College Algebra

John W. Coburn, Jeremy P. Coffelt

Chapter 1

Equations and Inequalities - all with Video Answers

Educators

Section 1

Linear Equations, Formulas, and Problem Solving

Problem 1

For inequalities, the three ways of writing a solution set are notation, a(n) graph, and

Elizabeth Xu

Elizabeth Xu

Numerade Educator

Problem 2

The intersection of set $A$ with set $B$ is the set of elements in $A$ $B$. The union of set $A$ with set $B$ is the set of

Himanshu Gupta

Numerade Educator

Problem 3

Discuss/Explain how the concept of allowable values relates to rational and radical expressions. Include a few examples.

Khanh Ha

Numerade Educator

Problem 4

Discuss/Explain the use of the words "and" and "or" in the statement of compound inequalities. Include a few examples to illustrate.

Leanne Hockman

Numerade Educator

Problem 5

To qualify for a secretarial position, a person must type at least 45 words per minute.

John Mcalister

Numerade Educator

Problem 6

The balance in a checking account must remain above $\$ 1000$ or a fee is charged.

Liuxi Sun

Numerade Educator

Problem 7

To bake properly, a turkey must be kept at temperatures above $250^{\circ}$, but below $450^{\circ}$.

Caleb Miller

Numerade Educator

Problem 8

To fly efficiently, the airliner must cruise at or between altitudes of 30,000 and $35,000 \mathrm{ft}$.

Tiffany Tran

Numerade Educator

Problem 9

Graph each inequality on a number line.
$y<3$

Elizabeth Xu

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Problem 10

Graph each inequality on a number line.
$x>-2$

Elizabeth Xu

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Problem 11

Graph each inequality on a number line.
$m \leq 5$

Elizabeth Xu

Numerade Educator

Problem 12

Graph each inequality on a number line.
$n \geq-4$

Elizabeth Xu

Numerade Educator

Problem 13

Graph each inequality on a number line.
$x \neq 1$

Elizabeth Xu

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Problem 14

Graph each inequality on a number line.
$x \neq-3$

Elizabeth Xu

Numerade Educator

Problem 15

Graph each inequality on a number line.
$5>x>2$

Elizabeth Xu

Numerade Educator

Problem 16

Graph each inequality on a number line.
$-3<y \leq 4$

Elizabeth Xu

Numerade Educator

Problem 17

Write the solution set illustrated on each graph in set notation and interval notation.

Julie Silva

Numerade Educator

Problem 18

Write the solution set illustrated on each graph in set notation and interval notation.

Julie Silva

Numerade Educator

Problem 19

Write the solution set illustrated on each graph in set notation and interval notation.

Julie Silva

Numerade Educator

Problem 20

Write the solution set illustrated on each graph in set notation and interval notation.

Julie Silva

Numerade Educator

Problem 21

Solve the inequality and write the solution in set notation. Then graph the solution and write it in interval notation.
$5 a-11 \geq 2 a-5$

Elizabeth Xu

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Problem 22

Solve the inequality and write the solution in set notation. Then graph the solution and write it in interval notation.
$-8 n+5>-2 n-12$

Elizabeth Xu

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Problem 23

Solve the inequality and write the solution in set notation. Then graph the solution and write it in interval notation.
$2(n+3)-4 \leq 5 n-1$

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Problem 24

Solve the inequality and write the solution in set notation. Then graph the solution and write it in interval notation.
$-5(x+2)-3<3 x+11$

Elizabeth Xu

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Problem 25

Solve the inequality and write the solution in set notation. Then graph the solution and write it in interval notation.
$\frac{3 x}{8}+\frac{x}{4}<-4$

Elizabeth Xu

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Problem 26

Solve the inequality and write the solution in set notation. Then graph the solution and write it in interval notation.
$\frac{2 y}{5}+\frac{y}{10}<-2$

Elizabeth Xu

Numerade Educator

Problem 27

Solve each inequality and write the solution in set notation.
$7-2(x+3) \geq 4 x-6(x-3)$

Elizabeth Xu

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Problem 28

Solve each inequality and write the solution in set notation.
$-3-6(x-5) \leq 2(7-3 x)+1$

Elizabeth Xu

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Problem 29

Solve each inequality and write the solution in set notation.
$4(3 x-5)+18<2(5 x+1)+2 x$

Elizabeth Xu

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Problem 30

Solve each inequality and write the solution in set notation.
$8-(6+5 m)>-9 m-(3-4 m)$

Elizabeth Xu

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Problem 31

Solve each inequality and write the solution in set notation.
$-6(p-1)+2 p \leq-2(2 p-3)$

Elizabeth Xu

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Problem 32

Solve each inequality and write the solution in set notation.
$9(w-1)-3 w \geq-2(5-3 w)+1$

Elizabeth Xu

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Problem 33

Solve each inequality and write the solution in set notation.
Determine the intersection and union of sets $A, B, C$, and $D$ as indicated, given $A=\{-3,-2,-1,0,1,2,3\}$, $B=\{2,4,6,8\}, C=\{-4,-2,0,2,4\}$, and $D=\{4,5,6,7\}$.
$A \cap B$ and $A \cup B$

Jason Taylor-Pestell

Jason Taylor-Pestell

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Problem 34

Determine the intersection and union of sets $A, B, C$, and $D$ as indicated, given $A=\{-3,-2,-1,0,1,2,3\}$, $B=\{2,4,6,8\}, C=\{-4,-2,0,2,4\}$, and $D=\{4,5,6,7\}$.
$A \cap C$ and $A \cup C$

Elizabeth Xu

Numerade Educator

Problem 35

Determine the intersection and union of sets $A, B, C$, and $D$ as indicated, given $A=\{-3,-2,-1,0,1,2,3\}$, $B=\{2,4,6,8\}, C=\{-4,-2,0,2,4\}$, and $D=\{4,5,6,7\}$.
$A \cap D$ and $A \cup D$

Elizabeth Xu

Numerade Educator

Problem 36

Determine the intersection and union of sets $A, B, C$, and $D$ as indicated, given $A=\{-3,-2,-1,0,1,2,3\}$, $B=\{2,4,6,8\}, C=\{-4,-2,0,2,4\}$, and $D=\{4,5,6,7\}$.
$B \cap C$ and $B \cup C$

Elizabeth Xu

Numerade Educator

Problem 37

Determine the intersection and union of sets $A, B, C$, and $D$ as indicated, given $A=\{-3,-2,-1,0,1,2,3\}$, $B=\{2,4,6,8\}, C=\{-4,-2,0,2,4\}$, and $D=\{4,5,6,7\}$.
$B \cap D$ and $B \cup D$

Elizabeth Xu

Numerade Educator

Problem 38

Determine the intersection and union of sets $A, B, C$, and $D$ as indicated, given $A=\{-3,-2,-1,0,1,2,3\}$, $B=\{2,4,6,8\}, C=\{-4,-2,0,2,4\}$, and $D=\{4,5,6,7\}$.
$C \cap D$ and $C \cup D$

Elizabeth Xu

Numerade Educator

Problem 39

Express the compound inequalities graphically and in interval notation.
$x<-2$ or $x>1$

Elizabeth Xu

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Problem 40

Express the compound inequalities graphically and in interval notation.
$x<-5$ or $x>5$

Elizabeth Xu

Numerade Educator

Problem 41

Express the compound inequalities graphically and in interval notation.
$x<5$ and $x \geq-2$

Elizabeth Xu

Numerade Educator

Problem 42

Express the compound inequalities graphically and in interval notation.
$x \geq-4$ and $x<3$

Elizabeth Xu

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Problem 43

Express the compound inequalities graphically and in interval notation.
$x \geq 3$ and $x \leq 1$

Elizabeth Xu

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Problem 44

Express the compound inequalities graphically and in interval notation.
$x \leq-5$ or $x \geq-7$

Elizabeth Xu

Numerade Educator

Problem 45

Solve the compound and joint inequalities, then graph the solution set.
$4(x-1) \leq 20$ or $x+6>9$

Elizabeth Xu

Numerade Educator

Problem 46

Solve the compound and joint inequalities, then graph the solution set.
$-3(x+2)>15$ or $x-3 \leq-1$

Elizabeth Xu

Numerade Educator

Problem 47

Solve the compound and joint inequalities, then graph the solution set.
$-2 x-7 \leq 3$ and $2 x \leq 0$

Elizabeth Xu

Numerade Educator

Problem 48

Solve the compound and joint inequalities, then graph the solution set.
$-3 x+5 \leq 17$ and $5 x \leq 0$

Elizabeth Xu

Numerade Educator

Problem 49

Solve the compound and joint inequalities, then graph the solution set.
$\frac{3}{5} x+\frac{1}{2}>\frac{3}{10}$ and $-4 x>1$

Elizabeth Xu

Numerade Educator

Problem 50

Solve the compound and joint inequalities, then graph the solution set.
$\frac{2}{3} x-\frac{5}{6} \leq 0$ and $-3 x<-2$

Elizabeth Xu

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Problem 51

Solve the compound and joint inequalities, then graph the solution set.
$\frac{3 x}{8}+\frac{x}{4}<-3$ or $x+1>-5$

Elizabeth Xu

Numerade Educator

Problem 52

Solve the compound and joint inequalities, then graph the solution set.
$\frac{2 x}{5}+\frac{x}{10}<-2$ or $x-3>2$

Elizabeth Xu

Numerade Educator

Problem 53

Solve the compound and joint inequalities, then graph the solution set.
$-3 \leq 2 x+5<7$

Elizabeth Xu

Numerade Educator

Problem 54

Solve the compound and joint inequalities, then graph the solution set.
$2<3 x-4 \leq 19$

Elizabeth Xu

Numerade Educator

Problem 55

Solve the compound and joint inequalities, then graph the solution set.
$-0.5 \leq 0.3-x \leq 1.7$

Elizabeth Xu

Numerade Educator

Problem 56

Solve the compound and joint inequalities, then graph the solution set.
$-8.2<1.4-x<-0.9$

Elizabeth Xu

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Problem 57

Solve the compound and joint inequalities, then graph the solution set.
$-7<-\frac{3}{4} x-1 \leq 11$

Elizabeth Xu

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Problem 58

Solve the compound and joint inequalities, then graph the solution set.
$-21 \leq-\frac{2}{3} x+9<7$

Elizabeth Xu

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Problem 59

Determine the allowable values in each expression.
$\frac{12}{m}$

Julie Silva

Numerade Educator

Problem 60

Determine the allowable values in each expression.
$\frac{-6}{n}$

Charles Machakwa

Charles Machakwa

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Problem 61

Determine the allowable values in each expression.
$\frac{5}{y+7}$

Nick Johnson

Numerade Educator

Problem 62

Determine the allowable values in each expression.
$\frac{4}{x-3}$

Joshua Fischbach

Joshua Fischbach

Numerade Educator

Problem 63

Determine the allowable values in each expression.
$\frac{a+5}{6 a-3}$

Shahab Ullah

Numerade Educator

Problem 64

Determine the allowable values in each expression.
$\frac{m+5}{8 m+4}$

J Alessandro Briseño-Tapia

J Alessandro Briseño-Tapia

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Problem 65

Determine the allowable values in each expression.
$\frac{15}{3 x-12}$

Karly Williams

Numerade Educator

Problem 66

Determine the allowable values in each expression.
$\frac{7}{2 x+6}$

AG

Ankit Gupta

Numerade Educator

Problem 67

Determine the allowable values in each expression.
$\sqrt{x-2}$

AG

Ankit Gupta

Numerade Educator

Problem 68

Determine the allowable values in each expression.
$\sqrt{y+7}$

Erika Bustos

Numerade Educator

Problem 69

Determine the allowable values in each expression.
$\sqrt{3 n-12}$

James Kiss

Numerade Educator

Problem 70

Determine the allowable values in each expression.
$\sqrt{2 m+5}$

Melissa Barry

Numerade Educator

Problem 71

Determine the allowable values in each expression.
$\sqrt{b-\frac{4}{3}}$

Amy Jiang

Numerade Educator

Problem 72

Determine the allowable values in each expression.
$\sqrt{a+\frac{3}{4}}$

Amrita Bhasin

Numerade Educator

Problem 73

Determine the allowable values in each expression.
$\sqrt{8-4 y}$

Narayan Hari

Numerade Educator

Problem 74

Determine the allowable values in each expression.
$\sqrt{12-2 x}$

Jennifer Stoner

Jennifer Stoner

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Problem 75

Body mass index: $B=\frac{704 W}{H^{2}}$
The U.S. government publishes a body mass index formula to help people consider the risk of heart diseas An index $B$ of 27 or more means that a person is at risk Here $W$ represents weight in pounds and $H$ represents height in inches. (a) Solve the formula for $W$. (b) While dominating as a competitive bodybuilder, Arnold Schwarzenegger typically weighed at least $235 \mathrm{lb}$. If he is $6^{\prime} 2^{\prime \prime}$, what was his body mass index back then? (A person with an index above 30 is typically considered obese.)

Elizabeth Xu

Numerade Educator

Problem 76

Lift capacity: $75 S+125 B \leq 750$
The capacity in pounds of the lift used by a roofing company to place roofing shingles and buckets of roofing nails on rooftops is modeled by the formula shown, where $S$ represents packs of shingles and $B$ represents buckets of nails. Solve the formula for (a) $S$ in terms of $B$ and (b) $B$ in terms of $S$. Then use the formula to find (c) the largest number of shingle packs that can be lifted, (d) the largest number of nail buckets that can be lifted, and (e) the largest number of shingle packs that can be lifted along with three nail buckets.

Elizabeth Xu

Numerade Educator

Problem 77

Exam scores: Jacques is going to college on an academic scholarship that requires him to maintain at least a $75 \%$ average in all of his classes. So far he has scored $82 \%, 76 \%, 65 \%$, and $71 \%$ on four exams. What scores are possible on his last exam that will enable him to keep his scholarship?

Breanna Ollech

Numerade Educator

Problem 78

Timed trials: In the first three trials of the $100-\mathrm{m}$ butterfly, Johann had times of $50.2,49.8$, and $50.9 \mathrm{sec}$. How fast must he swim the final timed trial to have an average time of at most $50 \mathrm{sec}$ ?

Nicholas Majtenyi

Nicholas Majtenyi

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Problem 79

Checking account balance: If the average daily balance in a certain checking account drops below $\$ 1000$, the bank charges the customer a $\$ 7.50$ service fee. The table gives the daily balance for one customer. What must the daily balance be for Friday to avoid a service charge?

Melissa Stefan

Numerade Educator

Problem 80

Average weight: In the National Football League, many consider an offensive line to be "small" if the average weight of the five down linemen is less than 325 lb. Using the table, what must the weight of the right tackle be so that the line will not be considered small?

Elizabeth Xu

Numerade Educator

Problem 81

Area of a rectangle: Given the rectangle shown, what is the range of values for the width in order to keep the area less than $150 \mathrm{~m}^{2}$ ?

Elizabeth Xu

Numerade Educator

Problem 82

Area of a triangle: Using the triangle shown, find the height that will guarantee an area equal to or greater than $48 \mathrm{in}^{2}$.

Elizabeth Xu

Numerade Educator

Problem 83

Heating and cooling subsidies: As long as the outside temperature is over $45^{\circ} \mathrm{F}$ and less than $85^{\circ} \mathrm{F}$ $(45<F<85)$, the city does not issue heating or cooling subsidies for low-income families. What is the corresponding range of Celsius temperatures $C$ ? Recall that $F=\frac{9}{5} C+32$.

Elizabeth Xu

Numerade Educator

Problem 84

U.S. and European shoe sizes: To convert a European male shoe size ""E" to an American male shoe size "A," the formula $A=0.76 E-23$ can be used. Lillian has five sons in the U.S. military, with shoe sizes ranging from size 9 to size $14(9 \leq A \leq 14)$. What is the corresponding range of European sizes? Round to the nearest half-size.

Elizabeth Xu

Numerade Educator

Problem 85

Power tool rentals: Sunshine Equipment Co. rents its power tools for a $\$ 20$ fee, plus $\$ 4.50 / \mathrm{hr}$. Kealoha's Rentals offers the same tools for an $\$ 11$ fee plus $\$ 6.00 / \mathrm{hr}$. How many hours $h$ must a tool be rented to make the cost at Sunshine a better deal?

Elizabeth Xu

Numerade Educator

Problem 86

Moving van rentals: Stringer Truck Rentals will rent a moving van for $\$ 15.75 /$ day plus $\$ 0.35 / \mathrm{mi}$. Bertz Van Rentals will rent the same van for $\$ 25 /$ day plus $\$ 0.30 / \mathrm{mi}$. How many miles $m$ must the van be driven to make the cost at Bertz a better deal on a one-day rental?

Elizabeth Xu

Numerade Educator

Problem 87

Jeff had an old ' 63 Chrysler that got $11 \mathrm{mpg}$ back when gas was only $\$ 0.99 / \mathrm{gal}$. He now owns a Honda Civic that gets $32 \mathrm{mpg}$, but today gas costs $\$ 3.50 / \mathrm{gal}$.
(a) If he commutes to work five days a week, $50 \mathrm{mi}$ each way, what are the current weekly fuel costs in the Honda? (b) What were the weekly fuel costs when he was making the same commute in the Chrysler? (c) How much closer would he need to move to make the weekly commute cheaper now than it was then?

DL

Debra Leonard

Numerade Educator

Problem 88

By transmitting data on the same lines that deliver cable to a TV, cable modems are able to download data at rates in excess of $54 \mathrm{Mbit} / \mathrm{sec}$. Dial-up modems, which send and receive data through telephone lines, are typically limited to download speeds below $56 \mathrm{kbit} / \mathrm{sec}$.
(a) How long would it take to download an averagelength (12-Gbit) movie using a cable modem? (b) How long would it take to download an average-quality (24-Mbit) photograph using dial-up? (c) In 1 hr, how many more movies can be downloaded via cable than photos via dial-up? (Hint: 1 Gbit $=1000$ Mbit, $1 \mathrm{Mbit}=1000 \mathrm{kbit}$.)

Shelby Mohamed

Numerade Educator

Problem 89

Place the correct inequality symbol in the blank to make the statement true.
a. If $m \leq n$ and $p<0$, then $m p$ $n p .$
b. If $m>n$, then $-m$ $-n$.
c. If $m<n$, then $\frac{1}{m}$
$-\frac{1}{n}$
d. If $m<0$ and $n>0$, then $m^{3}$ $n^{2}$

Elizabeth Xu

Numerade Educator

Problem 90

What can be said about $x, y$, and $z$ if the following statements are assumed true?
a. $x y z<0$
b. $x^{2} y^{3} z^{4}>0$
c. $x^{2}+y^{2}+z^{2} \leq 0$
d. $\frac{x^{2}}{y z} \geq 0$

Ahmad Reda

Numerade Educator

Problem 91

The following argument leads to a false statement.
Explain why the final statement is false and then find the error that led to it.
$$
\begin{array}{cl}
a<b & \text { initial assumption } \\
a(a-b)<b(a-b) & \text { multiply by }(a-b) \\
a^{2}-a b<a b-b^{2} & \text { distributel } \\
a^{2}-2 a b+b^{2}<0 & \text { subtract ab, add } b^{2} \\
(a-b)^{2}<0 & \text { factor, false statement }
\end{array}
$$

Robert S.

Numerade Educator

Problem 92

(R.1) Translate into an algebraic expression: eight subtracted from twice a number.

Elizabeth Xu

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Problem 93

(R.2) Simplify the algebraic expression:
$$
2\left(\frac{5}{4} x-1\right)-\left(\frac{1}{6} x+3\right)
$$

Elizabeth Xu

Numerade Educator

Problem 94

Solve: $-4(x-7)-3=2 x+1$

Elizabeth Xu

Numerade Educator

Problem 95

Solve: $\frac{4}{5} m+\frac{2}{3}=\frac{1}{2}$

Elizabeth Xu

Numerade Educator