Intermediate Algebra

Alan S. Tussy, R. David Gustafson

Chapter 4

Inequalities - all with Video Answers

Educators

Section 1

Solving Linear Inequalities in One Variable

Problem 1

Fill in the blanks.
$<,>, \leq,$ and $\geq$ are _____ symbols.

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 2

Fill in the blanks.
$3 x+2 \geq 7$ is an example of a _____ inequality in one variable.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 3

Fill in the blanks.
The graph of a set of real numbers that is a portion of a number line is called an _____.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 4

Fill in the blanks.
In $(-\infty, 5),$ the right _____ is used to show that 5 is not included in the interval. $\operatorname{In}[12, \infty),$ the left _____ is used to show that 12 is included in the interval.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 5

Fill in the blanks.
We read the set- _____ notation $\{x | x<1\}$ as "the set of all real numbers $x$ _____ _____ $x$ is less than $1 . "$

Linh Vu

Numerade Educator

Problem 6

Fill in the blanks.
To _____ an inequality means to find all values of the variable that make the inequality true.

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 7

Which of the following are inequalities?
$$
6-x=8 \quad 5+a \quad 7 t-5>4 \quad \frac{x}{2} \leq-1
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 8

Perform each step listed below on the inequality $4>-2$ and give the resulting true inequality.
a. Add 2 to both sides.
b. Subtract 4 from both sides.
c. Multiply both sides by 4.
d. Divide both sides by $-2$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 9

Use a check to determine whether each number is a solution of $3 x+6 \leq 6$.
a. 0
b. $\frac{2}{3}$
c. $-10$
d. 1.5

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 10

The solution set of a linear inequality in $x$ is graphed on the right. Determine whether a true or false statement results when
a. $-4$ is substituted for $x$
b. $-3$ is substituted for $x$
c. 0 is substituted for $x$.

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 11

a. Suppose that when solving a linear inequality, the variable drops out, and the result is $6 \leq 10 .$ Write the solution set in interval notation and graph it.
b. Suppose that when solving a linear inequality, the variable drops out, and the result is $7<-1 .$ What symbol is used to represent the solution set?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 12

Insert the correct symbol, $<, \leq,>,$ or $\geq,$ in each blank.
a. As many as 16 people were seriously injured: The number of people seriously injured $\square$ 16.
b. There were no fewer than 8 references to taxes in the speech: The number of tax references $\square$ 8.
c. The weight $w$ of the roast is at most 8 pounds: $w$ $\square$ 8.
d. The temperature $t$ exceeded $100^{\circ}: t \square 100$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 13

Complete the solution to solve the inequality.
$$
\begin{array}{c}
-5 x-1 \geq-11 \\
-5 x \geq \square \\
\frac{-5 x}{-5} \square \frac{-10}{\square}\\
x \leq \square
\end{array}
$$
The solution set is $(\square, 2] .$ Using set-builder notation, it is $\{x | \square\}$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 14

Match each interval with its graph. (GRAPH CANNOT COPY).
a. $(-\infty,-1]$
b. $(-\infty, 1)$
c. $[-1, \infty)$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 15

Complete the solution to solve the inequality.
Fill in the blank: If $-10>x$ then $x \square-10 .$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 16

Complete the solution to solve the inequality.
Fill in the blank: $\infty$ is a symbol representing positive _____.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 17

Represent each set using a graph, interval notation, and set-builder notation. See Example 1.
The set of real numbers less than 14.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 18

Represent each set using a graph, interval notation, and set-builder notation. See Example 1.
The set of real numbers greater than 6.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 19

Represent each set using a graph, interval notation, and set-builder notation. See Example 1.
The set of real numbers greater than or equal to $-2$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 20

Represent each set using a graph, interval notation, and set-builder notation. See Example 1.
The set of real numbers less than or equal to $-7$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 21

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2.
$$
x+4<5
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 22

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2.
$$
x-5>2
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 23

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2.
$$
3 x>-9
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 24

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2.
$$
4 x<-36
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 25

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2.
$$
2 x-7 \geq-29
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 26

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2.
$$
6 x+8 \leq-16
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 27

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2.
$$
9 a+11 \leq 29
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 28

Solve each inequality. Graph the solution set and write it using interval notation. See Example 2.
$$
3 b-26 \geq 4
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 29

Solve each inequality. Graph the solution set and write it using interval notation. See Example 3.
$$
2 x+4+6 x>2-3 x+2
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 30

Solve each inequality. Graph the solution set and write it using interval notation. See Example 3.
$$
5 x+6+2 x \geq 2-x+4
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 31

Solve each inequality. Graph the solution set and write it using interval notation. See Example 3.
$$
t+1-3 t \geq t-20
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 32

Solve each inequality. Graph the solution set and write it using interval notation. See Example 3.
$$
a+4-10 a>a-16
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 33

Solve each inequality. Graph the solution set and write it using interval notation. See Example 4.
$$
4 \leq \frac{9}{10} x+1
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 34

Solve each inequality. Graph the solution set and write it using interval notation. See Example 4.
$$
2 \leq \frac{9}{4} x+8
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 35

Solve each inequality. Graph the solution set and write it using interval notation. See Example 4.
$$
-3>\frac{7}{8} x-1
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 36

Solve each inequality. Graph the solution set and write it using interval notation. See Example 4.
$$
-10>\frac{11}{2} x-6
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 37

Solve each inequality. Graph the solution set and write it using interval notation. See Example 5.
$$
\frac{3}{4}(x-3)<\frac{1}{3}(x-4)
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 38

Solve each inequality. Graph the solution set and write it using interval notation. See Example 5.
$$
\frac{5}{16}(x+1) \geq \frac{1}{4}(x-3)
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 39

Solve each inequality. Graph the solution set and write it using interval notation. See Example 5.
$$
\frac{2}{5}(3-2 n) \geq \frac{3}{8}(2-3 n)
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 40

Solve each inequality. Graph the solution set and write it using interval notation. See Example 5.
$$
\frac{1}{5}(1-n)<\frac{1}{3}(2-n)
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 41

Solve each inequality. Graph the solution set and write it using interval notation, if possible. See Example 6.
$$
2(5 x-6)>4 x-15+6 x
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 42

Solve each inequality. Graph the solution set and write it using interval notation, if possible. See Example 6.
$$
3(4 x-2)>14 x-7-2 x
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 43

Solve each inequality. Graph the solution set and write it using interval notation, if possible. See Example 6.
$$
\frac{5 x+2}{-4}>\frac{5 x+1}{-4}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 44

Solve each inequality. Graph the solution set and write it using interval notation, if possible. See Example 6.
$$
\frac{7-n}{-6}>\frac{1-n}{-6}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 45

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-5 t+3 \leq 5
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 46

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-9 t+6 \geq 16
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 47

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{2}{5}>\frac{4}{5} x
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 48

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{5}{9}<\frac{11}{9} x
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 49

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-0.6 x \leq-36
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 50

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-0.2 x>-8
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 51

Solve each inequality. Graph the solution set and write it using interval notation.
$$
7<\frac{5}{3} a-3
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 52

Solve each inequality. Graph the solution set and write it using interval notation.
$$
5>\frac{7}{2} a-9
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 53

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-7 y+5>-5 y-1
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 54

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-2 s-105 \leq-7 s-205
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 55

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{6-d}{-2} \leq-6
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 56

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{9-3 b}{-8}<3
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 57

Solve each inequality. Graph the solution set and write it using interval notation.
$$
0.4 x+0.4 \leq 0.1 x+0.85
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 58

Solve each inequality. Graph the solution set and write it using interval notation.
$$
0.05+0.8 x \leq 0.5 x-0.7
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 59

Solve each inequality. Graph the solution set and write it using interval notation.
$$
3(z-2) \leq 2(z+7)
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 60

Solve each inequality. Graph the solution set and write it using interval notation.
$$
5(3+z)>-3(z+3)
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 61

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{3 b+7}{3} \leq \frac{2 b-9}{2}
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 62

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-\frac{5 x}{4}>\frac{3-5 x}{4}
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 63

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{x-7}{2}-\frac{x-1}{5} \geq-\frac{x}{4}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 64

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{3 a+1}{3}-\frac{4-3 a}{5} \geq-\frac{1}{15}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 65

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{1}{2} x+6 \geq 4+2 x
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 66

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{1}{3} x+1<4+5 x
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 67

Solve each inequality. Graph the solution set and write it using interval notation.
$$
5(2 n+2)-n>3 n-3(1-2 n)
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 68

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-1+4(y-1)+2 y \leq \frac{1}{2}(12 y-30)+15
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 69

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{1}{2} y+2 \geq \frac{1}{3} y-4
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 70

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{1}{4} x-\frac{1}{3} \leq x+2
$$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 71

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-11(2-b)<4(2 b+2)
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 72

Solve each inequality. Graph the solution set and write it using interval notation.
$$
-9(h-3)+2 h \leq 8(4-h)
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 73

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{2}{3} x+\frac{3}{2}(x-5) \leq x
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 74

Solve each inequality. Graph the solution set and write it using interval notation.
$$
\frac{5}{9}(x+3)-\frac{4}{3}(x-3) \geq x-1
$$

Nicole Hoffman

Numerade Educator

Problem 75

Solve each inequality. Graph the solution set and write it using interval notation.
$$
5[3 t-(t-4)]-11 \leq-12(t-6)-(-t)
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 76

Solve each inequality. Graph the solution set and write it using interval notation.
$$
2-2[3 h-(7-h)]>6[-(19+h)-(1-h)]
$$

Nicole Hoffman

Numerade Educator

Problem 77

Let $f(x)=\frac{1}{2} x-\frac{2}{3}$ and $g(x)=x+\frac{4}{3} .$ Find all values of $x$ for which $f(x)<g(x)$.

Nicole Hoffman

Numerade Educator

Problem 78

Let $s(x)=\frac{1}{4} x-\frac{1}{2}$ and $g(x)=\frac{1}{2} x-\frac{2}{3} .$ Find all values of $x$ for which $s(x) \geq g(x)$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 79

Let $y_{1}=0.7 x-0.15$ and $y_{2}=x+0.3 .$ Find all values of $x$ for which $y_{2}$ exceeds $y_{1}$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 80

Let $y_{1}=0.8 x-1.1$ and $y_{2}=3.1-0.6 x .$ Find all values of $x$ for which $y_{1}$ does not exceed $y_{2}$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 81

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your answer to part a to determine the solution set for the inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation.
a. $12 x-33.16 \leq 5.84$
b. $12 x-33.16>5.84$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 82

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your answer to part a to determine the solution set for the inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation.
a. $-\frac{3}{4} x>-\frac{21}{32}$
b. $-\frac{3}{4} x \leq-\frac{21}{32}$

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 83

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your answer to part a to determine the solution set for the inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation.
a. $3(2 x+2)>5(x-1)+3 x$
b. $3(2 x+2)<5(x-1)+3 x$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 84

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your answer to part a to determine the solution set for the inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation.
a. $\frac{x-3}{2} \leq \frac{1}{2}-\frac{x-5}{4}$
b. $\frac{x-3}{2}>\frac{1}{2}-\frac{x-5}{4}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 85

Use a graphing calculator to solve each inequality. Write the solution set using interval notation. See Using Your Calculator: Solving Linear Inequalities in One Variable.
$$
2 x+3<5
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 86

Use a graphing calculator to solve each inequality. Write the solution set using interval notation. See Using Your Calculator: Solving Linear Inequalities in One Variable.
$$
5 x+2 \geq 4 x-2
$$

Nicole Hoffman

Numerade Educator

Problem 87

Use a graphing calculator to solve each inequality. Write the solution set using interval notation. See Using Your Calculator: Solving Linear Inequalities in One Variable.
$$
3 x-2>4
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 88

Use a graphing calculator to solve each inequality. Write the solution set using interval notation. See Using Your Calculator: Solving Linear Inequalities in One Variable.
$$
3 x-4 \leq 2 x+4
$$

Nicole Hoffman

Numerade Educator

Problem 89

Real Estate. Refer to the graph below. For which regions of the country were the following inequalities true in August, $2010 ?$
a. Median sales price < U.S. median price
b. Median sales price $\geq$ U.S. median price

Carson Merrill

Numerade Educator

Problem 90

Public Education. For which years shown in the graph below is the inequality true?
a. Grade 4 enrollment $\geq$ Grade 1 enrollment
b. Projected grade 1 enrollment $>$ Projected grade 4 enrollment
(GRAPH CANNOT COPY)

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 91

The percent of the air-borne particles in a room that a Climatec furnace filter can remove is approximated by the linear function $p(m)=\frac{6}{5} m$ where $m$ is the time in minutes that the furnace has been operating. Use an inequality to determine the time after which at least $60 \%$ of the air-borne particles in the room will have been removed. (Source: climatec.com)

Narayan Hari

Numerade Educator

Problem 92

Wikipedia. The number of articles $a(t)$ in millions in the Englishlanguage edition of Wikipedia can be approximated by the function $a(t)=0.54 t+0.5,$ where $t$ is the number of years since $2005 .$ Use an inequality to determine those years for which the number of articles surpassed 2.5 million. (Source: Wikipedia article: Size of Wikipedia)

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 93

National Parks. The number of visitors $v(t)$ to U.S. national parks can be approximated by the function $v(t)=-100,000 t+3,600,000,$ where $t$ is the number of years after $1990 .$ Use an inequality to determine those years for which the number of visitors fell below $2,400,000 .$ (Source: National Park Service Stats)

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 94

Medicare. The number of workers $w(t)$ for each Medicare beneficiary (each person receiving Medicare benefits) is approximated by the function $w(t)=-0.05 t+4,$ where $t$ is the number of years after $2000 .$ Use an inequality to determine those years for which there will be less than 2 workers for each Medicare beneficiary. (Source: Kaiser Family Foundation)

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 95

Moving Day. Valley Truck Rentals charges 25.50 dollars per day and 0.75 dollars per mile to rent a 14-foot truck. Nationwide Truck Rentals' daily charge for the same vehicle is 36.75 dollars and 0.60 dollars per mile. If the truck is rented for one day, for what range of miles driven is Nationwide's plan better?

Elizabeth Xu

Numerade Educator

Problem 96

Telephone Service. A telephone company offers two long-distance calling plans.
Plan 1: 17 dollars per month and $2 \not c$ per minute
Plan 2: 8 dollars per month and $8 \not c$ per minute
For how many minutes of long distance calls would Plan 2 save the caller money?

Nicole Hoffman

Numerade Educator

Problem 97

Business Losses. It costs a company 1,400 dollars to set up the necessary machinery and then 5 dollars each to manufacture bath towels. They sell the towels for 8.50 dollars each. Up to this point, the company has lost money on the towels. What is the greatest number of towels they could have sold for this to happen?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 98

Job Offers. A company has offered a newly hired salesperson her choice of compensation packages:
Package 1: 2,500 dollars salary per month and a $5 \%$ sales commission
Package 2: 3,500 dollars salary per month and a $3 \%$ sales commission
What amount of sales per month would the salesperson have to make so that Package 1 is more profitable for her?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 99

Averaging Grades. A student has scores of $70,77$ and 85 on three government exams. Use an inequality to determine the score she needs on a fourth exam to give her an average of 80 or better.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 100

Video Game Systems. A student who can afford to spend up to 1,000 dollars sees the ad shown in the illustration and decides to buy the video game system. Use an inequality to find the greatest number of video games that she can also purchase. (Disregard sales tax.) (IMAGE CANNOT COPY).

Melissa Lupinacci

Melissa Lupinacci

Numerade Educator

Problem 101

Work Schedules. A student works two part-time jobs. He earns 8 dollars an hour for working at the college library and 15 dollars an hour for construction work. To save time for study, he limits his work to 25 hours a week. If he enjoys the work at the library more, what is the greatest number of hours he can work at the library and still earn at least 300 dollars a week?

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 102

Scheduling Equipment. An excavating company charges 300 dollars an hour for the use of a backhoe and 500 dollars an hour for the use of a bulldozer. (Part of an hour counts as a full hour.) The company employs one operator for 40 hours per week to operate the machinery. If the company wants to bring in at least 18,500 dollars each week from equipment rental, how many hours per week can it schedule the operator to use a backhoe?

Aman Gupta

Numerade Educator

Problem 103

Fundraising. A school PTA wants to rent a dunking tank for its annual school fundraising carnival. The cost is 85.00 dollars for the first three hours and then 19.50 dollars for each additional hour or part thereof. How long can the tank be rented if up to 185 dollars is budgeted for this expense?

Jeff Harris

Numerade Educator

Problem 104

Investments. If a woman has invested 10,000 dollars at $8 \%$ annual interest, how much more must she invest at $9 \%$ so that her annual income will exceed 1,250 dollars?

James Macpherson

James Macpherson

Numerade Educator

Problem 105

How are the methods for solving linear equations and linear inequalities similar? How are they different?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 106

Explain how the symbol $\infty$ is used in this section. Is $\infty$ a real number?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 107

Explain what is wrong with the following statement:
When solving inequalities involving negative numbers,
the direction of the inequality symbol must be reversed.

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 108

In each case, determine what is wrong with the interval notation.
a. $(\infty,-3)$
b. $[-\infty,-3)$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 109

Geometry. The triangle inequality states an important relationship between the sides of any triangle:
$\begin{array}{l}\text { The sum of the lengths } \\ \text { of two sides of a triangle }\end{array}>$
$\begin{array}{l}\text { the length of } \\ \text { the third side. }\end{array}$
Use the triangle inequality to explain why the dimensions of the triangle shown here must be mislabeled.

Amy Jiang

Numerade Educator

Problem 110

Explain how to use the graph to solve $2 x+1<3$.
(GRAPH CANNOT COPY)

Jerelyn Nevil

Numerade Educator

Problem 111

Use the graph of the function to find $f(-1), f(0),$ and $f(2)$.
(GRAPH CANNOT COPY)

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 112

Use the graph of the function to find $f(-1), f(0),$ and $f(2)$.
(GRAPH CANNOT COPY)

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 113

In the illustration, which of the following are true?
i. $-a \geq 0 \quad$ ii. $a-b<0 \quad$ iii. $b-a<0 \quad$ iv. $a b>0$
(GRAPH CANNOT COPY)

AG

Ankit Gupta

Numerade Educator

Problem 114

Consider the following "solution" of the inequality $\frac{1}{3}>\frac{1}{x}$ where it appears that the solution set is the interval $(3, \infty)$.
$$
\frac{1}{3}>\frac{1}{x}
$$
$$
3 x\left(\frac{1}{3}\right)>3 x\left(\frac{1}{x}\right)
$$
$$
x>3
$$
a. Show that $x=-1$ makes the original inequality true.
b. If $x=-1$ makes the original inequality true, there must be an error in the solution. Where is it?

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 115

Medical Plans. A college provides its employees with a choice of the two medical plans shown in the following table. For what hospital bill dollar amount is Plan 2 better for the employee than Plan 1? (Hint: The cost to the employee includes both the deductible payment and the employee's coinsurance payment.) (GRAPH CANNOT COPY).

Catheryn Taylor

Catheryn Taylor

Numerade Educator

Problem 116

Medical Plans. To save costs, the college in Exercise 115 raised the employee deductible, as shown in the following table. For what hospital bill dollar amount is Plan 2 better for the employee than Plan 1? (Hint: The cost to the employee includes both the deductible payment and the employee's coinsurance payment.) (GRAPH CANNOT COPY).

Hollyann Mccann

Hollyann Mccann

Numerade Educator