 Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator.

−1

x − 48( ) x − 4( ) +

x − 6

88( ) x − 4( )

x − 5

x − 48( ) x − 4( )

The LCD is 8 x − 4( ).

= −1 8( ) + x − 6( ) x − 4( )

8 x − 5( )Cancel common factors.

= x

2− 10x + 16

8 x − 5( )or

x2

− 10x + 16

8x − 40Simplify.

ID: A

10

25. Zeros at −4 and −1.

Vertical asymptote: x = −2

Factor the numerator.

g(x) =(x − (−4))(x − (−1))

x − (−2)

The zeros are the values that make the numerator zero, x = −4 and x = −1.The vertical asymptote is where the denominator is zero, x = −2.Plot the zeros and draw the asymptote, then make a table of values to fill in missing points.

ID: A

11

26. Zeros: −2 and 2

Vertical asymptotes: x = −4, x = 4Horizontal asymptote: y = 3

f(x) =3 x + 2( ) x − 2( )

x + 4( ) x − 4( )Factor the numerator and denominator.

Zeros: −2 and 2 The numerator is 0 when x = −2 or x = 2.

Vertical asymptotes: x = −4, x = 4 The denominator is 0 when x = −4 or x = 4.

Horizontal asymptote: y = 3 Both p and q have the same degree: 2. The horizontal asymptote is

y =leading coefficient of p

leading coefficient of q=

3

1= 3

.

ID: A

12

27. There is a hole in the graph at x = −2.

f x( ) =x

2+ 8x + 12

x + 2

=x + 2( ) x + 6( )

x + 2

Factor the numerator. x + 2 is a factor in both the numerator and the denominator, so there is a hole at x = −2.

= x + 6 Divide out common factors.Except for the hole at x = −2, the graph of f is the same as y = x + 6. On the graph, indicate the hole with an

open circle. The domain of f is x x ≠ −2|ÏÌÓ

ÔÔÔÔ

¸˝˛

ÔÔÔÔ .

28. Vertical asymptote: x = −2Domain: {x x ≠ −2| }

Horizontal asymptote: y = −3

Range: {y y ≠ −3|| }

Write the function in the form g(x) =1

x − h+ k where x = h is the vertical asymptote and helps find the

domain, and y = k is the horizontal asymptote and helps find the range.

g(x) =1

x − (2)− 3, so h = −2 and k = −3.

Vertical asymptote: x = −2Domain: {x x ≠ −2| }

Horizontal asymptote: y = −3

Range: {y y ≠ −3|| }

ID: A

13

29. x = 3 or x = −1

x x( ) − 2 x( ) =3

xx( ) Multiply each term by the LCD.

x2

− 2x = 3 Simplify. Note x ≠ 0

x2

− 2x − 3 = 0 Write in standard form.

x − 3( ) x + 1( ) = 0 Factor.

x − 3 = 0 or x + 1 = 0 Apply the Zero-Product Property.

x = 3 or x = −1 Solve for x.

Check:

x − 2 =3

x

3 − 23

3

1 1

x − 2 =3

x

−1 − 23

−1

−3 −3

30. There is no solution.

3x

x − 2(x − 2) =

x + 4

x − 2(x − 2) Multiply each term by the LCD, (x – 2).

3x = x + 4 Simplify. Note that x g 2.2x = 4 Solve for x.x = 2

The solution x = 2 is extraneous because it makes the denominators of the original equation equal to 0. Therefore the equation has no solution.

31. x = 0 or x = 132x

x2

− 7x − 18=

6x

x2

+ x − 2

2x

(x + 2)(x − 9)=

6x

(x + 2)(x − 1)Factor the denominator.

2x(x − 1) = 6x(x − 9)Multiply each term by the LCD (x + 2)(x − 9)(x − 1)

and simplify. Note that x ≠ −2, x ≠ 9, and x ≠ 1.

2x2

− 2x = 6x2

− 54x Use the Distributive Property.

4x2

− 52x = 0 Write in standard form.

4x(x − 13) = 0 Factor.

4x = 0 or x − 13 = 0 Use the Zero-Product Property.

x = 0 or x = 13 Solve for x.

ID: A

14

32. x ≤ 3 or x > 6

Use a graphing calculator. Let Y1 =x

x − 6 and Y2 = −1

X Y1 Y2

2 –0.5 –1

3 –1 –1

4 –2 –15 –5 –16 ERROR –1

7 7 –1

8 4 –1

The graph of x

x − 6 is greater or equal to –1 for values of x that are less than or equal to 3 or greater than 6.

Also notice in the table that y =x

x − 6 is undefined when x = 6.

ID: A

15

33.

The domain is the set of all real numbers. The range is also the set of real numbers.Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x.

x 5 x + 33 (x, f(x))

–11 5 −11 + 33= 5 −83

= −10 (–11, –10)

–4 5 −4 + 33= 5 −13

= −5 (–4, –5)

–3 5 −3 + 33= 5 03

= 0 (–3, 0)

–2 5 −2 + 33= 5 13

= 5 (–2, 5)

5 5 5 + 33= 5 83

= 10 (5, 10)

The domain is the set of all real numbers. The range is also the set of all real numbers.

34. Stretch f vertically by a factor of 2 and translate it left 5 units.

Write g x( ) in the form g x( ) = a1

bx − h( ) + k .

g x( ) = 21

1x − (−5)ÊËÁÁ ˆ

¯˜̃ + 0

Thus a = 2 and h = −5. Stretch f vertically by a factor of 2 and translate it left 5 units.

ID: A

16

35. 3x3

81x124

= 81 · x4 · x

4 · x44 Factor into perfect powers of four.

= 3 · x · x · x Use the Product Property of Roots.

= 3x3 Simplify.

36. ( 164 )3; 8

16

3

4

= ( 164 )3 Write with a radical.

= (2)3 Evaluate the root.

= 8 Evaluate the power.

37. g(x) = x + 3 − 7

x + 7 ⇒ (x − 4) + 7 = x + 3 Replace f(x) with f(x – h) and simplify.

x + 3 ⇒ x + 3 − 7 Replace f(x – h) with f(x – h) + k.

38. 27

(27)

1

3⋅ (27)

2

3

= (27)

1 + 2

3 Product of Powers

= (27)1 Simplify.

= 27

39. x = 103

x − 3 = 10 Subtract –5 from both sides.

x − 3 = 100 Square both sides.x = 103 Simplify.

Check

−5 + 103 − 3 = 5

−5 + 100 = 5−5 + 10 = 55 = 5 OK

40. x = 4

( 9x )2= (2 x + 5 )2 Square both sides.

9x = 4 x + 5( ) Simplify.

9x = 4x + 20 Distribute 4.5x = 20 Solve for x.x = 4

ID: A

17

41. x = 4

Step 1 Solve for x.

2x + 8( )1

2 = x

2x + 8( )1

2

È

Î

ÍÍÍÍÍÍÍÍÍ

˘

˚

˙̇˙̇˙̇˙̇˙

2

= x2 Raise both sides to the reciprocal power.

2x + 8 = x2 Simplify.

x2

− 2x − 8 = 0 Write in standard form.

x + 2( ) x − 4( ) = 0 Factor.

x + 2 = 0 or x − 4 = 0 Use the Zero-Product Property.

x = −2 or x = 4 Solve for x.

Step 2 Use substitution to check for extraneous solutions.

2x + 8( )1

2 = x

2( ) −2( ) + 8ÈÎÍÍÍ

˘˚˙̇˙

1

2−2

4( )1

2 −2

2 −2

2x + 8( )1

2 = x

2( ) 4( ) + 8ÈÎÍÍÍ

˘˚˙̇˙

1

2 4

16( )1

2 4

4 4

Because x = −2 does not satisfy the original equation, it is extraneous. The only solution is x = 4.

42. 4

5 ≤ x ≤

68

5

Step 1 Solve for x.

( 5x − 4)2≤ (8)2 Square both sides.

5x − 4 ≤ 64 Simplify.5x ≤ 68 Solve for x.

x ≤ 68

5

Step 2 Consider the radicand.5x − 4 ≥ 0 The radicand cannot be negative.5x ≥ 4 Solve for x.

x ≥ 4

5

The solution to 5x − 4 ≤ 8 is x ≤ 68

5 and x ≥

4

5 or

4

5 ≤ x ≤

68

5

43. 7

11

When the spinner does not land on orange, it must land on yellow or purple.

experimental probability =number of times the event occurs

number of trials=

9 + 5

22=

14

22=

7

11

ID: A

18

44. 85.7%P(different days) = 1 − P(Monday) Use the complement.

= 1 − (1

7) There are 7 days in the week.

= 85.7%

45. 0.121

Use the Female row. Of 66,033,000 female labor force, 8,004,000 work in the Industry field.

P(Industry Female| ) =8,004,000

66,033,000≈ 0.121

46. independent; 0.24One outcome does not affect the other, so the events are independent.To find the probability that A and B both happen, multiply the probabilities.

P(A and B) = P(A) • P(B) = 0.6 • 0.4 = 0.24.

47. 0.67

48.

Brother(s) No Brothers Total

Sister(s) 0.18 0.32 0.5

No Sisters 0.31 0.19 0.5

Total 0.49 0.51 1

49. 792 ways

Step 1 Determine whether the problem represents a combination or a permutation.The order does not matter because choosing a green shirt, a blue shirt, and a red shirt is the same as choosing a red shirt, a blue shirt, and a green shirt. It is a combination.

Step 2 Use the formula for combinations.

The number of combinations of n items taken r at a time is n Cr =n!

r! (n − r)!.

12 C5 =12!

5! (12 − 5)!n = 12 and r = 5

12C5 =12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1

5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 ⋅ (7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1)Expand.

12C5 =12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1

5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 ⋅ (7 + 6 + 5 + 4 + 3 + 2 + 1)=

12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8

5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1Divide out common factors.

12C5 =12 ⋅ 11 ⋅ 10 ⋅ 9 ⋅ 8

5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1=

12 ⋅ 11 ⋅ 10 ⋅ 9

5 ⋅ 3= 12 ⋅ 11 ⋅ 2 ⋅ 3 = 729 Simplify.

There are 729 ways to select a group of 5 shirts from 12.

50. 120 ways

Since the order matters, use the formula for permutations.

5!P5!

5!

(5 − 5)!=

5!

0!

Since 0! = 1, the number of ways is 5! = 120.

ID: A

19

51. 1

2

P(5 or odd)

= P(5) + P(odd) − P(5 and odd)

= 1

6 +

3

6 −

1

65 is also an odd number.

= 1

2

52. 1

3

There are six possible outcomes when a fair number cube is rolled. Because the number cube is fair, all outcomes are equally likely. There are two numbers greater than 4 on the number cube: 5 and 6. So the

probability of rolling one of these numbers is 2

6=

1

3.

53. The data appear to be normally distributed. The actual number of data values for each z-value is close to the expected number.

z Area below z x Projected values less than x Actual values less than x

−2 0.2 1.90 0 0

−1 0.16 1.95 3 3

0 0.5 2.0 9 9

1 0.84 2.05 15 15

2 0.98 2.10 18 17

54. mean = 24.8; median = 23;

mode = 29To find the mean, add all the values in the list and divide by 5.To find the median, sort the values in ascending order and choose the third value, which is the middle number, in the sorted list.To find the mode, look for the value that appears the most times in the list.

55. 3.5The expected value is the weighted average of all the outcomes of the study.

Expected value = 2(.20) + 3(.32) + 4(.288) + 5(.1536) + 6(.0384) = 3.5104 ≈ 3.5

ID: A

20

56. The mean is 2, and the standard deviation is about 2.1.

Step 1 Find the mean.

x =5 + 1 + 0 + 4 + 0

5= 2

Step 2 Find the difference between the mean and each data value, and square it.

Data Value 5 1 0 4 0

x − x 3 –1 –2 2 –2

x − xÊËÁÁÁ

ˆ¯˜̃̃

2

9 1 4 4 4

Step 3 Find the variance. Find the average of the last row of the table.

σ2

=9 + 1 + 4 + 4 + 4

5= 4.4

Step 4 Find the standard deviation. The standard deviation is the square root of the variance.

σ = 4.4 ≈ 2.1

The mean is 2, and the standard deviation is about 2.1.

57. 0.69

58.

Interquartile range: 5.5Order the data from least to greatest.

Find the minimum, maximum, median, and quartiles.Minimum = 7Maximum = 18Median = 12Lower Quartile = 9.5Upper Quartile = 15Interquartile range = 15 – 9.5 = 5.5

59. 39.49

60. 0.43