Solution manual for college algebra 10th edition by larson

Section P.1 Review of Real Numbers and Their Properties 3 18.. a The interval [ 4,∞ denotes the set of all real numbers greater than or equal to 4.. a The interval [−5, 2 denotes the s

C H A P T E R P Prerequisites

Section P.1 Review of Real Numbers and Their Properties 2

Section P.2 Exponents and Radicals 5

Section P.3 Polynomials and Special Products 9

Section P.4 Factoring Polynomials 16

Section P.5 Rational Expressions 22

Section P.6 The Rectangular Coordinate System and Graphs 31

Review Exercises 37

Problem Solving 44

Practice Test 48

(e) Irrational numbers: 2

8 5, 7,− −73, 0, 3.14, , 3, 12, 554 −

(a) Natural numbers: 12, 5 (b) Whole numbers: 0, 12, 5 (c) Integers: 7, 0, 3, 12, 5− −(d) Rational numbers: − −7, 73, 0, 3.14, , 3, 12, 554 −

(e) Irrational numbers: 5

9 2.01, 0.6, 13, 0.010110111 , 1, 6− −

(a) Natural numbers: 1 (b) Whole numbers: 1 (c) Integers: 13, 1, 6− − (d) Rational numbers: 2.01, 0.6, 13, 1, 6− − (e) Irrational numbers: 0.010110111

10 25, 17,− −125, 9, 3.12,12π, 7, 11.1, 13−

(a) Natural numbers: 25, 9, 7, 13 (b) Whole numbers: 25, 9, 7, 13 (c) Integers: 25, 17,− 9, 7, 13(d) Rational numbers:

12 5

25, 17,− − , 9, 3.12, 7, 11.1, 13−

(e) Irrational numbers: 12π

11 (a)

(b) (c)

(d)

12 (a)

(b)

(c) (d)

13 4− > − 8

− 8 − 7 − 6 − 5 − 4 x

16 3

Section P.1 Review of Real Numbers and Their Properties 3

18 (a) The inequality x < denotes the set of all real 0

numbers less than zero

(b) (c) The interval is unbounded

19 (a) The inequality 2− < x < denotes the set of all 2

real numbers greater than 2− and less than 2

(b) (c) The interval is bounded

20 (a) The inequality 0 < x ≤ denotes the set of all real 6

numbers greater than zero and less than or equal to 6

(b) (c) The interval is bounded

21 (a) The interval [ )4,∞ denotes the set of all real

numbers greater than or equal to 4

(b) (c) The interval is unbounded

22 (a) (−∞, 2)denotes the set of all real numbers less

(b) (c) The interval is unbounded

23 (a) The interval [−5, 2) denotes the set of all real

numbers greater than or equal to 5− and less than 2

(b) (c) The interval is bounded

24 (a) The interval (−1, 2]denotes the set of all real

numbers greater than − and less than or equal to 2 1(b)

(c) The interval is bounded

Receipts, R Expenditures, E R− E

60 9− 7x

(a) 9− 7 3( )− = 9+ 21 = 30(b) 9− 7 3( ) = 9 −21 = − 12

61 x2 −3x + 2(a) ( )2 ( )

−+

66 (x + 3) (− x+ 3) = 0 Additive Inverse Property

( )

3 Commutative Property of Multiplication

x y

=

Section P.2 Exponents and Radicals 5

68 1( ) ( )1

7 7 12 7 7 12 Associative Property of Multiplication

73 False Because zero is nonnegative but not positive, not

every nonnegative number is positive

74 False Two numbers with different signs will always

have a product less than zero

75 The product of two negative numbers is positive

76 (a) Because the price can only be a positive rational

number with at most two decimal places, the description matches graph (ii)

(b) Because the distance is a positive real number, the description matches graph (i)

A range of prices can only include zero and positive numbers with at most two decimal places So, a range of prices can be represented by whole numbers and some noninteger positive fractions

A range of lengths can only include positive numbers So, a range of lengths can be represented

by positive real numbers

77 (a)

(b) (i) As n approaches 0, the value of 5 n increases

without bound (approaches infinity)

(ii) As n increases without bound (approaches

infinity), the value of 5 n approaches 0

Section P.2 Exponents and Radicals

15 When x = 2,

( )3 3

16 When x = 4,

( ) 2 2

4 4

x y x y

x

x x

Section P.2 Exponents and Radicals 7

a b

( ) ( )

2

4 2 4 2 2 2

2

7575

33

x x x x

Section P.3 Polynomials and Special Products 9

x x

x x

=

So, 2x = 2 6.3( ) =12.6 in

Because 7.9 <12.6,the length of a side of package A is

less than twice the length of a side of package B

71 False When x = 0,the expressions are not equal

72. False When a power is raised to a power, you multiply the exponents: ( )a n k = a nk

73 False When a sum is raised to a power, you multiply the sum by itself using the Distributive Property

6

101010

6 (a) Standard form: 3

(b) Degree: 0 Leading coefficient: 3 (c) Monomial

7 (a) Standard form: 1 5

2x 14x

(b) Degree: 5 Leading coefficient: −12

(c) Binomial

8 (a) Standard form: 2x + 3

(b) Degree: 1 Leading coefficient: 2 (c) Binomial

9 (a) Standard form: −4x5 + 6x4 + 1

(b) Degree: 5 Leading coefficient: −4 (c) Trinomial

10 (a) Standard form: 25y2 − y+ 1

(b) Degree: 2 Leading coefficient: 25 (c) Trinomial

11 2x− 3x3 + is a polynomial 8

Standard form: −3x3 + 2x + 8

12 5x4 − 2x2 + x− 2is not a polynomial because it includes

a term with a negative exponent

16 y4 − y is not a polynomial because it includes a term

with a square root

Section P.3 Polynomials and Special Products 11

9 4 9 4

3030

Section P.3 Polynomials and Special Products 13

(c) As r increases, the amount increases

73 (a) The possible gene combinations of an offspring with albino coloring is 14, or 25%

(b) As x increases, the area of the foundation decreases

(c) When x = 21 feet,the area of the new foundation is

r

77. Area of shaded region = Area of larger square− Area of smaller square

1 2 1 2 3 2

( )3

Section P.3 Polynomials and Special Products 15

81 (a) Approximations will vary Actual safe loads for x = 12:

the sum of the polynomials is n

89. The middle term was omitted when squaring the binomial

( )2 ( )2 ( )( ) ( )2

2 2

90. (a) The cardboard was 52 inches long and 24 inches wide The box was constructed by cutting a square of x inches by x inches

from each corner of the rectangular piece of cardboard and folding the side pieces up

(b) The polynomial is a 3rd degree polynomial because the length, width, and height of the box are expressions in the

The maximum volume occurs for a value of x between 7 and 8

The maximum volume occurs when x ≈ 7.7 inches

mi hr

( ) ft

91 The unknown polynomial may be found by adding −x3 + 3x2 + 2x − and 1 5x2 + 8:

Section P.4 Factoring Polynomials 17

52 a c⋅ = ( )( )12 1 = 12 Rewrite the middle term,

Section P.4 Factoring Polynomials 19

80 3x2 +7x +2 = (3x+1)(x+2)

ππππ

2 (average radius)(thickness of shell)

h p w pwh

h

πππππ

85. For x2 + bx−15to be factorable, b must equal m + where n mn = −15

The possible b-values are 14, 14, 2,− − and 2

86. For x2 +bx+ 24 to be factorable, b must equal m+ where n mn = 24

The possible b-values are 25, 25, 14, 14,− − 11, 11, 10, 10.− −

Factors of 15− Sum of factors ( )( )15 −1 15+ − =( )1 14( )( )−15 1 −15+ = − 1 14( )( )3 −5 3+ − = − ( )5 2( )( )−3 5 − +3 5 = 2

Factors of 24 Sum of factors ( )( )1 24 1+ 24 = 25( )( )− −1 24 − + −1 ( )24 = − 25( )( )2 12 2+12 =14( )( )− −2 12 − + −2 ( )12 = − 14( )( )3 8 3+8 = 11( )( )− −3 8 − + − = − 3 ( )8 11( )( )4 6 4+6 =10( )( )− −4 6 − + − = − 4 ( )6 10

r

h

R

Section P.4 Factoring Polynomials 21

87. For 2x2 + 5x + to be factorable, the factors of 2c must add up to 5 c

These are a few possible c-values There are many correct answers

88. For 3x2 − x + to be factorable, the factors of 3c must add up to 1 c −

These are a few possible c-values There are many correct answers

2 2 2

91 3 should be factored out of both binomials to yield (3x + 6 3)( x −9) = 3(x + 2 3)( )(x− 3) = 9(x+ 2)(x −3)

92 No, (3x− 6)(x+ is not completely factored because 1) (3x− 6) = 3(x− 2 )

So, the completely factored form is 3(x −2)(x+1 )

− − 14 ( )( )7 −2 = − and 14 7+ − = ( )2 512

− − 24 ( )( )8 − = − and 3 24 8+ − = ( )3 5

Possible c-values 3c Factors of 3c must add up to 1− 2

− − 6 ( )( )2 − = − and 3 6 2+ − = − ( )3 14

− − 12 ( )( )3 − = − and 4 12 3+ −( )4 = − 110

− −30 ( )( )5 − = − and 6 30 5+ − = − ( )6 1

94 3n 3n ( ) ( ) (n 3 n 3 n n)( 2n n n 2n)

Depending on the value of n, this may factor further

95 Answers will vary Sample answer: x2 − 3

a a

b b

b b

a a

b a

a − b

b b

a

b

a b

a

a b

Section P.5 Rational Expressions 23

++ is the set of all real numbers x

set of all real numbers x such that x ≠ − 2 5,

13. The domain of x − is the set of all real numbers x7

x x

13

1

4 Undef

16

17

181

2

x +

12

13

14

15

16

1718

Section P.5 Rational Expressions 25

47 The minus sign should be distributed to each term in the numerator of the second fraction

11

x

x x

x x

2 2

2 2

Section P.5 Rational Expressions 27

2 2

2 2

2 2

2 2

22

+

t t

t t

Section P.5 Rational Expressions 29

Number of households using mobile bankingNumber of households using online banking

Section P.6 The Rectangular Coordinate System and Graphs 31

NM P

78 The bar graph indicates that the oxygen in the pond level

drops dramatically for the first week By the twelfth week, the oxygen levels in the pond are almost at the previous high

79 False In order for the simplified expression to be equivalent to the original expression, the domain of the

simplified expression needs to be restricted If n is even,

9. x > and 0 y < in Quadrant IV 0

10 x < and 0 y < in Quadrant III 0

11 x = − and 4 y > in Quadrant II 0

12 x < 0 and y = in Quadrant II 7

13 x + y = 0, 0, 0x ≠ y ≠ means x = −yor y = −x

This occurs in Quadrant II or IV

14 ( )x y xy, , > means x and y have the same signs 0

This occurs in Quadrant I or III

(2, 4) (−6, 2)

(−4, 0)

(3, −1) (1.5, −3.5) (−1, −8)

8000 9000 10,000

7500 8500 9500 10,500 11,000 11,500

40

0

y

Section P.6 The Rectangular Coordinate System and Graphs 33

Distance = 1−13 = −12 = 12(b) 52 +122 = 25 +144 =169 = 132

24 (a) The distance between (−1, 1) and ( )9, 1 is 10

The distance between ( )9, 1 and ( )9, 4 is 3

The distance between (−1, 1) and ( )9, 4 is ( )

y

Full file at https://TestbankDirect.eu/Solution-Manual-for-College-Algebra-10th-Edition-by-Larson

=

≈ The plane flies about 192 kilometers

y

x

2 4 6 8 10

−5

15 20

1 3 4

2

2 2 3

1

5 2

3 5

,

−

y

Section P.6 The Rectangular Coordinate System and Graphs 35

So, the minimum wage increased 53.7% from 1985 to 2000 and 40.8% from 2000 to 2015

So, the minimum wage will be about $10.21 in the year 2030

(d) Answers will vary Sample answer: Yes, the prediction is reasonable because the percent increase is over an equal time

− y < which is located in Quadrant III

48. False The Midpoint Formula would be used 15 times

49. True Two sides of the triangle have lengths 149 and the third side has a length of 18

50. False The polygon could be a rhombus For example, consider the points ( ) ( ) (4, 0 , 0, 6 , 4, 0 ,− ) and (0, 6 − )

51. Answers will vary Sample answer: When the x-values are much larger or smaller than the y-values, different

scales for the coordinate axes should be used

52. The y-coordinate of a point on the x-axis is 0 The

x-coordinates of a point on the y-axis is 0

40 60 80 100

58 (a) Because (x y lies in Quadrant II, 0, 0) (x0,−y0) must lie in Quadrant III Matches (ii)

(b) Because (x y lies in Quadrant II, 0, 0) (−2 ,x y0 0) must lie in Quadrant I Matches (iii)

(c) Because (x y lies in Quadrant II, 0, 0) ( 0 1 0)

2

,

x y must lie in Quadrant II Matches (iv)

(d) Because (x y lies in Quadrant II, 0, 0) (−x0,−y0) must lie in Quadrant IV Matches (i)

Review Exercises for Chapter P 37

60 (a) The point (x0, −y0)is reflected in the x-axis

(b) The point (−x0, −y0)is reflected in the x- and

9 2

11, 14,− − , , 0.4(e) Irrational numbers: 6

15, 22,− − , 0, 5.2, (a) Natural numbers: none (b) Whole numbers: 0 (c) Integers: 22, 0−

6 (a) 4− < x < denotes the set of all real numbers 4 greater than 4− and less than 4

(c) The set is bounded

7 (a) 3− ≤ x < denotes the set of all real numbers 4

greater than or equal to 3− and less than 4

(b) (c) The set is bounded

8 (a) x ≥ denotes the set of all real numbers greater 2

x

Full file at https://TestbankDirect.eu/Solution-Manual-for-College-Algebra-10th-Edition-by-Larson

+ Illustrates the Multiplicative Inverse Property

Review Exercises for Chapter P 39

49. These are not like terms Radicals cannot be combined

by addition or subtraction unless the index and the radicand are the same

Review Exercises for Chapter P 41

78 (a) The surface is the sum of the area of the side, 2πrh,and the areas of the top and bottom which are eachπr2

Full file at https://TestbankDirect.eu/Solution-Manual-for-College-Algebra-10th-Edition-by-Larson

99 2 6 2 6

43

6

(5, 5) (−3, 6)

y

x

(0, 6) (8, 1)

(5, −4) (−3, −3)

Review Exercises for Chapter P 43

107 x > and 0 y = − in Quadrant IV 2

108 xy = means x and y have the same signs This occurs 4

in Quadrants I and III

8 6

y

x

Actual temperature (in °F)

80 70 90 110 130 150

100 120 140

118. False, x n − y n = (x− y x) ( n− 1 +x n− 2y ++ y n− 1) for odd values of n, not for all values of n

n n n n n n

Problem Solving for Chapter P

−

6.31 10 kg mm1,150,347

−

5.74 10 kg mm696,910

−

(c) No The weight would be different Cork is much lighter than iron so it would have a much smaller density

2 Let a = and 5 b = − Then 3 a−b = 5− −( )3 = and 8 a − b = 5 − − =3 2 Thus, a−b > a − b

Let a = 11 and b = 3 Then a −b = 11−3 = and 8 a − b = 11 − 3 = 8 Thus, a−b = a − b

To prove a−b ≥ a − b for all a, b, consider the following cases

Therefore, a−b ≥ a − b for all a, b

3 To say that a number has n significant digits means that the number has n digits with the leftmost non-zero digit and ending

with the rightmost non-zero digit For example; 28,000, 1.400, 0.00079 each have two significant digits

Problem Solving for Chapter P 45

πππ

29.57 square feet

(b) From 1960 to 1970, the population increased by 203.30 – 179.32 = 23.98 million people

From 1970 to 1980, the population increased by 226.54 – 203.30 = 23.24 million people

From 1980 to 1990, the population increased by 248.72 – 226.54 = 22.18 million people

From 1990 to 2000, the population increased by 281.42 – 248.72 = 32.70 million people

From 2000 to 2010, the population increased by 308.75 – 281.42 = 27.33 million people

(c) The population increased the most from 1990 to 2000 The population increased the least from 1980 to 1990

(d) From 1960 to 1970, the percent increase in population was 23.98 million people 0.134 13.4%

7

1 43225

lw x

10 The distance between (x y and 1, 1) 21 2 21 2

Problem Solving for Chapter P 47

1, 2 and 4, 1The points of trisection are:

14 (a) Either graph could be misleading The scales on the vertical axes make it appear that the rise in profits is either dramatic or

small, but the total increase is only 2-6 units

(b) If the company wanted to gloss over the dip in profits during July, as at a stockholders meeting, the first graph could be used If the company wished to project an image of rapidly increasing profits, as to potential investors, the second graph could be used

Full file at https://TestbankDirect.eu/Solution-Manual-for-College-Algebra-10th-Edition-by-Larson