Elementary and Intermediate Algebra

Marvin L. Bittinger, David J. Ellenbogen,Barbara L. Johnson

Chapter 8

Inequalities

Educators


Problem 1

match the set with the most appropriate choice from the column on the right.
$$
\{x | x<-2 \text { or } x>2\}
$$

Darian K.
Numerade Educator

Problem 2

match the set with the most appropriate choice from the column on the right.
$$
\{x | x<-2 \text { and } x>2\}
$$

Darian K.
Numerade Educator

Problem 3

match the set with the most appropriate choice from the column on the right.
$$
\{x | x>-2\} \cap\{x | x<2\}
$$

Darian K.
Numerade Educator

Problem 4

match the set with the most appropriate choice from the column on the right.
$$
\{x | x \leq-2\} \cup\{x | x \geq 2\}
$$

Darian K.
Numerade Educator

Problem 5

match the set with the most appropriate choice from the column on the right.
$$
\{x | x \leq-2\} \cup\{x | x \leq 2\}
$$

Darian K.
Numerade Educator

Problem 6

match the set with the most appropriate choice from the column on the right.
$$
\{x | x \leq-2\} \cap\{x | x \leq 2\}
$$

Darian K.
Numerade Educator

Problem 7

match the set with the most appropriate choice from the column on the right.
$$
\{x | x \geq-2\} \cap\{x | x \geq 2\}
$$

Darian K.
Numerade Educator

Problem 8

match the set with the most appropriate choice from the column on the right.
$$
\{x | x \geq-2\} \cup\{x | x \geq 2\}
$$

Darian K.
Numerade Educator

Problem 9

match the set with the most appropriate choice from the column on the right.
$$
\{x | x \leq 2\} \text { and }\{x | x \geq-2\}
$$

Darian K.
Numerade Educator

Problem 10

match the set with the most appropriate choice from the column on the right.
$$
\{x | x \leq 2\} \text { or }\{x | x \geq-2\}
$$

Darian K.
Numerade Educator

Problem 11

Solve each inequality using the given graph.
$$
f(x) \geq g(x)
$$

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Numerade Educator

Problem 12

Solve each inequality using the given graph.
$$
f(x)<g(x)
$$

Darian K.
Numerade Educator

Problem 13

Solve each inequality using the given graph.
$$
y_{1}<y_{2}
$$

Darian K.
Numerade Educator

Problem 14

Solve each inequality using the given graph.
$$
y_{1} \geq y_{2}
$$

Darian K.
Numerade Educator

Problem 15

The graphs of $f(x)=2 x+1, g(x)=-\frac{1}{2} x+3$
and $h(x)=x-1$ are as shown below. Solve each inequality, referring only to the figure.
a) $2 x+1 \leq x-1$
b) $x-1>-\frac{1}{2} x+3$
c) $-\frac{1}{2} x+3<2 x+1$

Darian K.
Numerade Educator

Problem 16

The graphs of $y_{1}=-\frac{1}{2} x+5, y_{2}=x-1,$ and $y_{3}=2 x-3$ are as shown below. Solve each inequality, referring only to the figure.
a) $-\frac{1}{2} x+5>x-1$
b) $x-1 \leq 2 x-3$
c) $2 x-3 \geq-\frac{1}{2} x+5$

Darian K.
Numerade Educator

Problem 17

Solve graphically.
$$
x-3<4
$$

Darian K.
Numerade Educator

Problem 18

Solve graphically.
$$
x+4 \geq 6
$$

Darian K.
Numerade Educator

Problem 19

Solve graphically.
$$
2 x-3 \geq 1
$$

Darian K.
Numerade Educator

Problem 20

Solve graphically.
$$
3 x+1<1
$$

Darian K.
Numerade Educator

Problem 21

Solve graphically.
$$
x+3>2 x-5
$$

Darian K.
Numerade Educator

Problem 22

Solve graphically.
$$
3 x-5 \leq 3-x
$$

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Numerade Educator

Problem 23

Solve graphically.
$$
\frac{1}{2} x-2 \leq 1-x
$$

Darian K.
Numerade Educator

Problem 24

Solve graphically.
$$
x+5>\frac{1}{3} x-1
$$

Darian K.
Numerade Educator

Problem 25

Solve graphically.
$$
4 x+7 \leq 3-5 x
$$

Darian K.
Numerade Educator

Problem 26

Solve graphically.
$$
5 x+6<8 x-11
$$

Darian K.
Numerade Educator

Problem 27

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge if graphing by hand.
Show Business. A band receives $\$ 750$ plus $15 \%$ of receipts over $\$ 750$ for playing a club date. If a club charges a $\$ 6$ cover charge, how many people must attend in order for the band to receive at least $\$ 1200 ?$

Darian K.
Numerade Educator

Problem 28

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge if graphing by hand.
Temperature Conversion. The function
$C(F)=\frac{5}{9}(F-32)$
can be used to find the Celsius temperature $C(F)$ that corresponds to $F^{\circ}$ Fahrenheit.
a) Gold is solid at Celsius temperatures less than $1063^{\circ} \mathrm{C} .$ Find the Fahrenheit temperatures for which gold is solid.
b) Silver is solid at Celsius temperatures less than $960.8^{\circ} \mathrm{C} .$ Find the Fahrenheit temperatures for which silver is solid.

Darian K.
Numerade Educator

Problem 29

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge if graphing by hand.
A musement Park Attendance. The table below lists the estimated annual ridership at amusement parks for various years. Use linear regression to find a linear function that can be used to predict the ridership $r(x),$ in billions, where $x$ is the number of years after $2000 .$ Then predict those years in which there will be fewer than 1.5 billion riders.

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Problem 30

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge if graphing by hand.
Smoking. The table below lists the percent of teenagers in the United States who smoked cigarettes during various years. Use linear regression to find a linear function that can be used to predict the percent $p(x)$ of teenagers who smoked, where $x$ is the number of years after 2000 . Then predict those years in which the Healthy People 2010 goal of no more than $16 \%$ teenage smokers will be met.

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Problem 31

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge if graphing by hand.
Advertising. The table below lists advertising revenue for newspapers and the Internet for various years. Use linear regression to find two linear functions that can be used to estimate advertising revenue $n(x),$ in billions of dollars, for newspapers and $t(x)$ for the Internet, where $x$ is the number of years after 2006 . Then predict the years for which advertising revenue for the Internet will exceed that for newspapers.

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Problem 32

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge if graphing by hand.
Income. The table below lists the median adjusted household income for unmarried and married men for various years. Use linear regression to find two linear functions that can be used to estimate the median household earnings $u(x)$ for unmarried men and $m(x)$ for married men, where $x$ is the number of years after $1970 .$ Then estimate the years for which the household earnings for married men exceeds that for unmarried men.

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Problem 33

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge if graphing by hand.
100-Meter Freestyle. The table below lists the world record for the women's 100 -meter freestyle long-course swim for various years. Use linear regression to find a linear function that can be used to predict the world record $x$ years after $1990 .$ Then predict those years in which the world record will be less than 50 sec.

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Problem 34

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge if graphing by hand.
Mile Run. The table below lists the world record for the mile run for various years. Use linear regression to find a linear function that can be used to predict the world record for the mile run $x$ years after $1954 .$ Then predict those years in which the world record will be less than $3.5 \mathrm{min}$.

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Problem 35

Find each indicated intersection or union.
$\{5,9,11\} \cap\{9,11,18\}$

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Problem 36

Find each indicated intersection or union.
$$
\{2,4,8\} \cup\{8,9,10\}
$$

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Problem 37

Find each indicated intersection or union.
$$
\{0,5,10,15\} \cup\{5,15,20\}
$$

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Problem 38

Find each indicated intersection or union.
$$
\{2,5,9,13\} \cap\{5,8,10\}
$$

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Problem 39

Find each indicated intersection or union.
$$
\{a, b, c, d, e, f\} \cap\{b, d, f\}
$$

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Problem 40

Find each indicated intersection or union.
$$
\{a, b, c\} \cup\{a, c\}
$$

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Problem 41

Find each indicated intersection or union.
$$
\{r, s, t\} \cup\{r, u, t, s, v\}
$$

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Problem 42

Find each indicated intersection or union.
$$
\{m, n, o, p\} \cap\{m, o, p\}
$$

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Problem 43

Find each indicated intersection or union.
$$
\{3,6,9,12\} \cap\{5,10,15\}
$$

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Problem 44

Find each indicated intersection or union.
$$
\{1,5,9\} \cup\{4,6,8\}
$$

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Problem 45

Find each indicated intersection or union.
$$
\{3,5,7\} \cup \varnothing
$$

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Problem 46

Find each indicated intersection or union.
$$
\{3,5,7\} \cap \varnothing
$$

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Problem 47

Graph and write interval notation for each compound inequality.
$$
3<x<7
$$

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Problem 48

Graph and write interval notation for each compound inequality.
$$
0 \leq y \leq 4
$$

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Problem 49

Graph and write interval notation for each compound inequality.
$$
-6 \leq y \leq 0
$$

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Problem 50

Graph and write interval notation for each compound inequality.
$$
-9 \leq x<-5
$$

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Problem 51

Graph and write interval notation for each compound inequality.
$$
x<-1 \text { or } x>4
$$

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Problem 52

Graph and write interval notation for each compound inequality.
$$
x<-5 \text { or } x>1
$$

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Problem 53

Graph and write interval notation for each compound inequality.
$$
x \leq-2 \text { or } x>1
$$

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Problem 54

Graph and write interval notation for each compound inequality.
$$
x \leq-5 \text { or } x>2
$$

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Problem 55

Graph and write interval notation for each compound inequality.
$$
x>-2 \text { and } x<4
$$

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Problem 56

Graph and write interval notation for each compound inequality.
$$
x>-7 \text { and } x<-2
$$

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Problem 57

Graph and write interval notation for each compound inequality.
$$
-4 \leq-x<2
$$

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Problem 58

Graph and write interval notation for each compound inequality.
$$
3>-x \geq-1
$$

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Problem 59

Graph and write interval notation for each compound inequality.
$$ 5> a or a>7 $$

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Problem 60

Graph and write interval notation for each compound inequality.
$$
t \geq 2 \text { or }-3>t
$$

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Problem 61

Graph and write interval notation for each compound inequality.
$$
x \geq 5 \text { or }-x \geq 4
$$

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Problem 62

Graph and write interval notation for each compound inequality.
$$
-x<3 \text { or } x<-6
$$

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Problem 63

Graph and write interval notation for each compound inequality.
$$
7>y \text { and } y \geq-3
$$

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Problem 64

Graph and write interval notation for each compound inequality.
$$
6>-x \geq 0
$$

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Problem 65

Graph and write interval notation for each compound inequality.
$$
x<7 \text { and } x \geq 3
$$

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Problem 66

Graph and write interval notation for each compound inequality.
$$
x \geq-3 \text { and } x<3
$$

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Problem 67

Graph and write interval notation for each compound inequality.
$$
t<2 \text { or } t<5
$$

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Problem 68

Graph and write interval notation for each compound inequality.
$$
t>4 \text { or } t>-1
$$

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Problem 69

Solve and graph each solution set.
$$
-2<t+1<8
$$

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Problem 70

Solve and graph each solution set.
$$
-3<t+1 \leq 5
$$

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Problem 71

Solve and graph each solution set.
$$
4<x+4 \text { and } x-1<3
$$

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Problem 72

Solve and graph each solution set.
$$
-1<x+2 \text { and } x-4<3
$$

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Problem 73

Solve and graph each solution set.
$$
-7 \leq 2 a-3 \text { or } 3 a+1>7
$$

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Problem 74

Solve and graph each solution set.
$$
-4 \leq 3 n+5 \text { or } 2 n-3 \leq 7
$$

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Problem 75

Solve and graph each solution set.
$$
x+7 \leq-2 \text { or } x+7 \geq-3
$$

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Problem 76

Solve and graph each solution set.
$$
x+5<-3 \text { or } x+5 \geq 4
$$

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Problem 77

Solve and graph each solution set.
$$
-7 \leq 4 x+5 \leq 13
$$

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Problem 78

Solve and graph each solution set.
$$
-4 \leq 2 x+3 \leq 15
$$

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Problem 79

Solve and graph each solution set.
$$
5>\frac{x-3}{4}>1
$$

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Problem 80

Solve and graph each solution set.
$$
3 \geq \frac{x-1}{2} \geq-4
$$

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Problem 81

Solve and graph each solution set.
$$
-2 \leq \frac{x+2}{-5} \leq 6
$$

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Problem 82

Solve and graph each solution set.
$$
-10 \leq \frac{x+6}{-3} \leq-8
$$

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Problem 83

Solve and graph each solution set.
$$
2 \leq f(x) \leq 8, \text { where } f(x)=3 x-1
$$

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Problem 84

Solve and graph each solution set.
$$
7 \geq g(x) \geq-2, \text { where } g(x)=3 x-5
$$

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Problem 85

Solve and graph each solution set.
$$
-21 \leq f(x)<0, \text { where } f(x)=-2 x-7
$$

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Problem 86

Solve and graph each solution set.
$$
4>g(t) \geq 2, \text { where } g(t)=-3 t-8
$$

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Problem 87

Solve and graph each solution set.
$$
f(t)<3 \text { or } f(t)>8, \text { where } f(t)=5 t+3
$$

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Problem 88

Solve and graph each solution set.
$$
g(x) \leq-2 \text { or } g(x) \geq 10, \text { where } g(x)=3 x-5
$$

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Problem 89

Solve and graph each solution set.
$$
6>2 a-1 \text { or }-4 \leq-3 a+2
$$

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Problem 90

Solve and graph each solution set.
$$
3 a-7>-10 \text { or } 5 a+2 \leq 22
$$

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Problem 91

Solve and graph each solution set.
$$
1-a<-2 \text { and } 2 a+1>9
$$

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Problem 92

Solve and graph each solution set.
1-a<-2 \text { and } 2 a+1>9

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Problem 93

Solve and graph each solution set.
$$
3 x+2<2 \text { and } 3-x<1
$$

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Problem 94

Solve and graph each solution set.
$$
2 x-1>5 \text { and } 2-3 x>11
$$

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Problem 95

Solve and graph each solution set.
$$
2 t-7 \leq 5 \text { or } 5-2 t>3
$$

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Problem 96

Solve and graph each solution set.
$$
5-3 a \leq 8 \text { or } 2 a+1>7
$$

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Problem 97

Use the accompanying graph of $f(x)=2 x-5$ to solve $-7<2 x-5<7$

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Problem 98

Use the accompanying graph of $g(x)=4-x$ to solve $4-x<-2$ or $4-x>7$

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Problem 99

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\frac{9}{x+8}
$$

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Problem 100

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\frac{2}{x+3}
$$

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Problem 101

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\frac{-8}{x}
$$

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Problem 102

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\frac{x+3}{2 x-8}
$$

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Problem 103

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\sqrt{x-6}
$$

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Problem 104

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\sqrt{x-2}
$$

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Problem 105

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\sqrt{2 x+7}
$$

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Problem 106

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\sqrt{8-5 x}
$$

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Problem 107

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\sqrt{8-2 x}
$$

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Problem 108

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\sqrt{10-2 x}
$$

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Problem 109

Use interval notation to write each domain.
The domain of $f+g,$ if $f(x)=\sqrt{x-5}$ and $g(x)=\sqrt{\frac{1}{2} x+1}$

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Problem 110

Use interval notation to write each domain.
The domain of $f-g,$ if $f(x)=\sqrt{x+3}$ and $g(x)=\sqrt{2 x-1}$

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Problem 111

Use interval notation to write each domain.
The domain of $f \cdot g,$ if $f(x)=\sqrt{3-x}$ and $g(x)=\sqrt{3 x-2}$

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Problem 112

Use interval notation to write each domain.
The domain of $f+g,$ if $f(x)=\sqrt{3-4 x}$ and $g(x)=\sqrt{x+2}$

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Problem 113

Use interval notation to write each domain.
Why can the conjunction $2<x$ and $x<5$ be rewritten as $2<x<5,$ but the disjunction $2<x$ or $x<5$ cannot be rewritten as $2<x<5 ?$

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Problem 114

Use interval notation to write each domain.
Can the solution set of a disjunction be empty? Why or why not?

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Problem 115

Graph.
$$
g(x)=2 x
$$

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Problem 116

Graph.
$$
f(x)=4
$$

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Problem 117

Graph.
$$
g(x)=-3
$$

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Problem 118

Graph.
$$
f(x)=|x|
$$

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Problem 119

Solve by graphing.
$$
x+4=3
$$

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Problem 120

Solve by graphing.
$$
x-1=-5
$$

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Problem 121

What can you conclude about $a, b, c,$ and $d,$ if $[a, b] \cup[c, d]=[a, d] ?$ Why?

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Problem 122

What can you conclude about $a, b, c,$ and $d,$ if $[a, b] \cap[c, d]=[a, b] ?$ Why?

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Problem 123

Counseling. The function given by $s(t)=500 t+16,500$ can be used to estimate the number of student visits to Cornell University's counseling center $t$ years after 2000 . For what years is the number of student visits between $18,000$ and $21,000 ?$

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Problem 124

The function given by $P(d)=1+(d / 33)$ gives the pressure, in atmospheres (atm), at a depth of $d$ feet in the sea. For what depths $d$ is the pressure at least 1 atm and at most 7 atm?

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Problem 125

The function given by $f(x)=2(x+10)$ can be used to convert dress sizes $x$ in the United States to dress sizes $f(x)$ in Italy. For what dress sizes in the United States will dress sizes in Italy be between 32 and $46 ?$

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Problem 126

The function given by $w(t)=0.0125 t+4.525$ can be used to estimate the number of pounds of solid waste, $w(t),$ produced daily, on average, by each person in the United States, $t$ years after $2000 .$ For what years will waste production range from 4.6 lb to 4.8 lb per person per day?

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Problem 127

The function given by $w(t)=0.0125 t+4.525$ can be used to estimate the number of pounds of solid waste, $w(t),$ produced daily, on average, by each person in the United States, $t$ years after $2000 .$ For what years will waste production range from 4.6 lb to 4.8 lb per person per day?

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Problem 128

A $\$ 6.00$ toll is charged to cross the bridge from mainland Florida to Sanibel Island. A six-month reduced-fare pass, costing $\$ 50.00$ reduces the toll to $\$ 2.00 .$ A six-month unlimited-trip pass costs $\$ 300$ and allows free crossings. How many crossings in six months does it take for the reducedfare pass to be the more economical choice?

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Problem 129

Solve and graph.
$$
4 m-8>6 m+5 \text { or } 5 m-8<-2
$$

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Problem 130

Solve and graph.
$$
4 a-2 \leq a+1 \leq 3 a+4
$$

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Problem 131

Solve and graph.
$$
3 x<4-5 x<5+3 x
$$

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Problem 132

Solve and graph.
$$
x-10<5 x+6 \leq x+10
$$

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Problem 133

Determine whether each sentence is true or false for all real numbers $a, b,$ and $c$
If $-b<-a,$ then $a<b$

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Problem 134

Determine whether each sentence is true or false for all real numbers $a, b,$ and $c$
If $a \leq c$ and $c \leq b,$ then $b>a$

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Problem 135

Determine whether each sentence is true or false for all real numbers $a, b,$ and $c$
If $a<c$ and $b<c,$ then $a<b$

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Problem 136

Determine whether each sentence is true or false for all real numbers $a, b,$ and $c$
If $-a<c$ and $-c>b,$ then $a>b$

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Problem 137

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\frac{\sqrt{3-4 x}}{x+7}
$$

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Problem 138

For $f(x)$ as given, use interval notation to write the domain of $f$
$$
f(x)=\frac{\sqrt{5+2 x}}{x-1}
$$

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Problem 139

On many graphing calculators, the TEST key provides access to inequality symbols, while the LOGIC option of that same key accesses the conjunction and and the disjunction or. Thus, if $y_{1}=x>-2$ and $y_{2}=x<4,$ Exercise 55 can be checked by forming the expression $y_{3}=y_{1}$ and $y_{2} .$ The interval(s) in the solution set appears as a horizontal line 1 unit above the $x$ -axis. (Be careful to "deselect" $y_{1}$ and $y_{2}$ so that only $y_{3}$ is drawn.) Use the Test key to check Exercises $59,63,65,$ and 67

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