Intermediate Algebra : Concepts and Applications

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

Chapter 4

Inequalities and Problem Solving - all with Video Answers

Educators

AG

Section 1

Inequalities and Applications

Problem 1

Choose from the following list the word that best completes each statement. Not every word will be used.
closed
half-open
negative
open
positive
solution
Because $-8<-1$ is true, $-8$ is a(n) of $x<-1$

AG

Ankit Gupta

Numerade Educator

Problem 2

Choose from the following list the word that best completes each statement. Not every word will be used.
closed
half-open
negative
open
positive
solution
The interval $[4,9]$ is a(n) interval.

AG

Ankit Gupta

Numerade Educator

Problem 3

Choose from the following list the word that best completes each statement. Not every word will be used.
closed
half-open
negative
open
positive
solution
The interval $(-7,1]$ is a(n) interval.

AG

Ankit Gupta

Numerade Educator

Problem 4

Choose from the following list the word that best completes each statement. Not every word will be used.
closed
half-open
negative
open
positive
solution
We reverse the direction of the inequality symbol when we multiply both sides of an inequality by $\mathrm{a}(\mathrm{n})$ number.

AG

Ankit Gupta

Numerade Educator

Problem 5

Classify each of the following as either equivalent inequalities, equivalent equations, equivalent expressions, or not equivalent.
$$
5 x+7=6-3 x, 8 x+7=6
$$

AG

Ankit Gupta

Numerade Educator

Problem 6

Classify each of the following as either equivalent inequalities, equivalent equations, equivalent expressions, or not equivalent.
$$
2(4 x+1), 8 x+2
$$

AG

Ankit Gupta

Numerade Educator

Problem 7

Classify each of the following as either equivalent inequalities, equivalent equations, equivalent expressions, or not equivalent.
$$
x-7>-2, x>5
$$

AG

Ankit Gupta

Numerade Educator

Problem 8

Classify each of the following as either equivalent inequalities, equivalent equations, equivalent expressions, or not equivalent.
$$
-4 t \leq 12, t \leq-3
$$

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Ankit Gupta

Numerade Educator

Problem 9

Classify each of the following as either equivalent inequalities, equivalent equations, equivalent expressions, or not equivalent.
$$
\frac{3}{5} a+\frac{1}{5}=2,3 a+1=10
$$

AG

Ankit Gupta

Numerade Educator

Problem 10

Classify each of the following as either equivalent inequalities, equivalent equations, equivalent expressions, or not equivalent.
$$
-\frac{1}{3} t \leq-5, t \geq 15
$$

AG

Ankit Gupta

Numerade Educator

Problem 11

Determine whether the given numbers are solutions of the inequality.
$x-4 \geq 1$.
a) $-4$
b) 4
c) 5
d) 8

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Ankit Gupta

Numerade Educator

Problem 12

Determine whether the given numbers are solutions of the inequality.
$3 x+1 \leq-5$
a) $-5$
b) $-2$
c) 0
d) 3

AG

Ankit Gupta

Numerade Educator

Problem 13

Determine whether the given numbers are solutions of the inequality.
$2 y+3<6-y$
a) 0
b) 1
c) $-1$
d) 4

AG

Ankit Gupta

Numerade Educator

Problem 14

Determine whether the given numbers are solutions of the inequality.
$5 t-6>1-2 t$
a) 6
b) 0
c) $-3$
d) 1

AG

Ankit Gupta

Numerade Educator

Problem 15

Graph each inequality, and write the solution set using both set-builder notation and interval notation.
$$
y<6
$$

Charlene Bische

Charlene Bische

Numerade Educator

Problem 16

Graph each inequality, and write the solution set using both set-builder notation and interval notation.
$$
x>4
$$

Charlene Bische

Charlene Bische

Numerade Educator

Problem 17

Graph each inequality, and write the solution set using both set-builder notation and interval notation.
$$
x \geq-4
$$

Charlene Bische

Charlene Bische

Numerade Educator

Problem 18

Graph each inequality, and write the solution set using both set-builder notation and interval notation.
$$
t \leq 6
$$

Charlene Bische

Charlene Bische

Numerade Educator

Problem 19

Graph each inequality, and write the solution set using both set-builder notation and interval notation.
$$
t>-3
$$

Charlene Bische

Charlene Bische

Numerade Educator

Problem 20

Graph each inequality, and write the solution set using both set-builder notation and interval notation.
$$
y<-3
$$

Charlene Bische

Charlene Bische

Numerade Educator

Problem 21

Graph each inequality, and write the solution set using both set-builder notation and interval notation.
$$
x \leq-7
$$

Charlene Bische

Charlene Bische

Numerade Educator

Problem 22

Graph each inequality, and write the solution set using both set-builder notation and interval notation.
$$
x \geq-6
$$

Charlene Bische

Charlene Bische

Numerade Educator

Problem 23

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
x+2>1
$$

AG

Ankit Gupta

Numerade Educator

Problem 24

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
x+9>6
$$

AG

Ankit Gupta

Numerade Educator

Problem 25

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
t-6 \leq 4
$$

AG

Ankit Gupta

Numerade Educator

Problem 26

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
t-1 \geq 5
$$

AG

Ankit Gupta

Numerade Educator

Problem 27

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
x-12 \geq-11
$$

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Ankit Gupta

Numerade Educator

Problem 28

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
x-11 \leq-2
$$

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Ankit Gupta

Numerade Educator

Problem 29

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
9 t<-81
$$

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Ankit Gupta

Numerade Educator

Problem 30

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
8 x \geq 24
$$

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Ankit Gupta

Numerade Educator

Problem 31

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
-0.3 x>-15
$$

AG

Ankit Gupta

Numerade Educator

Problem 32

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
-0.5 x<-30
$$

AG

Ankit Gupta

Numerade Educator

Problem 33

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
-9 x \geq 8.1
$$

AG

Ankit Gupta

Numerade Educator

Problem 34

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
-8 y \leq 3.2
$$

AG

Ankit Gupta

Numerade Educator

Problem 35

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
\frac{3}{4} y \geq-\frac{5}{8}
$$

AG

Ankit Gupta

Numerade Educator

Problem 36

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
\frac{5}{6} x \leq-\frac{3}{4}
$$

AG

Ankit Gupta

Numerade Educator

Problem 37

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
3 x+1<7
$$

AG

Ankit Gupta

Numerade Educator

Problem 38

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
2 x-5 \geq 9
$$

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Ankit Gupta

Numerade Educator

Problem 39

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
3-x \geq 12
$$

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Ankit Gupta

Numerade Educator

Problem 40

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
8-x<15
$$

AG

Ankit Gupta

Numerade Educator

Problem 41

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
\frac{2 x+7}{5}<-9
$$

AG

Ankit Gupta

Numerade Educator

Problem 42

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
\frac{5 y+13}{4}>-2
$$

AG

Ankit Gupta

Numerade Educator

Problem 43

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
\frac{3 t-7}{-4} \leq 5
$$

AG

Ankit Gupta

Numerade Educator

Problem 44

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
\frac{2 t-9}{-3} \geq 7
$$

AG

Ankit Gupta

Numerade Educator

Problem 45

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
\frac{9-x}{-2} \geq-6
$$

AG

Ankit Gupta

Numerade Educator

Problem 46

Solve. Then graph. Write the solution set using both set-builder notation and interval notation.
$$
\frac{3-x}{-5}<-2
$$

AG

Ankit Gupta

Numerade Educator

Problem 47

Let $f(x)=7-3 x$ and $g(x)=2 x-3 .$ Find all values of $x$ for which $f(x) \leq g(x)$

AG

Ankit Gupta

Numerade Educator

Problem 48

Let $f(x)=8 x-9$ and $g(x)=3 x-11 .$ Find all values of $x$ for which $f(x) \leq g(x)$

AG

Ankit Gupta

Numerade Educator

Problem 49

Let $f(x)=2 x-7$ and $g(x)=5 x-9 .$ Find all values of $x$ for which $f(x)<g(x)$

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Ankit Gupta

Numerade Educator

Problem 50

Let $f(x)=0.4 x+5$ and $g(x)=1.2 x-4 .$ Find all values of $x$ for which $g(x) \geq f(x)$

AG

Ankit Gupta

Numerade Educator

Problem 51

Let $y_{1}=\frac{3}{8}+2 x$ and $y_{2}=3 x-\frac{1}{8} .$ Find all values of $x$ for which $y_{2} \geq y_{1}$

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Ankit Gupta

Numerade Educator

Problem 52

Let $y_{1}=2 x+1$ and $y_{2}=-\frac{1}{2} x+6 .$ Find all values of $x$ for which $y_{1}<y_{2}$

AG

Ankit Gupta

Numerade Educator

Problem 53

Solve. Write the solution set using both set-builder notation and interval notation.
$$
3-8 y \geq 9-4 y
$$

AG

Ankit Gupta

Numerade Educator

Problem 54

Solve. Write the solution set using both set-builder notation and interval notation.
$$
4 m+7 \geq 9 m-3
$$

Mukesh Devi

Numerade Educator

Problem 55

Solve. Write the solution set using both set-builder notation and interval notation.
$$
5(t-3)+4 t<2(7+2 t)
$$

AG

Ankit Gupta

Numerade Educator

Problem 56

Solve. Write the solution set using both set-builder notation and interval notation.
$$
2(4+2 x)>2 x+3(2-5 x)
$$

AG

Ankit Gupta

Numerade Educator

Problem 57

Solve. Write the solution set using both set-builder notation and interval notation.
$$
5[3 m-(m+4)]>-2(m-4)
$$

AG

Ankit Gupta

Numerade Educator

Problem 58

Solve. Write the solution set using both set-builder notation and interval notation.
$$
8 x-3(3 x+2)-5 \geq 3(x+4)-2 x
$$

AG

Ankit Gupta

Numerade Educator

Problem 59

Solve. Write the solution set using both set-builder notation and interval notation.
$$
19-(2 x+3) \leq 2(x+3)+x
$$

AG

Ankit Gupta

Numerade Educator

Problem 60

Solve. Write the solution set using both set-builder notation and interval notation.
$$
13-(2 c+2) \geq 2(c+2)+3 c
$$

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Ankit Gupta

Numerade Educator

Problem 61

Solve. Write the solution set using both set-builder notation and interval notation.
$$
\frac{1}{4}(8 y+4)-17<-\frac{1}{2}(4 y-8)
$$

AG

Ankit Gupta

Numerade Educator

Problem 62

Solve. Write the solution set using both set-builder notation and interval notation.
$$
\frac{1}{3}(6 x+24)-20>-\frac{1}{4}(12 x-72)
$$

AG

Ankit Gupta

Numerade Educator

Problem 63

Solve. Write the solution set using both set-builder notation and interval notation.
$$
2[8-4(3-x)]-2 \geq 8[2(4 x-3)+7]-50
$$

AG

Ankit Gupta

Numerade Educator

Problem 64

Solve. Write the solution set using both set-builder notation and interval notation.
$$
5[3(7-t)-4(8+2 t)]-20 \leq-6[2(6+3 t)-4]
$$

AG

Ankit Gupta

Numerade Educator

Problem 65

Translate to an inequality.
A number is less than $10 .$

Emily Frampton

Numerade Educator

Problem 66

Translate to an inequality.
A number is greater than or equal to $4 .$

Emily Frampton

Numerade Educator

Problem 67

Translate to an inequality.
The temperature is at most $-3^{\circ} \mathrm{C}$

Emily Frampton

Numerade Educator

Problem 68

Translate to an inequality.
A full-time student must take at least 12 credits of classes.

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Ankit Gupta

Numerade Educator

Problem 69

Translate to an inequality.
The age of the Mayan altar exceeds 1200 years.

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Ankit Gupta

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Problem 70

Translate to an inequality.
The time of the test was between $45 \mathrm{min}$ and $55 \mathrm{min}$

Emily Frampton

Numerade Educator

Problem 71

Translate to an inequality.
Focus-group sessions should last no more than 2 hr.

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Ankit Gupta

Numerade Educator

Problem 72

Translate to an inequality.
Angenita earns no less than $\$ 12$ per hour.

Emily Frampton

Numerade Educator

Problem 73

Translate to an inequality.
To rent a car, a driver must have a minimum of 5 years of driving experience.

Ashley High

Numerade Educator

Problem 74

Translate to an inequality.
The maximum safe level for chronic inhalation of formaldehyde is 0.003 parts per million.

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Ankit Gupta

Numerade Educator

Problem 75

Translate to an inequality.
The costs of production of the software cannot exceed $\$ 12,500$

Emily Frampton

Numerade Educator

Problem 76

Translate to an inequality.
The number of volunteers was at most 20.

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Ankit Gupta

Numerade Educator

Problem 77

Solve.
Photography. Eli will photograph a wedding for a flat fee of $\$ 900$ or for an hourly rate of $\$ 120 .$ For what lengths of time would the hourly rate be less expensive?

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Ankit Gupta

Numerade Educator

Problem 78

Solve.
Jenn can rent a moving truck for either $\$ 99$ with unlimited mileage or $\$ 49$ plus $80 \mathfrak{Q}$ per mile. For what mileages would the unlimited mileage plan save money?

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Ankit Gupta

Numerade Educator

Problem 79

Solve.
Unconditional acceptance into the Master of Business Administration (MBA) program at the University of Arkansas at Little Rock is awarded to those students whose GMAT score plus 200 times their undergraduate grade point average is at least $1020 .$ Chloe's GMAT score was $500 .$ What must her grade point average be in order that she be unconditionally accepted into the program?

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Ankit Gupta

Numerade Educator

Problem 80

Solve.
As a rule of thumb, debt payments (other than mortgages) should be less than $8 \%$ of a consumer's monthly gross income. Oliver makes $\$ 54,000$ per year and has a $\$ 100$ studentloan payment every month. What size car payment can he afford?

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Ankit Gupta

Numerade Educator

Problem 81

Solve.
There are 80 questions on a college entrance examination. Two points are awarded for each correct answer, and one-half point is deducted for each incorrect answer. How many questions does Tami need to answer correctly in order to score at least 100 on the test? Assume that Tami answers every question.

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Ankit Gupta

Numerade Educator

Problem 82

Solve.
After a serious automobile accident, most insurance companies will replace the damaged car with a new one if repair costs exceed $80 \%$ of the NADA, or "blue-book,"," value of the car. Lorenzo's car recently sustained $\$ 9200$ worth of damage but was not replaced. What was the blue-book value of his car?

AG

Ankit Gupta

Numerade Educator

Problem 83

Solve.
Star Well Drilling offers two plans. Under the "pay-as-you-go" plan, they charge $\$ 500$ plus $\$ 8$ per foot for a well of any depth. Under their "guaranteed-water" plan, they charge a flat fee of $\$ 4000$ for a well that is guaranteed to provide adequate water for a household. For what depths would it save a customer money to use the pay-asyou-go plan?

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Ankit Gupta

Numerade Educator

Problem 84

Solve.
Bridgewater Legal Offices charges a $\$ 250$ retainer fee for real estate transactions plus $\$ 180$ per hour. Dockside Legal charges a $\$ 100$ retainer fee plus $\$ 230$ per hour. For what number of hours does Bridgewater charge more?

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Ankit Gupta

Numerade Educator

Problem 85

Solve.
Toni can be paid in one of two ways: Plan $A:$ A salary of $\$ 400$ per month, plus a commission of $8 \%$ of gross sales; Plan $B:$ A salary of $\$ 610$ per month, plus a commission of $5 \%$ of gross sales. For what amount of gross sales should Toni select plan A?

AG

Ankit Gupta

Numerade Educator

Problem 86

Solve.
Wages. Eric can be paid for his masonry work in one of two ways: Plan $A: \$ 300$ plus $\$ 15.00$ per hour Plan $B:$ Straight $\$ 17.50$ per hour. Suppose that the job takes $n$ hours. For what values of $n$ is plan $\mathrm{B}$ better for Eric?

AG

Ankit Gupta

Numerade Educator

Problem 87

Solve.
Green Village offers its residents two recycling plans. Their Purple Plan charges a $\$ 5$ monthly service fee plus $\$ 3$ for every bin collected. Their Blue Plan charges a $\$ 15$ monthly service fee plus $\$ 1.75$ for every bin collected. For what number of bins per month will the Blue Plan cost less?

AG

Ankit Gupta

Numerade Educator

Problem 88

Solve.
Checking Accounts. North Bank charges a monthly fee of $\$ 9$ for a business checking account. The first 200 transactions are free, and each additional transaction costs $\$ 0.75 .$ South Bank offers a business checking account with no monthly charge. Again, the first 200 transactions are free, and each additional transaction costs $\$ 0.90 .$ For what numbers of transactions is the South Bank plan more expensive? (Assume that the business will always have more than 200 transactions.)

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Ankit Gupta

Numerade Educator

Problem 89

Solve.
Solar Power. The cost per watt, in dollars, of installed solar panels in the United States $t$ years after 2000 can be approximated by
$$
c(t)=-0.42 t+11
$$
In $2011,$ the cost per watt of installed solar panels in Germany was $\$ 3.42 .$ Using an inequality, determine those years for which the cost in the United States will be less than the 2011 cost in Germany.

AG

Ankit Gupta

Numerade Educator

Problem 90

College Degrees. The percentage $B(t)$ of women ages 25 and older in the United States who hold a bachelor's degree or higher $t$ years after 1990 can be approximated by
$$
B(t)=0.48 t+18
$$
Using an inequality, determine those years for which more than $40 \%$ of women ages 25 and older in the United States will hold a bachelor's degree or higher.

Donna Densmore

Numerade Educator

Problem 91

The function given by
$$
F(d)=(4.95 / d-4.50) \times 100
$$
can be used to estimate the body fat percentage $F(d)$ of a person with an average body density $d$ in kilograms per liter.
a) A man is considered obese if his body fat percentage is at least $25 \% .$ Find the body densities of an obese man.
b) A woman is considered obese if her body fat percentage is at least $32 \% .$ Find the body densities of an obese woman.

AG

Ankit Gupta

Numerade Educator

Problem 92

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.

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Ankit Gupta

Numerade Educator

Problem 93

College Faculty. The number of part-time instructional faculty in U.S. postsecondary institutions is growing at a greater rate than the number of full-time faculty. The number of parttime faculty, in thousands, is approximated by
$$
p(t)=27 t+325
$$
and the number of full-time faculty, in thousands, is approximated by
$$
f(t)=16 t+500
$$
For both functions, $t$ represents the number of years after $1995 .$ Using an inequality, determine those years for which there were more part-time faculty than full-time faculty.

AG

Ankit Gupta

Numerade Educator

Problem 94

Beverages. As sales of soft drinks decrease in the United States, sales of coffee are increasing. The revenue from sales of soft drinks, in billions of dollars, is approximated by
$$
s(t)=0.33 t+17.1
$$
and the revenue from the sales of coffee, in billions of dollars, is approximated by
$$
c(t)=0.6 t+9.3
$$
For both functions, $t$ represents the number of years after $2010 .$ Using an inequality, determine those years for which there will be more revenue from the sale of coffee than from soft drinks.

AG

Ankit Gupta

Numerade Educator

Problem 95

Bright Ideas is planning to make a new kind of lamp. Fixed costs are $\$ 90,000,$ and variable costs are $\$ 25$ per lamp. The total-cost function for $x$ lamps is
$$
C(x)=90,000+25 x
$$
The company makes $\$ 48$ in revenue for each lamp sold. The total-revenue function for $x$ lamps is
$$
R(x)=48 x
$$
a) When $R(x)<C(x)$, the company loses money. Find those values of $x$ for which the company loses money.
b) When $R(x)>C(x),$ the company makes a profit. Find those values of $x$ for which the company makes a profit.

AG

Ankit Gupta

Numerade Educator

Problem 96

Publishing. The demand and supply functions for a locally produced poetry book are approximated by
$$
\begin{aligned}
&D(p)=2000-60 p \quad \text { and }\\
&S(p)=460+94 p
\end{aligned}
$$
where $p$ is the price, in dollars.
a) Find those values of $p$ for which demand exceeds supply.
b) Find those values of $p$ for which demand is less than supply.

AG

Ankit Gupta

Numerade Educator

Problem 97

How is the solution of $x+3=8$ related to the solution sets of
$$
x+3>8 \text { and } x+3<8 ?
$$

AG

Ankit Gupta

Numerade Educator

Problem 98

Why isn't roster notation used to write solutions of inequalities?

AG

Ankit Gupta

Numerade Educator

Problem 99

Solve.
$$
x-(9-x)=-3(x+5)
$$

AG

Ankit Gupta

Numerade Educator

Problem 100

Solve.
$$
\frac{2}{3} y-1=\frac{1}{4}
$$

AG

Ankit Gupta

Numerade Educator

Problem 101

Solve.
$$
\begin{aligned}
2 x-3 y &=5 \\
x+3 y &=-1
\end{aligned}
$$

AG

Ankit Gupta

Numerade Educator

Problem 102

Solve.
$$
\begin{aligned}
&4 x-y=1\\
&y=7-x
\end{aligned}
$$

AG

Ankit Gupta

Numerade Educator

Problem 103

Solve.
Solve $a r=b-c r$ for $r$

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Ankit Gupta

Numerade Educator

Problem 104

Solve.
$$
\text { Solve } y=\frac{a+b n}{t} \text { for } n
$$

AG

Ankit Gupta

Numerade Educator

Problem 105

The cost of solar panels cannot be less than 0. How should the domain of the function in Exercise 89 be adjusted to reflect this?
$$
3 a x+2 x \geq 5 a x-4 ; \text { assume } a>1
$$

AG

Ankit Gupta

Numerade Educator

Problem 106

Explain how the addition principle can be used to avoid ever needing to multiply or divide both sides of an inequality by a negative number.

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Ankit Gupta

Numerade Educator

Problem 107

Solve for x and y. Assume that a, b, c, d, and m are positive constants.
$$
3 a x+2 x \geq 5 a x-4 ; \text { assume } a>1
$$

AG

Ankit Gupta

Numerade Educator

Problem 108

Solve for x and y. Assume that a, b, c, d, and m are positive constants.
$$
6 b y-4 y \leq 7 b y+10
$$

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Ankit Gupta

Numerade Educator

Problem 109

Solve for x and y. Assume that a, b, c, d, and m are positive constants.
$$
a(b y-2) \geq b(2 y+5) ; \text { assume } a>2
$$

AG

Ankit Gupta

Numerade Educator

Problem 110

Solve for x and y. Assume that a, b, c, d, and m are positive constants.
$$
c(6 x-4)<d(3+2 x) ; \text { assume } 3 c>d
$$

AG

Ankit Gupta

Numerade Educator

Problem 111

Solve for x and y. Assume that a, b, c, d, and m are positive constants.
$c(2-5 x)+d x>m(4+2 x)$
assume $5 c+2 m<d$

AG

Ankit Gupta

Numerade Educator

Problem 112

Solve for x and y. Assume that a, b, c, d, and m are positive constants.
$a(3-4 x)+c x<d(5 x+2)$
assume $c>4 a+5 d$

AG

Ankit Gupta

Numerade Educator

Problem 113

Determine whether each statement is true or false. If false, give an example that shows this.
For any real numbers $a, b, c,$ and $d,$ if $a<b$ and $c<d,$ then $a-c<b-d$

AG

Ankit Gupta

Numerade Educator

Problem 114

Determine whether each statement is true or false. If false, give an example that shows this.
For all real numbers $x$ and $y,$ if $x<y,$ then $x^{2}<y^{2}$

AG

Ankit Gupta

Numerade Educator

Problem 115

Determine whether each statement is true or false. If false, give an example that shows this.
Are the inequalities..
$$
x<3 \text { and } x+\frac{1}{x}<3+\frac{1}{x}
$$
equivalent? Why or why not?

AG

Ankit Gupta

Numerade Educator

Problem 116

Determine whether each statement is true or false. If false, give an example that shows this.
Are the inequalities.
$x<3$ and $0 \cdot x<0 \cdot 3$
equivalent? Why or why not?

AG

Ankit Gupta

Numerade Educator

Problem 117

Solve. Then graph.
$$
x+5 \leq 5+x
$$

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Ankit Gupta

Numerade Educator

Problem 118

Solve. Then graph.
$$
x+8<3+x
$$

AG

Ankit Gupta

Numerade Educator

Problem 119

Solve. Then graph.
$$
x^{2}>0
$$

AG

Ankit Gupta

Numerade Educator

Problem 120

Solve. Then graph.
Abriana rented a compact car for a business trip. At the time of rental, she was given the option of prepaying for an entire tank of gasoline at $\$ 4.099$ per gallon, or waiting until her return and paying $\$ 7.34$ per gallon for enough gasoline to fill the tank. If the tank holds 14 gal, how many gallons can she use and still save money by choosing the second option? (Assume that Abriana does not put any gasoline in the car.)

AG

Ankit Gupta

Numerade Educator

Problem 121

Refer to Exercise $120 .$ If Abriana's rental car gets $30 \mathrm{mpg}$, how many miles must she drive in order to make the first option more economical?

AG

Ankit Gupta

Numerade Educator

Problem 122

Fundraising. Michelle is planning a fundraising dinner for Happy Hollow Children's Camp. The banquet facility charges a rental fee of $\$ 1500,$ but will waive the rental fee if more than $\$ 6000$ is spent on catering. Michelle knows that 150 people will attend the dinner.
a) How much should each dinner cost in order for the rental fee to be waived?
b) For what costs per person will the total cost (including the rental fee) exceed $\$ 6000 ?$
c) For some meal costs, it would be more economical to choose a more expensive meal because the rental fee would be waived. What are those meal costs?

AG

Ankit Gupta

Numerade Educator

Problem 123

Assume that 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 of the following inequalities, referring only to the figure.
a) $-\frac{1}{2} x+5>x-1$
b) $x-1 \leq 2 x-3$
c) $2 x-3 \geq-\frac{1}{2} x+5$
(GRAPH CANT COPY)

AG

Ankit Gupta

Numerade Educator

Problem 124

Use a graphing calculator to check your answers to Exercises 23, 47, and 63.

AG

Ankit Gupta

Numerade Educator