A Graphical Approach to College Algebra

John Hornsby,Margaret L. Lial,Gary Rockswold

Chapter 1

Linear Functions, Equations, and Inequalities - all with Video Answers

Educators

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Section 1

Real Numbers and the Rectangular Coordinate System

Problem 1

For each set, list all elements that belong to the
(a) natural numbers
(b) whole numbers
(c) integers
(d) rational numbers
(e) irrational numbers
(f) real numbers
$$\left\{-6,-\frac{12}{4},-\frac{5}{8},-\sqrt{3}, 0,0.31,0 . \overline{3}, 2 \pi, 10, \sqrt{17}\right\}$$

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Ankit Gupta

Numerade Educator

Problem 2

For each set, list all elements that belong to the
(a) natural numbers
(b) whole numbers
(c) integers
(d) rational numbers
(e) irrational numbers
(f) real numbers
$$\left\{-8,-\frac{14}{7},-0.245,0, \frac{6}{2}, 8, \sqrt{81}, \sqrt{12}\right\}$$

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Ankit Gupta

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Problem 3

For each set, list all elements that belong to the
(a) natural numbers
(b) whole numbers
(c) integers
(d) rational numbers
(e) irrational numbers
(f) real numbers
$$\left\{-\sqrt{100},-\frac{13}{6},-1,5.23,9 . \overline{14}, 3.14, \frac{22 {7}\right\}$$

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Ankit Gupta

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Problem 4

For each set, list all elements that belong to the
(a) natural numbers
(b) whole numbers
(c) integers
(d) rational numbers
(e) irrational numbers
(f) real numbers
$$\{-\sqrt{49},-0.405,-0 . \overline{3}, 0.1,3,18,6 \pi, 56\}$$

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Ankit Gupta

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Problem 5

Classify each number as one or more of the following: natural number, integer, rational number, or real number.
$16,351,000,000,000$ (The federal debt in dollars in January 2013 )

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Ankit Gupta

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Problem 6

Classify each number as one or more of the following: natural number, integer, rational number, or real number.
$700,000,000,000$ (The federal 2008 bailout fund in dollars)

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Ankit Gupta

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Problem 7

Classify each number as one or more of the following: natural number, integer, rational number, or real number.
$-25$ (The percent change in the number of Yahoo searches from 2011 to 2012 )

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Ankit Gupta

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Problem 8

Classify each number as one or more of the following: natural number, integer, rational number, or real number.
$-3$ (The annual percent change in the area of tropical rain forests)

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Problem 9

Classify each number as one or more of the following: natural number, integer, rational number, or real number.
$\frac{7}{3}$ (The fractional increase in online sales on Thanksgiving Day from 2006 to 2011 )

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Ankit Gupta

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Problem 10

Classify each number as one or more of the following: natural number, integer, rational number, or real number.
$-3.5$ (The amount in billions of dollars that the Motion Picture Association of America estimates is lost annually due to piracy)

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Ankit Gupta

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Problem 11

Classify each number as one or more of the following: natural number, integer, rational number, or real number.
$5 \sqrt{2}$ (The length of the diagonal of a square measuring 5 units on each side)

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Problem 12

Classify each number as one or more of the following: natural number, integer, rational number, or real number.
$\pi$ (The ratio of the circumference of a circle to its diameter)

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Problem 13

For each measured quantity, state the set of numbers that is most appropriate to describe it. Choose from the natural numbers, integers, and rational numbers.
Populations of cities

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Problem 14

For each measured quantity, state the set of numbers that is most appropriate to describe it. Choose from the natural numbers, integers, and rational numbers.
Distances to nearby cities on road signs

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Problem 15

For each measured quantity, state the set of numbers that is most appropriate to describe it. Choose from the natural numbers, integers, and rational numbers.
Shoe sizes

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Problem 16

For each measured quantity, state the set of numbers that is most appropriate to describe it. Choose from the natural numbers, integers, and rational numbers.
Prices paid (in dollars and cents) for gasoline tank fill-ups

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Problem 17

For each measured quantity, state the set of numbers that is most appropriate to describe it. Choose from the natural numbers, integers, and rational numbers.
Daily low winter temperatures in U.S. cities

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Problem 18

For each measured quantity, state the set of numbers that is most appropriate to describe it. Choose from the natural numbers, integers, and rational numbers.
Golf scores relative to par

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Problem 19

Graph each set of numbers on a number line.
$$\{-4,-3,-2,-1,0,1\}$$

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Problem 20

Graph each set of numbers on a number line.
$$\{-6,-5,-4,-3,-2\}$$

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Problem 21

Graph each set of numbers on a number line.
$$\left\{-0.5,0.75, \frac{5}{3}, 3.5\right\}$$

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Problem 22

Graph each set of numbers on a number line.
$$\left\{-0.6, \frac{9}{8}, 2.5, \frac{13}{4}\right\}$$

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Problem 23

Graph each set of numbers on a number line.
Explain the distinction between a rational number and an irrational number.

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Problem 24

Using her calculator, a student found the decimal 1.414213562 when she evaluated $\sqrt{2} .$ Is this decimal the exact value of $\sqrt{2}$ or just an approximation of $\sqrt{2} ?$ Should she write $\sqrt{2}=1.414213562$ or $\sqrt{2} \approx 1.414213562 ?$

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Problem 25

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(2,3)$$

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Problem 26

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(-1,2)$$

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Problem 27

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(-3,-2)$$

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Problem 28

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(1,-4)$$

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Problem 29

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(0,5)$$

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Problem 30

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(-2,-4)$$

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Problem 31

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(-2,4)$$

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Problem 32

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(3,0)$$

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Problem 33

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(-2,0)$$

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Problem 34

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies.
$$(3,-3)$$

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Problem 35

Name the possible quadrants in which the point ( $x, y$ ) can lie if the given condition is true.
$$x y>0$$

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Problem 36

Name the possible quadrants in which the point ( $x, y$ ) can lie if the given condition is true.
$$x y<0$$

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Problem 37

Name the possible quadrants in which the point ( $x, y$ ) can lie if the given condition is true.
$$\frac{x}{y}<0$$

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Problem 38

Name the possible quadrants in which the point ( $x, y$ ) can lie if the given condition is true.
$$\frac{x}{y}>0$$

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Problem 39

If the $x$ -coordinate of a point is $0,$ the point must lie on which axis?

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Problem 40

If the $y$ -coordinate of a point is $0,$ the point must lie on which axis?

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Problem 41

Give the values of $\mathrm{Xmin}, \mathrm{Xmax}, \mathrm{Ymin},$ and $\mathrm{Ymax}$ for each screen, given the values for $\mathrm{Xscl}$ and $\mathrm{Yscl}$. Use the notation described in this section.
$$X s c 1=1, Y s c l=5$$
(THE IMAGES CANNOT COPY)

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Ankit Gupta

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Problem 42

Give the values of $\mathrm{Xmin}, \mathrm{Xmax}, \mathrm{Ymin},$ and $\mathrm{Ymax}$ for each screen, given the values for $\mathrm{Xscl}$ and $\mathrm{Yscl}$. Use the notation described in this section.
$X$ scl $=5, Y$ scl $=1$

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Ankit Gupta

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Problem 43

Give the values of $\mathrm{Xmin}, \mathrm{Xmax}, \mathrm{Ymin},$ and $\mathrm{Ymax}$ for each screen, given the values for $\mathrm{Xscl}$ and $\mathrm{Yscl}$. Use the notation described in this section.
$$\mathrm{Xscl}=10, \mathrm{Yscl}=50$$
(THE IMAGES CANNOT COPY)

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Ankit Gupta

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Problem 44

Give the values of $\mathrm{Xmin}, \mathrm{Xmax}, \mathrm{Ymin},$ and $\mathrm{Ymax}$ for each screen, given the values for $\mathrm{Xscl}$ and $\mathrm{Yscl}$. Use the notation described in this section.
$$\mathrm{Xscl}=50, \mathrm{Yscl}=10$$

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Ankit Gupta

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Problem 45

Give the values of $\mathrm{Xmin}, \mathrm{Xmax}, \mathrm{Ymin},$ and $\mathrm{Ymax}$ for each screen, given the values for $\mathrm{Xscl}$ and $\mathrm{Yscl}$. Use the notation described in this section.
$$\mathrm{Xscl}=100, \mathrm{Yscl}=100$$
(THE IMAGES CANNOT COPY)

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Ankit Gupta

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Problem 46

Give the values of $\mathrm{Xmin}, \mathrm{Xmax}, \mathrm{Ymin},$ and $\mathrm{Ymax}$ for each screen, given the values for $\mathrm{Xscl}$ and $\mathrm{Yscl}$. Use the notation described in this section.
$$\mathbf{X} \text { scl }=75, Y \text { scl }=75$$

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Ankit Gupta

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Problem 47

Set the viewing window of your calculator to the given specifications. Make a sketch of your window.
$$\begin{aligned}
&[-10,10] \text { by }[-10,10]\\
&\mathrm{Xscl}=1 \quad \mathrm{Yscl}=1
\end{aligned}$$

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Problem 48

Set the viewing window of your calculator to the given specifications. Make a sketch of your window.
$$\begin{aligned}
&[-40,40] \text { by }[-30,30]\\
&\mathrm{Xscl}=5 \quad \mathrm{Yscl}=5
\end{aligned}$$

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Problem 49

Set the viewing window of your calculator to the given specifications. Make a sketch of your window.
$$\begin{aligned}
&[-5,10] \text { by }[-5,10]\\
&\mathrm{Xscl}=3 \quad \mathrm{Yscl}=3
\end{aligned}$$

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Problem 50

Set the viewing window of your calculator to the given specifications. Make a sketch of your window.
$$\begin{aligned}
&[-3.5,3.5] \text { by }[-4,10]\\
&\mathrm{Xscl}=1 \quad \mathrm{Yscl}=1
\end{aligned}$$

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Problem 51

Set the viewing window of your calculator to the given specifications. Make a sketch of your window.
$$\begin{aligned}
&[-100,100] \text { by }[-50,50]\\
&\mathrm{Xscl}=20 \quad \mathrm{Yscl}=25
\end{aligned}$$

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Ankit Gupta

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Problem 52

Set the viewing window of your calculator to the given specifications. Make a sketch of your window.
$$\begin{aligned}
&[-4.7,4.7] \text { by }[-3.1,3.1]\\
&\mathbf{X s c l}=1 \quad \mathbf{Y s c l}=1
\end{aligned}$$

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Ankit Gupta

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Problem 53

Set your viewing window to $[-10,10]$ by $[-10,10]$ and then set Xscl to 0 and $Y$ scl to $0 .$ What do you notice? Make a conjecture as to how to set the screen with no tick marks on the axes.

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Problem 54

Set your viewing window to $[-50,50]$ by $[-50,50]$ Xscl to $1,$ and Yscl to $1 .$ Describe the appearance of the axes compared with those seen in the standard window. Why do you think they appear this way? How can you change your scale settings so that this "problem" is alleviated?

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Problem 55

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$\sqrt{58}$$

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Problem 56

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$\sqrt{97}$$

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Problem 57

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$\sqrt[3]{33}$$

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Problem 58

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$\sqrt[3]{91}$$

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Problem 59

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$\sqrt[4]{86}$$

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Problem 60

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$\sqrt[4]{123}$$

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Problem 61

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$19^{1 / 2}$$

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Problem 62

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$29^{1 / 3}$$

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Problem 63

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$46^{1.5}$$

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Problem 64

Find a decimal approximation of each root or power. Round answers to the nearest thousandth.
$$23^{2.75}$$

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Problem 65

Approximate each expression to the nearest hundredth.
$$\frac{5.6-3.1}{8.9+1.3}$$

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Problem 66

Approximate each expression to the nearest hundredth.
$$\frac{34+25}{23}$$

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Problem 67

Approximate each expression to the nearest hundredth.
$$\sqrt{\pi^{3}+1}$$

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Problem 68

Approximate each expression to the nearest hundredth.
$$\sqrt[3]{2.1-6^{2}}$$

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Problem 69

Approximate each expression to the nearest hundredth.
$$3(5.9)^{2}-2(5.9)+6$$

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Problem 70

Approximate each expression to the nearest hundredth.
$$2 \pi^{3}-5 \pi^{2}-3$$

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Problem 71

Approximate each expression to the nearest hundredth.
$$\sqrt{(4-6)^{2}+(7+1)^{2}}$$

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Problem 72

Approximate each expression to the nearest hundredth.
$$\sqrt{[-1-(-3)]^{2}+(-5-3)^{2}}$$

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Problem 73

Approximate each expression to the nearest hundredth.
$$\text { 73. } \frac{\sqrt{\pi-1}}{\sqrt{1+\pi}}$$

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Problem 74

Approximate each expression to the nearest hundredth.
$$\sqrt[3]{4.5 \times 10^{5}+3.7 \times 10^{2}}$$

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Problem 75

Approximate each expression to the nearest hundredth.
$$\frac{2}{1-\sqrt[3]{5}}$$

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Problem 76

Approximate each expression to the nearest hundredth.
$$1-\frac{4.5}{3-\sqrt{2}}$$

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Problem 77

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse.
$$a=8, b=15 ; \text { find } c$$

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Problem 78

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse.
$$a=7, b=24 ; \text { find } c$$

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Problem 79

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse.
$$a=13, c=85 ; \text { find } b$$

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Problem 80

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse.
$$a=14, c=50 ; \text { find } b$$

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Problem 81

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse.
$$a=5, b=8 ; \text { find } c$$

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Problem 82

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse.
$$a=9, b=10 ; \text { find } c$$

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Problem 83

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse.
$$b=\sqrt{13}, c=\sqrt{29} ; \text { find } a$$

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Problem 84

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse.
$$b=\sqrt{7}, c=\sqrt{11} ; \text { find } a$$

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Problem 85

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q.
(THE GRAPH CANNOT COPY)

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Problem 86

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q.

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Problem 87

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q.
(THE GRAPH CANNOT COPY)

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Problem 88

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q.

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Problem 89

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q
$$P(5,7), Q(2,11)$$

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Problem 90

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q
$$P(-2,5), Q(4,-3)$$

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Problem 91

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q
$$P(-8,-2), Q(-3,-5)$$

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Problem 92

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q
$$P(-6,-10), Q(6,5)$$

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Problem 93

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q
$$P(9.2,3.4), Q(6.2,7.4)$$

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Problem 94

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q
$$P(8.9,1.6), Q(3.9,13.6)$$

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Problem 95

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q.
$$P(13 x,-23 x), Q(6 x, x), \quad x>0$$

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Problem 96

Find
(a) the distance between P and Q and
(b) the coordinates of the midpoint M of the segment joining P and Q.
$$P(12 y,-3 y), Q(20 y, 12 y), \quad y>0$$

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Problem 97

Suppose that P is an endpoint of a segment PQ and M is the midpoint of $P Q .$ Find the coordinates of endpoint Q.
$$P(7,-4), M(8,5)$$

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Problem 98

Suppose that P is an endpoint of a segment PQ and M is the midpoint of $P Q . Find the coordinates of endpoint Q.
$$P(13,5), M(-2,-4)$$

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Problem 99

Suppose that P is an endpoint of a segment PQ and M is the midpoint of $P Q . Find the coordinates of endpoint Q.
$$P(5.64,8.21), M(-4.04,1.60)$$

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Problem 100

Suppose that P is an endpoint of a segment PQ and M is the midpoint of $P Q . Find the coordinates of endpoint Q.
$$P(-10.32,8.55), M(1.55,-2.75)$$

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Problem 101

Solve each problem.
From 2007 to 2011, worldwide Google advertising revenue (in billions of dollars) rose in an approximately linear fashion. The graph depicts this growth with a line segment. Use the midpoint formula to approximate this revenue for 2009.
(THE GRAPH CANNOT COPY)

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Problem 102

Solve each problem.
Estimated and projected enrollments in two-year colleges for 2012 , $2014,$ and 2016 are shown in the table. Use the midpoint formula to estimate the enrollments to the nearest thousand for 2013 and 2015.
(THE IMAGES CANNOT COPY)

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Problem 103

Solve each problem.
The table lists poverty-level income cutoffs for a family of four for selected years. Use the midpoint formula to estimate the poverty-level cutoffs (rounded to the nearest dollar) in 2005 and 2009.
(THE IMAGES CANNOT COPY)

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Problem 104

Solve each problem.
Triangles can be classified by their sides.
(a) An isosceles triangle has at least two sides of equal length. Determine whether the triangle with vertices (0,0),(3,4), and (7,1) is isosceles.
(b) An equilateral triangle has all sides of equal length. Determine whether the triangle with vertices (-1,-1), (2,3) , and (-4,3) is equilateral.
(c) Determine whether a triangle having vertices (-1,0)
(1,0) and (0, \sqrt{3}) is isosceles, equilateral, or neither.
(d) Determine whether a triangle having vertices (-3,3) (-2,5) and (-1,3) is isosceles, equilateral, or neither.

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Problem 105

Solve each problem.
At 9: 00 A.M., Car A is traveling north at 50 mph and is located 50 miles south of Car $B$. Car B is traveling west at 20 mph.
(a) Let (0,0) be the initial coordinates of Car B in the xy-plane, where units are in miles. Plot the locations of each car at 9: 00 A.M. and at 11: 00 A.M.
(b) Find the distance between the cars at 11: 00 A.M.

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Problem 106

Solve each problem.
Two ships leave the same harbor at the same time. The first ship heads north at
20 mph and the second ship heads west at 15 mph.
(a) Draw a sketch depicting their positions after t hours.
(b) Write an expression that gives the distance between the ships after t hours.

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Problem 107

One of the most popular proofs of the Pythagorean theorem uses the figure shown here. Determine the area of the figure in two ways. First, find the area of the large square, using the formula for the area of a square. Then, find its area as the sum of the areas of the smaller square and the four right triangles. Set these expressions equal to each other and simplify to obtain
$$a^{2}+b^{2}=c^{2}$$
(THE IMAGES CANNOT COPY)

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Problem 108

Solve each problem.
Prove that the midpoint $M$ of the line segment joining endpoints $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right) has coordinates
$$
\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)
$$
by showing that the distance between P and M is equal to the distance between M and Q and that the sum of these distances is equal to the distance between P and Q.

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